Athermal dynamics of artificial spin ice:disorder, edge and field protocol effects

Zoe Budrikis, BSc. (Hons)

This thesis is presented for the degree of

Doctor of Philosophy of

The University of Western Australia

School of Physics

2012

Abstract

Artificial spin ice consists of nano-patterned arrays of magnetic material, and is designed

as a two-dimensional experimental model of pyrochlore spin ices such as Dy2Ti2O7. The

fundamental components are elongated islands of magnetic material. These are coupled

by dipolar interactions, which are frustrated by geometry. The islands are small enough

to be single-domain – that is, to good approximation, the magnetic moment is aligned

throughout the island – and their elongated shape constrains the magnetisation to point

along the island long axis, so that the islands act as Ising macrospins with two states.

The islands are large enough that their magnetisation reversal is inaccessible to thermal

dynamics, and the system behaves as if at zero temperature. Accordingly, dynamics must

be induced by external fields which act uniformly on all spins. These global dynamics are

highly constrained and many states are inaccessible, regardless of their energy.

This Thesis is a theoretical study of the athermal, driven dynamics of square artificial

spin ice. The aim is to understand how dynamics are affected not only by the complex

energy landscape generated by the interactions between spins, but also disorder in the

system. The work can be divided into three main strands.

The first is to treat the vertices of the spin array as objects, and to consider how these

interact with each other. Analysis of energetically-allowed vertex processes provides an

explanation of how in an ideal system, array edge geometry and the sequence of applied

fields controls dynamics. It is found that in perfect systems low-energy states can be

generated by careful selection of dynamical processes, but also by the introduction of

randomness into the sequence of driving fields. The treatment of vertices as particles also

makes it possible to write down a set of equations describing how the population fractions

of these vertex objects changes when an external magnetic field is applied, under the

assumption that vertices are well mixed and interact with their nearest neighbours only.

The solution of these equations agrees well with simulations in the field regime where the

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assumptions are expected to hold.

The second strand of the work is a study of how quenched disorder affects vertex

dynamics. Intriguingly, simulations show that different sources of disorder lead to similar

effects on the dynamics, and the strength of disorder rather than its origin is the best way

to characterise it. There are two disorder regimes. In the weak disorder regime, dynamics

are only slightly perturbed from those of the ideal system. In the strong disorder regime

dynamics are fundamentally altered as new dynamical pathways are opened. Comparison

with experimental results shows that real systems are typically in the strong disorder

regime.

The final part of this work is to map the field driven dynamics onto a directed net-

work. In the network, nodes represent spin configurations and two nodes are connected

by a directed link if one configuration can relax into the other under some applied field.

Accordingly, the network contains information about dynamics under any sequence of ap-

plied fields, and analysis of the network allows the extraction of general information about

the systems behaviour. A systematic study of the effects of the width of the switching

field distribution and also the applied field strength is presented. Disorder dramatically

increases the accessibility of states and the reversibility of dynamics by allowing transi-

tions that are forbidden in the perfect system. We also discuss how disorder affects the

accessibility of the system’s ground state.

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Contents

1 Introduction 1

1.1 Background: geometrical frustration and the ice rules . . . . . . . . . . . . 1

1.2 Artificial spin ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The model of interactions and dynamics . . . . . . . . . . . . . . . . . . . . 13

1.4 Overview of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Spin ice dynamics: vertex populations and processes 23

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Vertex types in square artificial spin ice . . . . . . . . . . . . . . . . . . . . 24

2.3 Vertex configurations and dynamics . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Vertex dynamics of rotating field protocols . . . . . . . . . . . . . . . . . . . 30

2.5 Vertex population dynamics: an analytical model . . . . . . . . . . . . . . . 38

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Selecting vertex processes: field protocols and edge effects in ideal sys-

tems 51

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Array edge geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Response to a field at 180 to initial net magnetisation . . . . . . . . . . . . 55

3.4 Response to a field at 90 to initial net magnetisation . . . . . . . . . . . . 60

3.5 Rotating protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Random and large ∆θ protocols . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Disorder effects 77

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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4.2 Disorder types and regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Single direction field protocols . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Rotating field protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 A note on vertex population dynamics . . . . . . . . . . . . . . . . . . . . . 89

4.6 Random θ and large ∆θ protocols . . . . . . . . . . . . . . . . . . . . . . . 91

4.7 Symmetry breaking by disorder . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.8 Inaccessibility of the ground state for disordered systems . . . . . . . . . . . 96

4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Network representation of dynamics 105

5.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Network construction and properties . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Node degrees and energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.5 Accessibility of states and reversibility of dynamics . . . . . . . . . . . . . . 120

5.6 A note on field protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.7 Network structure and ‘rewiring’ . . . . . . . . . . . . . . . . . . . . . . . . 128

5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Summary and outlook 133

6.1 Summary of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Bibliography 143

A Numerical simulation methods 155

A.1 Code used in simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.2 Switching criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.3 Range of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.4 Comparison of periodic and free boundary conditions . . . . . . . . . . . . . 168

A.5 Field pulse ramp rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B Vertex dynamics 173

B.1 All vertex processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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B.2 Vertex population equations from random initial configuration . . . . . . . 175

C Results relating to disorder 177

C.1 Disordered array with field applied at 90 to initial magnetisation . . . . . . 177

C.2 Orientational disorder reduces nearest-neighbour coupling . . . . . . . . . . 177

D Methods and results relating to networks 181

D.1 Simulation results for a 4× 4 array . . . . . . . . . . . . . . . . . . . . . . . 181

D.2 Choice of field angle step for networks . . . . . . . . . . . . . . . . . . . . . 181

List of figures 185

List of tables 191

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Acknowledgements

First, I’d like to thank my two supervisors, Bob Stamps and Paolo Politi. You have

contributed so much of your time and expertise to this project, and it really has been a

privilege and a pleasure to collaborate with both of you. Thank you for the many cups

of tea/coffee (depending on which country I was in!) and the many conversations that

started with “I wonder if. . . ” or “So, I had an idea. . . ” And thank you also for the

conversations that started with “This would be better if. . . ” – you have both pushed me

to exceed my own expectations. Thank you Bob for encouraging me to do a PhD in the

first place, and for having the confidence in me to give me the opportunity to follow my

research ideas and interests (Me: “Do you know much about complex networks?” Bob:

“Are they networks that are. . . complicated?” Me: “Yeah, as far as I can tell. I was

reading a paper that applies them to a problem a bit like spin ice.” Bob: “Sounds fun!

Let me know how it goes.”) Thank you Paolo for making so much time for your students.

One of the highlights of my PhD has been the many conversations where we went over

details of calculations, and turned my handwaving arguments into rigorous calculations.

Thank you also for your hospitality over the years – I hope to have many opportunities

to return the favour in future!

My time over the past four years has been split between three research groups, plus

visits elsewhere, and there are many people to thank.

To the Condensed Matter group at UWA: thank you for being such a fun group of

people to work – and eat cake – with. Thanks to Matthew Ambrose, Chunlian Hu, Julian

de Jong and Rhet Magaraggia for bringing a bit of Perth to Glasgow; Rhet, thank you for

dragging me away from my thesis to go and take advantage of being in an amazing part of

the world. Thank you to Crosby Chang for all your hard work with me on the Singaporean

student outreach – it was really fun! Thank you also to the former members who showed

me the ropes when I was first starting out and who have taught me so much about the

vii

practicalities of being a physicist – Karen Livesey, Peter Metaxas and Kim Kennewell.

Karen, we only managed to collaborate on a project briefly, but maybe in the future. . . ?

And thank you to the non-Condensed Matter group people I spent many a morning tea

or beers on the roof with: Anna Lurie, Simon Tyler and Daniel Creedon, among many

others.

Thank you to everyone in the SSP/MCMP group at the University of Glasgow for

your hospitality over the past 11 months – it was partly the welcome I received that

convinced me to stay so much longer than my original short visit. Learning about all your

fun research projects was a welcome distraction from thesis-writing, as of course were the

pub trips, where I learned the concept of “emergency whisky” (“What’s the emergency?”

“That you don’t have any whisky.”) Thanks to Sam McFadzean for help with IT issues

and poster printing, and to Lucy Murray for much help with all the miscellaneous admin

issues that arise for a visiting researcher.

And of course, thanks to everyone at the Istituto dei Sistemi Complessi in Sesto

Fiorentino for making me feel at home and for patiently explaining many linguistic and

cultural differences, and of course for the many, many lunches at the mensa – who knew

you could get so much food for 5 euro? And thank you to Sofia Biagi for the conversational

practice and the introduction to Italian pop music.

I’ve also spent a few weeks in Leeds, where again I was made to feel so much at home

that I find it hard to believe I wasn’t there longer. A huge thanks to Chris Marrows and

Jason Morgan (and their collaborators outside of Leeds – Sean Langridge, Aaron Stein

and Joanna Akerman) for keeping me focussed on reality by making some really nice

experimental tests of some of my results. Jason, thank you for your comments on a draft

of this thesis and on several papers; they have been incredibly helpful for improving the

clarity of presentation.

Thanks to Giovanni Carlotti, Gianluca Gubbiotti, Marco Madami and everyone else

in the optical spectroscopy group at the University of Perugia for hosting me for a day

and for our interesting discussions about artificial spin ice Brillouin light scattering.

I’ve been fortunate to work in a research area that’s full of friendly, helpful people:

thank you to the many artificial spin ice researchers I’ve met who have been so happy

to share ideas and many helpful suggestions for this work. Thank you especially to Sean

Pollard and Yimei Zhu for helpful discussions about switching field distributions. And to

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non-spin ice people who have also made conferences so worthwhile: Thank you to Barbara

Kaeswurm for showing me around York and for the birthday (magnetic) lobster, and to

Ondrej Hovorka for pointing out the references to Bertotti et al’s network model for the

random field Ising model.

Thank you to Paul Abbot for your hard work as graduate coordinator in the School

of Physics at UWA, and also for your help with efficiently implementing the methods of

Section 5.3, without which I would probably still be waiting for calculations to finish.

Thanks to Jay Jay, Jeff Pollard, Micah Foster, Lee Triplett and Gay Hollister in

the admin team at the UWA School of Physics for your cheerful help with navigating

university bureaucracy. Thank you especially to Micah for all your help with purchasing,

and apologies for always having complicated bookings to make!

Of course, nothing could have happened without funding, and I gratefully acknowledge

support from the Hackett foundation, ARC, ARNAM, IEEE, INFN and WUN.

Finally, a huge thanks to my family and friends. Thank you to my parents for telling me

to follow my interests, and meaning it. And thank you to my sister Amy for much sensible

advice rendered in all-caps, such as “LEECHBLOCK IT AND DO YOUR THESIS”.

Words of wisdom for us all.

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x

Publications arising from this

work

Sections of this Thesis describe work that has previously been published in jointly authored

manuscripts. In all cases, the text in the relevant Section(s) of the Thesis has been written

entirely by Zoe Budrikis.

• Zoe Budrikis, Paolo Politi, and R L Stamps. Vertex dynamics in finite two-dimensional

square spin ices, Physical Review Letters 105, 017201 (2010). (Sections 2.4, 2.5,

3.5.) Z.B. developed and implemented numerical simulations, performed analy-

sis, conceived and implemented the population dynamics equations, and wrote and

edited the manuscript. P.P. and R.L.S. provided guidance and wrote and edited the

manuscript.

• Zoe Budrikis, Paolo Politi, and R L Stamps. Disorder types and equivalence of

disorder regimes in artificial spin ice, Journal of Applied Physics 111, 07E109 (2012).

(Section 4.2.) Z.B. performed calculations, developed and implemented numerical

simulations, and wrote and edited the manuscript. P.P. and R.L.S. provided guidance

and edited the manuscript.

• Zoe Budrikis, J P Morgan, J Akerman, A Stein, Paolo Politi, S Langridge, C H

Marrows and R L Stamps. Disorder strength and field-driven ground state domain

formation in artificial spin ice: experiment, simulation and theory, Physical Review

Letters 109, 037203 (2012). (Sections 4.4, 4.8.) Z.B. developed and implemented

numerical simulations, conceived and implemented the probability argument, and

wrote and edited the manuscript. J.P.M., J.A., A.S., S.L. and C.H.M. were respon-

sible for experimental work and wrote and edited the manuscript. P.P. and R.L.S.

provided guidance and edited the manuscript.

xi

• Zoe Budrikis, Paolo Politi, and R L Stamps. Diversity enabling equilibration: disor-

der and the ground state in artificial spin ice, Physical Review Letters 107, 217204

(2011) (Chapter 5.) Z.B. conceived and implemented the networks analysis, and

wrote and edited the manuscript. P.P. and R.L.S. provided guidance and wrote and

edited the manuscript.

• Zoe Budrikis, Paolo Politi, and R L Stamps. A network model for field and quenched

disorder effects in artificial spin ice, New Journal of Physics 14, 045008 (2012).

(Chapter 5.) Z.B. conceived and implemented the networks analysis, and wrote and

edited the manuscript. P.P. and R.L.S. provided guidance and edited the manuscript.

Some experimental data and microscopy images appear in Sections 1.2, 4.4 and 6.2.

These were all supplied by Jason Morgan of the University of Leeds, and come from

work done by him in conjunction with Christopher Marrows (University of Leeds), Sean

Langridge (ISIS), Aaron Stein (Brookhaven National Laboratory) and Joanna Akerman

(University of Leeds and Universidad Politecnica de Madrid).

Also published is the paper:

• Z. Budrikis, K L Livesey, J P Morgan, J Akerman, A Stein, S Langridge, C H

Marrows and R L Stamps. Domain dynamics and fluctuations in artificial square

ice at finite temperatures , New Journal of Physics 14, 035014 (2012).

None of the material in this paper is included in this Thesis.

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Chapter 1

Introduction

1.1 Background: geometrical frustration and the ice rules

The hexagonal phase of water ice exhibits geometrical frustration, that is, it is unable to

minimise all its interaction energies. The hydrogen ions sit on the oxygen-oxygen bonds,

each taking a position near to either one oxygen or the other, as illustrated in Figure 1.1(a).

As pointed out by Bernal and Fowler [1] in 1933, the electrostatic interactions between

the hydrogen ions cannot all be minimised: for any single oxygen ion, the four hydrogens

are repelled from one another and the energy would be minimised by all four hydrogens

taking positions far from their shared oxygen. However, this configuration cannot be tiled

over the whole tetrahedral lattice of oxygens, and instead the global energy is minimised

when each oxygen has two close and two distant hydrogen ions, a condition that has come

to be known as the ‘two-in, two-out ice rules’.

Around the same time as Bernal and Fowler’s work, Giauque and co-workers [2, 3]

discovered a striking consequence of this geometrical frustration. They measured the

residual entropy of water ice, and found a non-zero value of S0 = 0.82±0.05 Cal/deg·mol.

At first glance, this result is a violation of the third law of thermodynamics, and indeed,

it caused some puzzlement at the time. However, an elegant solution was pointed out by

Pauling in 1935 [4]. When Bernal and Fowler proposed their ice rules, they had in mind

that the hydrogen ions would take a unique, regular configuration of positions. However,

Pauling made the significant observation that, in fact, a large number of configurational

states obey the ice rules. The number Ω of these can be estimated by taking the number

of configurations allowed under the restriction of one hydrogen per bond – namely 22N ,

1

where N is the number of oxygen ions – and reducing it by a factor of 6/16 per oxygen

ion, to account for the fact that of the sixteen ways to arrange four hydrogens around an

oxygen, only six obey the ice rules. This gives Ω = 22N × (6/16)N = (3/2)N , leading to a

residual entropy S0 = kBln(Ω) = 0.81 Cal/deg·mol (where kB is the Boltzmann constant),

a value that agrees beautifully with experiment.

Figure 1.1: The ice rules. (a) In frozen water, the ice rules are for each oxygen ion (largeopen circle) to have two near and two distant hydrogen ions (small closed circles). (b) Inthree dimensional spin ices, the ice rules are two spins pointing in, and two out. Greenlines indicate satisfied interactions; red lines indicate frustrated interactions. Observe thatany single spin flip that causes a frustrated interaction to become satisfied will also causesatisfied interactions to become frustrated, because the ice rules describe the lowest energystates. (Image modified from that appearing in Harris et al.’s original paper [5].)

Some six decades after the ice rules were first applied to water ice, it was realised

that they described also the low temperature states certain rare earth titanates, which

are now known as ‘spin ices’: Ho2Ti2O7 [5, 6], Dy2Ti2O7 [7] and Ho2Sn2O7 [8]. In spin

ices, the magnetic rare earth ions sit on the corners of tetrahedra arranged in a pyrochlore

lattice. High crystal field energies constrain each magnetic moment to a single axis, and

in the relevant temperature range the system can be described in terms of classical Ising

spins which point either into or out of the centre of each tetrahedron, as illustrated in

Figure 1.1(b). The spins can be mapped onto the hydrogen ion positions in water ice: if

the centre of the tetrahedron corresponds to an oxgyen, then the inward directed spins

correspond to near hydrogens and the outward directed spins correspond to far hydrogens.

The interactions between spins within a tetrahedron are such that if one spin points into

the centre, its three neighbours ‘want’ to point out of the centre. These interactions are

frustrated, and the lowest energy states are given by the ice rules. As in water ice, the

ice rules allow for an extensive number of degenerate states, as was shown by Ramirez et

al. [7], who demonstrated that the measured zero-point entropy of the spin ice Dy2Ti2O7

2

agrees well with Pauling’s calculated value for water ice.

In fact, it is surprising that spin ice has an extensive set of degenerate states described

by the ice rules, given that long-range dipolar interactions are dominant in these sub-

stances [9], and it is not clear a priori that such interactions should not lift degeneracies

of different configurational states. Indeed, Melko et al. [10] have shown in numerical sim-

ulations that an ordered ground state is expected at sufficiently low temperatures, but

such a state is not seen experimentally, presumably because equilibration time scales be-

come longer than measurement timescales below a freezing transition temperature [11, 12].

However, apart from the issue of time scales, it has also been shown that long range dipolar

interactions still lead to quasi degeneracy of ice rules states for a wide range of param-

eter values due to ‘self screening’ [13], and furthermore that there exists a model dipole

interaction that differs from the physical dipole interactions only by fast-decaying terms

can give the same ground state manifold as the ice rules, even if other nearest-neighbour

interactions are included [14]. 1

One aspect of spin ices that has received a great deal of attention has been the

‘monopole’ excitations they support. It was pointed out by Castelnovo et al. [16] that

if one takes an ice rule obeying configuration and flips one of the spins, a pair of ice rule

violating defects is generated, which act as a source and a sink of the magnetic H-field

(note, however, that the magnetic B-field, B = µ0(H+M), does not have a source or sink,

and Maxwell’s equations still hold). The defects can be propagated and the ‘Dirac string’

between them lengthened by flipping a chain of spins, the key point being that along the

chain the ice rules are obeyed so there is no domain wall cost associated with separating

the monopoles. The monopoles are therefore deconfined, and one can also show that they

interact with an effective Coulombic 1/r interaction. The monopole model appears to be

able to describe a variety of experimental results [16–20], such as the heat capacity of spin

ice and the temperature dependence of relaxation dynamics, though not without some

controversy [21, 22].

1This lack of degeneracy lifting has been referred to as the ‘second magic’ of spin ice [15] (the ‘firstmagic’ being that the effectively antiferromagnetic nearest-neighbour interactions can be obtained fromferromagnetic couplings).

3

1.2 Artificial spin ice

It was in the context of growing interest in three dimensional spin ices that artificial spin ice

was developed as a two dimensional model of spin ice. Artificial spin ice was first named as

such by Wang et al. [23] in 2006, although honeycomb lattices that could also be classified

as artificial kagome spin ices were reported on by Tanaka and co-workers as early as

2003 [24–29] and it was noted that these systems tended to obey ‘ice rules’. Artificial spin

ice consists of nano-patterned arrays of magnetic material, typically Permalloy (Ni80Fe20).

The Ising macrospins can be realised either as separated elongated islands of magnetic

material, coupled by dipolar interactions, or as bars of a connected lattice, in which case,

macrospins are separated by domain walls and exchange interactions enter, at least on the

microscopic level. In both types of system, the dipolar interactions between macrospins

are frustrated by geometry, and the lowest-energy states are described by ice rules. A

scanning electron micrograph of a typical system is shown in Figure 1.2. As is typical

in artificial spin ices, the islands are small enough to be single-domain, that is, to good

approximation, the magnetic moment is aligned throughout the island. The elongated

shape of the islands constrains the magnetisation to point along the island long axis, so

that the islands act as Ising macrospins with two states. The islands are large enough

that switching between the two states cannot be driven by temperature – for example, the

islands studied by Wang et al. [23], the temperature required for thermal demagnetisation

is estimated to be ∼ 104 K, well above the melting point for the Permalloy islands. In

effect, therefore, the system is athermal, and dynamics must be driven by an external

magnetic field. Analogous nonmagnetic systems have also been proposed using colloids in

optical traps [30–32] and vortices in nanopatterned superconductors [33]. These alternative

systems are also frustrated and, in general, do not respond to thermal driving at room

temperature.

A key motivation for the development of artificial spin ice was the increasing capa-

bilities of micro- and nano-scale engineering to construct model physical systems. This

opens the possibility of studying a range of physical phenomena – not only ice models

– using routine experimental techniques. Examples of such ‘experimental simulations’

include quasi two-dimensional systems of colloids that can be made to undergo glass tran-

sitions [34], arrays of superconducting rings that have magnetic moments analogous to

4

Figure 1.2: A scanning electron micrograph of a square artificial spin ice. Theimage has been taken near the edge of an array, so the lower-left islands have fewer nearneighbours than the others. Image courtesy of Jason Morgan.

Ising spins [35, 36], controllable atomic-scale antiferromagnets [37], self-organised arrays

of nanoparticles [38] that act as dipolar ferromagnets [39], and nanopatterned magnetic

systems such as artificial spin ice. These systems are – at least to some extent – exper-

imentally controllable, and have the additional advantage that the state of each of their

components can be measured directly. For example, in the case of artificial spin ice, the

strength of interactions can be varied by changing island volumes (and hence magnetic

moments) or spacings, since the dipolar energy of a pair of islands goes as M1M2/r3 in a

point-dipole approximation, where the Mi are the island magnetic moments and r is the

island centre-centre distance. The spin configurations of artificial spin ice can be measured

using techniques such as magnetic force microscopy [23, 40–43], x-ray photoemission elec-

tron microscopy [44–46], and Lorentz transmission electron microscopy [47, 48]. This is

an important difference between artifical spin ice and 3D spin ice; in the latter, the spin

configuration cannot be probed on a single-spin scale without disturbing the state of the

system.

Although artificial spin ice was originally introduced as a model for three dimensional

spin ice, it has rapidly developed into a system that is studied in its own right. Over

the past few years, several key issues have arisen, which are quite distinct from the issues

discussed in the context of three dimensional ices. We discuss these briefly, and describe

how they relate to the themes of this Thesis.

5

Lattice geometry

The two geometries most commonly studied are square and kagome, which represent two

different approaches to transforming three dimensional ice into a two dimensional system.

Like three dimensional spin ice, square artificial spin ice has four spins per vertex and the

ice rules are two-in, two-out. However, unlike in the tetrahedra of three dimensional spin

ice where all pairwise interactions are of equal strength, square artificial spin ice features an

inequivalence between nearest neighbour and next-nearest neighbour interactions within a

vertex2. This is illustrated in Figure 1.3. This lifting of degeneracy leads to a well-defined

ground state in which the spins are arranged in a microvortex configuration, as illustrated

in Figure 1.4(a). The ground state is two-fold degenerate, with the two equivalent states

linked by a global spin flip. In other words, the extensive entropy of three dimensional ice

is lost.

Figure 1.3: The inequivalence of nearest-neighbour and next-nearest-neighbourinteractions within a square ice vertex. Nearest-neighbour (diagonal) pairs of spinshave a centre-centre distance that is a factor of

√2 shorter than the distance between

next-nearest-neighbour pairs. Accordingly, four of the six pairwise interactions within avertex are stronger than the other two.

On the other hand, kagome ice [29, 47], which has three spins meeting at each vertex,

as illustrated in Figure 1.4(b), does have an equivalence of pairwise interactions within

a vertex. The ice rules are modified to two-in, one-out or one-in, two-out, and in the

approximation that interactions are short-ranged only, there is an extensive number of

degenerate states that obey the kagome ice rules. In this sense, kagome ice is a better

2At least, in systems studied experimentally to date. Moller and Moessner [49] propose an intriguingvariation on square ice, in which islands are offset in height so that the equivalence of interactions arounda vertex is (approximately) restored.

6

model of three dimensional ice than square ice is. However, it should be noted that when

long-range interactions are included in kagome ice, an ordered ground state with low de-

generacy does emerge [50, 51], though such a ground state is not observed experimentally,

because the system is effectively frozen into a disordered configuration. Indeed, Mengotti

et al. [44] have experimentally studied systems of up to three connected hexagonal rings

(the ‘building blocks’ of kagome ice), and shown that as the number of rings is increased

the system is less able to attain its lowest-energy state. They conclude that as the system

size is further increased, attaining the ground state should become impossible.

Because kagome ice vertices always have an imbalance of in and out spins, they always

have non-zero magnetic ‘charge’. As a result, the field-driven dynamics of kagome ice can

be cast in terms of emergent ‘monopoles’ [45, 52–55].3 Here we see a clear advantage of

being able to directly image the ‘microstate’ of an ice system with monopole dynamics:

whereas the individual magnetic moments of three dimensional spin ice are experimentally

inaccessible and the monopole dynamics must be inferred, in the artificial systems the

dynamics can be observed directly.

a) b) c) d)

Figure 1.4: The three spin ice geometries studied in the literature. (a) Squareice, in which four spins meet at each vertex, with unequal interactions. (b) Kagomeice, in which three spins meet at each vertex and all interactions within a vertex areequivalent. (c) Brickwork ice, which can be obtained from square ice by removing islands.(d) Triangular ice, in which each vertex has six spins.

Square and kagome ices differ in both their topology and symmetry, a fact which has

motivated Li et al. [56] to fabricate an ‘intermediate’ brickwork lattice, which is illustrated

in Figure 1.4(c). This lattice has the topology of kagome ice, having three spins meeting

at each vertex, but like square ice the interactions between spins are not all equivalent.

The authors demagnetised samples of square, kagome and brickwork ice using the same

procedure and compared the calculated energies of the configurations attained, as well

3Note that there are two definitions of ‘monopoles’ used in the literature. Mengotti et al. [45, 55]generate a ‘background’ state of NaCl-ordered ±1 charged vertices using an external magnetic field, anddefine monopoles as vertices whose charge differs from their original charge by ∆Q = ±2. Such verticesmay be ice-rule violating three-in/out vertices, but it is also possible for an ice-rule obeying vertex to be amonopole under this definition. On the other hand, Ladak et al. [52–54] refer to all vertices with ∆Q = ±2as ‘charge carriers’, but reserve the term ‘monopole’ for vertices that do not obey the ice rules.

7

as the pairwise correlations, as a function of lattice spacing. They found that brickwork

lattices behaved in a manner quite similar to square lattices, and fundamentally different to

kagome lattices, leading them to conclude that the symmetry of a lattice is more important

than its topology in determining behaviour of the magnetic configurations.

A fourth geometry has been recently proposed, namely ‘triangular ice’ [57]. As seen

in Figure 1.4(d), this lattice has six spins meeting at each vertex, and the ice rules are

three-in, three-out. Like square ice, triangular ice has a well-defined ground state even

when interactions are short-ranged. An analysis in terms of the energetics of excitations

above the ground state has been presented. Interestingly, the lowest-energy vertex that

breaks the ice rules actually has a lower energy than the second-lowest energy ice rule

vertex. This is fundamentally different to square and kagome ices, where the ice rules

define the lowest-energy vertex states.

A question that – prior to this work – had not been answered in the discussion of array

geometry in the literature was whether and how the arrangement of spins at the edges

of arrays affected their response to external driving. For example, Figure 1.5 illustrates

three possible edge geometries for square ice. In small arrays, especially, one might expect

edge effects to be rather important. We address this question in this Thesis, and find that

in ideal systems, array edges play a significant role in determining dynamics. Indeed, as

shown in Appendix A.4, ideal arrays with periodic boundary conditions have only trivial

responses to applied fields, and free edges are required for interesting dynamics. However,

we will also see that quenched disorder ‘washes out’ edge effects in typical experimental

systems.

Open edges 4-island edges Closed edges

Figure 1.5: The three edge geometries studied in this Thesis. The geometry inthe bulk is the same for all three arrays, but their differing edges can lead to differentresponses to applied fields.

8

Field-driven dynamics and square ice ground state accessibility

Although a well-defined ground state exists for square ices, attaining this ground state in

experimental systems has proved to be challenging. Because artificial spin ices are typically

constructed to be athermal at room temperature, with island magnetisation switching

barriers of ∼ 105K, thermal demagnetisation of fully-grown samples is not possible. As-

grown samples have been observed to display long-range ground state ordering [42], which

is believed to occur during sample deposition, while islands are sufficiently thin that the

volume-dependent energy barriers to their magnetisation reversal are small enough to

be overcome by thermal ‘kicks’ [42, 58, 59]. Excitations on the ground state background

appear in proportions consistent with Boltzmann statistics [42, 58]. However, the majority

of experimental studies focus on field-driven dynamics, which, as we will see, offer a rich

variety of interesting physics.

One of the simplest means of generating dynamics is by means of a field applied along

a single axis, ramped up and down to generate a hysteresis loop. Such protocols have

been studied experimentally in both square [43, 60] and kagome [45, 52, 61] ices; in both

these geometries interesting phenomena are seen. For example, Morgan et al. [43] prepare

a square ice into a polarised configuration by applying a field at approximately 45 to

the island axes, and then study the configurations attained when the system is subject

to reversal fields that are approximately 10 off-axis. The off-axis nature of the fields

means that the horizontal and vertical sublattices of spins are reversed independently of

one another. Intriguingly, during the reversals, vertices with a net ‘charge’ (that is, three-

in, one-out or one-in, three-out vertices) tend to be slightly correlated in position, with

pairs of unlike charges more likely to be nearest-neighbour vertices than one would expect

for a random configuration, and nearest-neighbour pairs of like charges correspondingly

suppressed. In kagome ices, single-axis field protocols can be analysed in terms of avalanche

dynamics [45, 52, 61]. In Mengotti et al.’s [45] studies of disconnected kagome lattices,

the distribution of avalanche sizes in experiments is found to be exponential. On the

other hand, in Shen et al.’s [62] theoretical studies of connected kagome lattices, the

distribution is found to follow a power law. However, an important difference between the

two systems is that the disconnected kagome lattices are seen to support 3-in and 3-out

vertices, whereas in the connected lattices such configurations are never observed [47] and

instead always decay into longer strings of reversed links with ice-rule obeying vertices at

9

their ends. This promotes longer avalanches.

However, much of the literature [23, 33, 40, 44, 46–48, 56, 63, 64] on field-driven dy-

namics focusses on field-driven demagnetisation, in which an external magnetic field is

used to generate states with low net magnetisation, and (preferably) low energy. Samples

are typically prepared in a polarised state by applying a strong field at 45 to the island

axes, and are then rotated continuously while subject to an in-plane field of varying am-

plitude, which may also reverse direction in the lab frame. Non-trivial dynamics occur

when the field strength is in the vicinity of the island coercive fields [40, 63].

In this Thesis, we present simulation results and analysis for fields applied along a

single axis, as well as a simplified version of the ac demagnetisation protocols, in which

the sample is initially polarised and then a rotating field with fixed amplitude close to

the island coercive fields is applied. The advantage of these simplified constant-amplitude

field protocols is that they are feasible to analyse in some depth, as we do in this Thesis.

We also propose alternative fixed-amplitude field protocols, in which the sequence of field

directions is either random, or increases in steps ∆θ incommensurate with 2π.

Effects of quenched disorder

The ‘effectiveness’ of field-driven demagnetisation can be measured in terms of the final

net magnetisation of the sample, or the proportion of vertices in a local ground state

configuration. Wang et al. [63] report that the most effective protocol of those they study,

in terms of both of these measures, involves a rotating field that steps down in amplitude

from well above the island coercive fields, and reverses its direction in the lab frame at

each amplitude step. The other two protocols they study are a field that steps down in

amplitude without reversals, and a continuously decreasing field, both applied to a rotating

sample. From the same research group, Ke et al. [40] report that smaller field steps result

in lower-energy final states. However, even the most effective protocol is reported to give

configurations with no more than around 50% of vertices in the ground state [65, 66].

In contrast, Morgan et al. [67] obtain a similar level of ground state ordering using a

rotating field with constant amplitude close to the island coercive fields. However, care

needs to be taken in comparing results from different experiments, because, as we will

see in Chapter 4, different levels of quenched disorder – as would arise from unavoidable

variations in sample fabrication – can give rise to different outcomes from the same field

10

protocol. Indeed, Kohli et al. [60] report quantitative (though not qualitative) differences

in behaviour between samples prepared in different processing runs, and Morgan et al. [42]

report non-reproducibility of magnetic ordering during sample growth, which they propose

to be attributable to quenched disorder.

The importance of disorder has been recognised in the literature, both of artificial spin

ice [31, 33, 45, 52, 60, 61] and other artificial systems such as superconductor rings [35, 36].

For example, Kohli et al. [60] used magneto-optical Kerr effect magnetometry to simul-

taneously characterise 5000–40, 000 islands (depending on lattice spacing) and construct

hysteresis loops. Comparison of angle dependence of experimental and simulated loops

revealed that the behaviour of the experimental system could only be reproduced in simula-

tions when disorder was present. In kagome artificial spin ices, quenched disorder strength

has been quantified, and some discussion has been made of how it affects avalanches during

array magnetisation reversal in those systems [45, 52, 61]. Disorder has also been discussed

in the context of artificial ‘spin’ ice in patterned superconductors [33], where simulations

suggest that disorder in the heights of barriers between pinning sites prevents the system

from reaching its ground state. An analysis of hysteresis loops generated in simulation

studies of a colloidal analogue of square artificial spin ice has also been performed by Libal

et al. [31], who analyse loops in terms of return point memory, that is, the tendency for

a hysteretic system to return to identically the same state after cycling through a mi-

nor loop. They show that disorder increases the amount of return point memory during

hysteresis, because of pinning.

However, a general understanding is still lacking of how exactly disorder affects dy-

namics and the accessibility of configurational states, especially in square artificial spin

ice. This is a gap that this Thesis seeks to partially fill. We use numerical simulations

to examine how quenched disorder affects the response of a system to external fields. We

find that different types of disorder have similar effects, and can be characterised in terms

of their strength, with two disorder regimes possible. We also discuss how disorder affects

the states available to the system – disorder increases the number of states accessible, but

in general makes the ground state inaccessible.

11

Theoretical approaches

As we have already seen, artificial spin ice is attractive because it can be used as a testing

ground for a wide variety of physics, beyond that of three dimensional spin ices. This has

lead to a variety of theoretical approaches being taken in the literature. For example, the

configurations attained by a square artificial spin ice after field-driven demagnetisation can

be analysed in terms of an effective temperature formalism [65, 66], an approach which

makes contact with work on granular systems [68–70]. The key idea of this approach is

that the system contains a fraction (1 − ρ) of ‘background’ vertices that remain in their

initial configuration and a fraction ρ of ‘defect’ vertices that are changed by the external

field. Under the assumption that demagnetisation maximises entropy, the final configura-

tion is described by a value of ρ that maximises the total entropy, and population fractions

of defect vertex configurations that maximise the entropy of that subsystem. These are

calculated under the experimentally-observed constraint that energy is (approximately)

conserved, although it is not clear why this occurs. This approach leads to surprisingly

good agreement with experimental results, and provides a way to define an effective tem-

perature: just as in standard thermodynamics, the Lagrange multiplier in the entropy

maximisation problem is the system’s temperature, here the Lagrange multiplier is the

effective temperature.

Alternatively, Lammert et al. [71] have analysed the same experiments in terms of the

Shannon entropy per spin, which they calculate using a conditional entropy argument that

allows them to make an accurate approximation of the entropy of a global configuration

based on the configuration of a small cluster. They find that demagnetised kagome ice

has slightly lower entropy density than an idealised uncorrelated system, indicating that

some correlations are present in that system. On the other hand, square ices are found to

have entropy densities significantly above zero, even in the best cases, indicating that the

system has not approached its low-degeneracy ground state.

Connections can also be made with the famous vertex models [72–79] which were

exactly solved in the 1960s. For example, if long-range interactions are neglected, a square

artificial spin ice can be viewed as a realisation of the sixteen vertex model, with vertex

weightings dictated by interactions, and a square ice configuration that fully obeys the ice

rules can be described by the six vertex model. Recent modelling of thermal dynamics of

a sixteen vertex model by Levis and Cugliandolo [79] shows that if the vertex weights are

12

consistent with those expected for artificial spin ice energetics, the outcome of a thermal

quench is consistent with results seen in Monte Carlo simulations of artificial spin ice [58].

The emphasis of the work in this Thesis is on developing simple models with wide

applicability, in order to understand the essential physics of the system and to relate it

to other phenomena. Two aspects of the work are particularly noteworthy. The first is

a description of dynamics under a rotating external field via equations for the evolution

of populations of different vertex types. Comparison of results from population dynam-

ics equations and simulations enables us to comment on the significance of the initial

configuration of the system and of correlations.

The second is a mapping of the field-driven transitions between configurational states

onto a directed network. In this approach, we ‘step back’ and view the applied field as

‘transporting’ the system from one spin configuration to another, in a path through the

phase space of all configurations. Such a phase space is discrete and can be enumerated

exactly, because each of the N spins can take 2 configurations, giving 2N configurations

total. The discrete phase space, and the pathways through it, can be mapped onto a

directed network – that is, a mathematical object consisting of a set of nodes connected

by links. In this mapping, network nodes are spin configurations and links represent

dynamical transitions.

1.3 The model of interactions and dynamics

One of the key methods used in this work is numerical simulations of how artificial spin

ice responds to external fields. In this Section, we discuss the choices made in developing

the model that is used in these simulations. The model also forms the basis for much of

the other work: for example, in Section 4.8 we describe how quenched disorder can block

dynamical pathways into the ground state, under assumptions about dynamics that are

described in this Section.

The guiding principle at each stage of developing the model has been to keep it as

simple as possible, in order to uncover the essential physics of the system. Of course,

simplifications are only useful inasmuch as they strip away inessential detail, and it is

important to justify the choices made, and to be aware of any limitations of the model

used.

13

The most accurate modelling method would be micromagnetic simulations, such as

OOMMF [80], Nmag [81] or Magpar [82]. In such simulations, the goal is to find a

magnetisation distribution that minimises the energy of the system, using either a regular

grid of volume elements (in finite difference schemes, such as OOMMF) or an adaptive

mesh (in finite element schemes, such as Magpar and Nmag). Time dependence is included,

generally via the Landau-Lifshitz-Gilbert equation [83, 84]

∂ ~M

∂t= −|γ| ~M × ~Heff −

|γ|αMS

~M × ( ~M × ~Heff), (1.1)

where γ is the Landau-Lifshitz gyromagnetic ratio, α is the Gilbert damping coefficient,

~M is the magnetisation at a point, and ~Heff is the effective field at a point, given by

~Heff = − 1

µ0

∂E

∂ ~M, (1.2)

where µ0 is the permeability of vacuum and E is the energy, generally a sum of exchange,

dipolar, anisotropy and Zeeman energies.

When implemented carefully, micromagnetic simulations offer accurate results [85], and

have proved to be a useful tool for studying problems such as the details of magnetisation

reversal in nanowires [86, 87]. However, they are computationally expensive, and not

feasible for spin ice systems of more than a few islands – a typical example of this being

the work of Kohli et al. [60] and Phatak et al. [48], who study systems of 24 islands. We

wish to study larger arrays with hundreds of islands, and micromagnetic simulations are

not feasible for this task.

Within a model that is simpler than micromagnetics, there are three issues that need

to be dealt with: first, what states are available to the island magnetisations; second –

and related to the first – what form the coupling between islands should take; and third,

how to implement dynamics. There are many options that are feasible to implement in a

simulation, but we will see that all the essential physics is captured by a minimal model

of Ising point dipoles that flip when the energy benefit from doing so is above a threshold,

with single spin flip dynamics.

As already mentioned, shape anisotropy constrains island magnetisations to be di-

rected approximately along the island long axes. This result has been well confirmed ex-

perimentally by several experimentalists studying islands with a range of dimensions, via

14

techniques such as magnetic force microscopy, Lorentz transmission electron microscopy,

and x-ray photoemission electron microscopy, see, for example [23, 41, 42, 44, 46, 48].

Indeed, the two-state nature of the island magnetic moments is a key feature of the design

of artificial spin ice, as it is designed to be an artificial Ising spin system. Therefore, since

the approximation of Ising macrospins is experimentally shown to be realistic, and any

small deviation from Ising-like behaviour is not of interest, we treat the islands as being

single-domain and having two states, rather than, for example, Heisenberg spins. We also

neglect any possibility of the island magnetic moments varying in magnitude.

Within the constraint that islands have two magnetisation states, we are still faced

with the question of how to model their interactions. The islands have a finite volume,

and – especially for close-spaced islands – the details of the islands’ shape affects their

interactions [46, 88]. Whereas the treatment of islands as having two magnetisation states

is widely accepted in the literature, there is no clear consensus about what form the

interactions should take.

For example, Rougemaille et al. [46] split the interactions of a kagome ice system into

short-range and long-range terms. The long-range terms account for all interactions other

than nearest-neighbours, and micromagnetic simulations show that these interactions are

well-approximated by interactions between point dipoles situated at the island centres, so

that the energy of island i is:

E(i)dip = −~h(i)

dip · ~Mi

= −D∑j 6=i

(~mi · ~mj

r3ij

− 3(~mi · ~rij)(~mi · ~rij)

r5ij

),

(1.3)

where ~M = M ~m is the island magnetic moment, where ~m is a dimensionless unit vector

in the direction of the island magnetisation. The inter-island distance r is given in units

of the lattice constant a. The constant D = µ0M2/(4πa3). The short-range terms in the

total interaction energy account for nearest-neighbour coupling, and in the kagome system

these are significantly stronger than predicted by a point dipole model, being increased

by a factor of approximately 5. This is why these interactions are treated separately.

An alternative, analytical, means of accounting for the effect of finite island sizes is

to assume that islands are in a single domain state, then perform a series expansion of

a sum over infinitesimal volume elements [88]. The first term of the series for a pair

15

of islands is the energy of a pair of point dipoles situated at the island centres, and

is proportional to r−3, where r is the island centre-centre distance. The second term

depends on a ‘moment of inertia’ of each island, to take into account the island shape,

and is proportional to r−5. A key result that can be obtained from this expansion is that,

in the case of the square artificial spin ice geometry, the perturbation on the point dipolar

energy serves to ‘reinforce’ the dipolar coupling. That is, although the relative strengths

of interactions are changed, their signs are not, a result consistent with Rougemaille et

al.’s [46] micromagnetic modelling.

Moving from a picture of islands with finite volumes to a picture of islands with finite

lengths [89], islands can be treated as consisting of ‘dumbells’ of magnetic ‘charge’ [66]. The

finite length of the islands can be important: for example, Moller and Moessner [89] show

that increasing the ratio of island length l to lattice constant a broadens the temperature

range for the ice rules-obeying regime in Monte Carlo simulations. However, as with finite

volume corrections to island interactions, the dumbell model does not qualitatively affect

the energetics of the system – for example, the energetic ordering of vertex types remains

unchanged [66]. It is therefore reasonable to expect that none of the physics that exists

in the point dipole approximation would be strongly affected by using a more complex

model for island interactions. Motivated by this, we neglect the island volumes in our

calculations.

The range of spin interactions, however, can be important. In Appendix A.3, we

demonstrate that in simulations, interactions between the closest spins in neighbouring

lines of spins in the square lattice must be accounted for in order to obtain qualitatively

similar simulation results to those obtained for simulations in which all spins interact

with all other spins. Unless stated otherwise, in this Thesis, interactions are taken to be

long-ranged so that every island interacts with every other island, and the island magnetic

moments are treated as Ising point dipoles situated at the island centres. The energy

of island i due to interactions with other islands is given by Equation (1.3). We set the

constant D = µ0M2/(4πa3) to unity, so that the nearest-neighbour coupling has strength

jN = 1.5. This gives the energy scale of the island-island interactions.

Although the interactions are long-ranged, the most important interactions are those

of the near neighbours, which can be divided into three types, using the same scheme

as Wang et al. [23]. These are illustrated in Figure 1.6. A bulk spin has four nearest

16

NN

L T

Figure 1.6: Types of near-neighbour interactions. The highlighted spin has fournearest neighbour (NN) interactions, two next-nearest neighbour L interactions, and twonext-nearest neighbour T interactions. We use the same scheme for labelling next-nearestneighbours as Wang et al. [23].

x

y

H

Figure 1.7: One of the four possible polarised configurations, with net momentin the +x direction. The field direction used to generate the state is also shown.

neighbours, two next-nearest neighbour L interactions and two next-nearest neighbour

T interactions. The difference between L and T next-nearest neighbours is the strength

of their coupling: although they are both the same distance from a spin, L neighbours

have a coupling of strength jL = 1/√

2 and T neighbours have a coupling of strength

jT = 1/(2√

2), in our reduced units.

The other field acting on each island is the external field ~h, which gives a Zeeman

contribution to the total energy

E(i)Z = −~h · ~Mi. (1.4)

When a field strong enough to overcome the dipolar interactions is applied at approxi-

mately 45 to the island axes, the system’s favoured state is a polarised configuration, in

which all spins have the same projection onto the field. One of the four possible polarised

configurations, with net moment in the +x direction, is illustrated for a 4× 4 spin array

in Figure 1.7. Polarised configurations are of particular interest because they are exper-

imentally reproducible, and the majority of this Thesis will deal with dynamics starting

from an initial polarised configuration.

In addition to the island interactions, a second energy scale in the system is a barrier

to island switching. How this is modelled is a factor in determining how the system’s

17

dynamics proceed. One point that is common to all models of switching is the ather-

mal, field-driven nature of island magnetisation reversal, due to the relatively large island

volumes used in experiments. Additionally, the island magnetisation reversal barrier is

high enough compared to interactions that switching must be driven by an external field

– using the system studied by Wang et al. as an example again, the nearest-neighbour

interactions are estimated to be ∼ 10 Oe, but the switching barriers are ∼ 770 Oe [40].

This point is important, because it means that if a spin configuration is prepared using

an external field and the field is switched off, the configuration observed at remanence is

the same as the configuration under the field.

The most accurate way to model island magnetisation switching is, as mentioned above,

to use micromagnetic simulations to study the response of an island magnetisation to ex-

ternal fields, mapping out the dependence on field amplitude and direction. Alternatively,

magnetisation switching can be modelled in an analytical approximation, such as the

Stoner-Wohlfarth model [90, 91], in which the moment takes a value that minimises the

sum of Zeeman and shape anisotropy energy for a uniformly magnetised ellipsoid in an

external magnetic field.

In the model we use, however, an island’s magnetisation can switch only if the compo-

nent of the total field antiparallel to an island’s magnetisation be greater than the island’s

intrinsic switching field h(i)c :

−(~h(i)dip + ~h) · Mi > h(i)

c , (1.5)

where M is the dimensionless unit vector along ~M . In this Thesis, we set the mean

switching field to hc = 11.25, a value outside of the range of dipolar coupling strengths, so

that an external field is required to drive dynamics. A similar threshold-based model for

switching has been used by other authors [41, 45, 49, 52], and remarkably, it gives results

that agree well with experiment [45, 52, 67], even though in reality the magnetisation

reversal of islands should be much more complex. In fact, we show in Appendix A.2 that

other switching criteria, such as Stoner-Wohlfarth switching, give qualitatively similar

system dynamics to the threshold model.

A model for dynamics requires not only a description of the reversal of an individual

island, but also a description of which islands can reverse, and when. One question

is whether to include time dependence explicitly. For example, there are Monte Carlo

18

schemes that allow for the determination of time dependence in a thermal system [92], and

as mentioned above, micromagnetic simulations can be used to model temporal evolution.

However, because in experiments the island magnetisation reversal happens very rapidly

(on a scale of nanoseconds [93, 94]) compared to the change in field angle or amplitude (for

example, in rotating field protocols, samples are typically rotated at ∼ 1000 rpm [23, 67]),

we are justified in neglecting time dependence and instead using a procedure where a field

is applied and the system is allowed to respond fully to that field.

Several spin updating algorithms exist that are based on collective spin flips, that is,

where more than one spin is simultaneously updated. For example, in ‘loop moves’ [95–

100], closed loops of spins in a configuration are identified. At each vertex the loop passes

through, one spin of the loop points in, and the other points out. If all of the spins in the

loop are flipped simultaneously, the configuration can be changed dramatically with only

a small concurrent change in energy, since the number of in and out spins at each vertex

is not changed. Loop moves allow simulations to escape energy minima in a reasonable

computational time, which is useful when searching for the ground state of an energy

landscape with many traps. However, our work is intended to model the behaviour of

real artificial spin systems, which are indeed unable to escape traps in experimental time

scales, and we instead use single spin flip dynamics. The choice of single spin flip dynamics

is common in the literature on artificial spin ice [45, 46, 52, 89, 101].

In our implementation of single spin flip dynamics, the set of all spins that satisfy the

criterion (1.5) is calculated, then one is chosen uniformly at random and flipped. The

set of spins satisfying (1.5) is re-calculated for the new configuration, and again one is

flipped at random. This process continues until no further spin flips are possible, at which

point a ‘final’, stationary configuration has been attained. The code for these simulations

is included in Appendix A.1. Because spin flips are selected at random and the set of

spins that can flip is recalculated at each step, more than one series of spin flips may be

possible under application of the same field. As a result, different simulation runs do not

necessarily have the same outcome, even in ideal systems without disorder. We average

over outcomes of different runs. In our network studies of Chapter 5, we instead enumerate

over all outcomes.

As seen in Equation (1.5), the response of an island to the external driving field is

controlled both by its interactions with other islands, and its intrinsic switching field,

19

unless the external field is strong enough to overcome these. Inter-island interactions bias

the response of the system towards low-energy states. At the same time, disorder, in the

form of a distribution of island properties, introduces randomness in the response to fields.

Different types of disorder are possible, such as disorder in island positions, or disorder in

the switching thresholds hc. In Chapter 4 we compare different disorder types and show

their effects on dynamics in our model are similar, and that all disorder types can be

characterised in terms of an effective switching field disorder.

1.4 Overview of this Thesis

This Thesis is a study of athermal, driven dynamics of square artificial spin ice. The aim

is to model how dynamics are affected not only by the complex energy landscape but also

disorder in the system. The main methods used are: numerical simulations, population

dynamics equations, arguments about the probability distribution of island characteristics,

and an analysis of dynamics in terms of networks.

The overall progression of this Thesis is from a study of an ideal system, via a discus-

sion of how the inequivalence of edge and bulk spins in an ideal system affects dynamics,

to a study of the effects of disorder – which can be thought of as making all spins inequiv-

alent to one another in a random way. We study disorder effects via simulations and an

enumeration of the configurational phase space of the system.

In Chapter 2, we describe the coarse-grained ‘vertex’ picture of square spin ice dy-

namics, which was first introduced by Wang et al. in the first reported study of square

artificial spin ice [23]. We use this picture to analyse simulation results, and to develop

vertex population dynamics equations that describe the evolution of square artificial spin

ice under a rotating applied field.

In Chapter 3, we explore how array edges control dynamics in ideal square spin ices.

We do this via numerical simulations of a variety of field protocols acting on systems

with similar sizes but different edge geometries. Comparison of the outcomes of different

field protocols also reveals how the sequence of fields applied to the system controls which

dynamical processes occur.

In Chapter 4, we examine disorder effects, using numerical simulations and analytical

arguments to show that the strength of disorder is more important than its origin, and

20

that strong disorder increases the number of states accessible to dynamics, but at the same

time makes the ground state inaccessible.

In Chapter 5, we continue the analysis of dynamics in terms of accessibility of states,

by constructing networks to represent the configurational phase space of square artificial

spin ice. The properties of networks constructed for different realisations of disorder can

be compared, to gain further understanding of how disorder affects dynamics. Networks

also provide a representation of dynamics that makes it straightforward to study the effects

of, for example, changing the field protocol, or controlling a single spin in the array.

Finally, in Chapter 6, we outline directions this work could take in the future.

21

22

Chapter 2

Spin ice dynamics: vertex

populations and processes

2.1 Overview

In this Chapter, we introduce a coarse-grained picture of spin ice configurations and dy-

namics, describing them in terms of vertices. This approach has been used extensively in

the literature [23, 30, 33, 40, 41, 43, 49, 65, 66].

We first define the four vertex types of square artificial spin ice in Section 2.2, before

discussing in Section 2.3 how vertices can be used to describe configurations and dynamics.

We discuss how this picture has been used in the literature, and describe its advantages

and also some of its pitfalls, such as cases where configurations that are close in vertex

population are dynamically ‘distant’.

We then describe the response of an ideal artificial spin ice to a constant-amplitude

rotating field in terms of the dynamics of vertices. As mentioned in the Introduction, a

rotating field with fixed amplitude serves as a simplified ‘model’ for the more complex ac

demagnetisation protocols that have been studied extensively in the literature [23, 33, 40,

44, 46–48, 56, 63, 64]. The benefit of studying the simplified protocol is that it is feasible to

describe the dynamics completely, as we do here in Section 2.4. This detailed description

is worth making because it will allow us in Chapter 4, where we study the dynamics of

a more realistic, non-ideal artificial spin ice, to see which dynamical processes have been

affected by quenched disorder in the system.

The picture of dynamics in terms of vertex processes is further formalised in Sec-

23

tion 2.5, where we write down and solve analytical equations for the evolution of vertex

populations in a ‘mean field’ picture of dynamics. The comparison between analytical

results and simulations gives information about how the key assumption of the analyt-

ical approach – namely, that vertices are well-mixed and their population fractions are

spatially independent – breaks down.

2.2 Vertex types in square artificial spin ice

The 24 = 16 vertex configurations of square artificial spin ice can be classified into groups

based on energy. The groupings depend in the energetics of the system. In magnetic

systems, which have received the most attention to date, island net moments preferen-

tially align to close flux loops, whereas in proposed systems of charged colloids in optical

traps [30–32] or vortices in patterned superconductors [33], colloids (or vortices) repel each

other. Thus, the lowest energy configuration of an isolated vertex is different for the two

types of system: in the magnetic system it is a type 1 vertex configuration, as illustrated

in Figure 2.1, whereas in the colloidal (or superconductor) systems it is a vertex with

colloids (or vortices) as far apart as possible, equivalent to the second type 4 vertex shown

in Figure 2.1, if arrows are taken to point towards colloid/vortex positions. (However, an

‘all out’ vertex cannot be tiled over an entire square ice array, and instead the ground

state of colloidal and superconductor square ices maps exactly onto that of a magnetic

system, namely, the microvortex configuration illustrated in Figure 1.4(a).)

The vertex configurations of a magnetic square ice can be divided into four groups

based on energy. Type 1 vertices have the lowest energy, and the ground state is a

chessboard tiling of the two type 1 vertices. The two-fold degeneracy of the ground state

can be thought of in terms of the two possible chessboard tilings. Type 2 vertices also obey

the ice rules, but have a net magnetic moment and higher energy than type 1 vertices.

There are four distinct type 2 vertices, and a tiling of a single one of these is known as a

‘polarised’ state, with maximum possible net moment. Type 3 and 4 vertices break the

ice rules. Type 3 vertices have both net magnetic moment and ‘charge’, with three spins

in (out) and one spin out (in). Type 4 vertices have all spins in or out and are doubly

charged. They are rarely observed in experiments or simulations of strongly interacting

islands (for example, in Nisoli et al.’s effective temperature studies, non-zero type 4 vertex

24

populations are only reported at high effective temperatures, for far-spaced islands [66]).

Type 1 Type 2 Type 3 Type 4

Figure 2.1: The square ice vertex types. The sixteen configurations are classed intofour types of equal energy, as described in the text.

2.3 Vertex configurations and dynamics

The use of vertex populations to describe square spin ice configurations dates back to the

original work of Wang et al. [23], who compared the number of ice-rule-obeying type 1 and

2 vertices found in their systems after ac demagnetisation to the populations that would

be expected if the configurations were random. In a random configuration, one expects

the population fractions of vertex types to follow their degeneracies, so that 2/16 = 12.5%

vertices are of type 1 and 4/16 = 25% vertices are of type 2. Instead, for the closest-spaced

lattices studied, Wang et al. found ∼ 35% of vertices in a type 1 configuration and ∼ 40%

of vertices in a type 2 configuration. The tendency for vertices to have ice-rule-obeying

configurations indicates that interactions between islands are important: the number of

energetically-favourable ice-rule-obeying vertices was lower for systems with larger lattice

constant (and hence weaker interactions), so that in the limit of non-interacting islands

the configurations were approximately random.

In much of the literature, vertex populations are used as an indication of how ‘close’ the

system is to its ground state. Two approaches have been taken. The first, used by Libal

et al. [30, 33], and which we will use in this Thesis, is to count the fraction of ground state

vertices. For example, in a system of colloids in optical traps, the state attained by thermal

annealing is random when the colloid-colloid interaction is zero, and is analogous to the

ground state of magnetic square artificial spin ice when the colloid-colloid interaction is

strong. The transition between disorder and order can be seen in the population of ground

state vertices.

Alternatively, one can calculate an approximate configuration energy for the whole

25

Figure 2.2: Energy vs type 1 population and magnetisation for random con-figurations. In both cases, the energy is the total inter-island dipole interaction energy(see Section 1.3). It is plotted against type 1 vertex population fraction n1 (left) andnormalised net magnetic moment m (right). The data are from 3000 random spin config-urations of a 20×20 island open edge array. While there is a trend in the Edip vs n1 data,no such trend is present in the Edip vs m data.

array [40, 65, 66] by assigning an energy Ei to each vertex type and then counting vertex

populations Ni, so that

E = E1N1 + E2N2 + E3N3 + E4N4. (2.1)

The energy can then be compared to E1N , the energy of the ground state (where N is

the total number of vertices). This approximation for the configuration energy is most

accurate for configurations with small local and global net magnetic moment, such as

demagnetised configurations, so that the contribution of long-range dipolar interactions is

minimised.

In fact, these two approaches are related, because, as we will show here, the type 1

vertex population fraction n1 is quite strongly correlated with dipolar energy, at least

for configurations accessed in our simulations. Even for random configurations, n1 is

more strongly correlated with the dipolar energy than are other quantities, such as the

net magnetisation. In our simulations, we can readily calculate the complete dipolar

energy of a configuration, involving summing over the interactions between all pairs of

spins, and given by Equation (1.3). Figure 2.2 shows this energy as a function of n1 and

m =√m2x +m2

y for a set of 3000 random configurations. There is a trend for Edip to

decrease with increasing n1, but no such correlation is seen in Edip vs m.

The correlation between energy and n1 is much stronger if we limit ourselves to the

configurations accessible from a polarised configuration using a rotating field. This subset

of configurations is relevant to our studies of rotating field protocols that we report later

26

a)

b)

Figure 2.3: Energy vs type 1 population and magnetisation for configurationsobtained by a rotating protocol acting on an initially polarised configuration.In both cases, the energy is the total inter-island dipole interaction energy (see Section 1.3).It is plotted against type 1 vertex population fraction n1 (top) and normalised net magneticmoment m (bottom). The energy is scaled to the energy of a ground state configuration.The data are from configurations of a 20×20 island open edge array, initially in a polarisedconfiguration and subject to a rotating applied field.

in this Chapter. Figure 2.3 shows the dipolar energy as a function of n1 and m for each

configuration accessed when an initially polarised array is driven by a rotating field of fixed

amplitude h, which takes values 10, 10.25, 10.5, . . . 13 (for more discussion of this field pro-

tocol, see Section 2.4). The strong correlation between Edip and n1, especially compared

to m, indicates the utility of the type 1 vertex population for describing configurations.

The second aspect of the vertex picture relates to dynamics. An advantage of the

vertex picture is that single spin flip dynamics – the type of dynamics studied in this

Thesis – usually modify vertex populations. For example, flipping a spin of an isolated

type 2 vertex will convert it to a type 3 vertex. This is illustrated in Figure 2.4. In the rest

of this Thesis, we describe vertex processes using a notation where i© refers to a vertex

of type i, so that a type 2 – type 3 conversion is written as 2© → 3© and a spin flip that

27

Figure 2.4: How a single vertex can evolve via single spin flips. The spin that hasflipped is indicated by a dotted arrow. The notation i© refers to a vertex of type i.

a) b)

Figure 2.5: Example vertex pair processes. a) Type 3 propagation with type 1creation. b) Type 3 propagation without type 1 creation. The spin that has flipped isindicated by a dotted arrow. The notation i© refers to a vertex of type i.

converts a type 2 – type 3 pair into a type 3 – type 1 pair is written as 2© 3© → 3© 1©,

where the different order of symbols indicates that the vertex types have swapped position.

Figure 2.5 shows these example processes. In the rest of this Chapter, we describe the

response of a square ice to a simple rotating field protocol using vertex processes, and

construct an analytical description of dynamics from them.

Casting dynamics in terms of vertices also allows us to connect with the ‘monopole’

picture of spin ice dynamics. Dynamics on a uniform type 1 or type 2 background always

involve type 3 vertex propagation [43, 102, 103], because flipping a single spin in a type

1 or 2 vertex always creates a type 3 vertex. Indeed, as illustrated in Figure 2.4, vertices

cannot pass directly between type 1, 2, or 4 configurations without taking an intermediate

type 3 configuration. This is because the number of in (or out) spins of a vertex can only

increase or decrease by 1 at a time – type 1 and 2 vertices have 2 spins in, and type 4

vertices have either 0 or 4 spins in, so the intermediate configuration must be a type 3

vertex with either 3 spins or 1 spin in. Type 3 vertices, with their imbalance of in and out

spins, carry a net magnetic charge q = ±1. On the other hand, type 1 and 2 vertices have

q = 0. Thus, type 3 propagation can be thought of as the motion of charges.

One of the values of a description of configurations and dynamics in terms of charges

28

lies in the possibility of describing configurations and energetics in terms of ‘monopoles’.

As mentioned in the Introduction, experimental evidence of interactions between charged

vertices (though not explicitly of Coulomb’s law) has been reported by Morgan et al. [43],

who find that in zero-net-magnetisation states that occur during magnetisation reversal,

neighbouring vertices have an enhanced density of pairs of unlike type 3 vertices, and a

suppressed density of like type 3 pairs, relative to a random allocation of type 3 vertex

positions. It should be noted though, that ‘monopoles’ in square artificial spin ice are not

deconfined, unlike those found in three-dimensional spin ice [16]: Mol et al. [102] have

shown that the separation of a pair of ‘monopoles’ has an associated energy cost that

grows with their separation, due to the topological necessity of the ‘string’ between the

pair having a higher energy per vertex than the ground state background 1.

We conclude this Section by noting some subtleties involved in describing states in

terms of vertices. The first is that two states with similar vertex populations may not be

dynamically accessible from one another. An extreme example of this is the two possible

type 1 tilings. Both have population fractions n1 = 1 but as we will see in subsequent

Chapters, it is very hard, if not impossible, to design a field protocol to take the system

from one to the other in a reproducible way. Alternatively, two states with very different

population fractions may be dynamically linked by a single applied field. For example, we

will see in Section 3.3 that when a field of strength 14.55 is applied in the −x direction to an

array in a +x polarised state the vertex population fractions jump from n1 = 0, n2 = 1

to n1 = 0.89, n2 = 0.11 in a single step. In Chapter 5, we will discuss the ‘connectedness’

of configuration states in more detail and show that complex networks provide a useful

framework for describing this.

There is also a loss of information associated with using the populations of four vertex

types rather than sixteen vertex configurations. For example, the process 2© 3© → 3© 2©,

shown in Figure 2.5(c), is one of the processes by which type 3 vertices propagate on a

type 2 background, and is common in dynamics that start from a polarised configuration.

In this process, the configuration is changed but the vertex populations are not, and a

description of dynamics in terms of global vertex populations – such as the one we develop

in this Chapter – cannot take this process into account. We note that a description in

1We note that in proposed systems with a height offset between sublattices [89] so that interactionswithin a vertex are equivalent, that is, jN = jL, the ‘string’ can have approximately the same energy asthe background, and the cost of separating a pair of charges disappears [50, 103].

29

x

y

ө

H

Figure 2.6: Definitions of the x and y directions, and the field angle θ. In allour simulation studies, the array is initially in a +x polarised configuration as shown. Instudies of rotating fields, the field is rotated from θ = 0 (the +x direction) in a counter-clockwise manner.

terms of spatially-dependent vertex populations can describe the process 2© 3© → 3© 2©,

but this requires the inclusion of extra information (spatial dependence) to make up for

the loss of information about vertex configurations.

To give a second example of information lost when using four vertex types: there are

six vertex pair processes that start with a type 3 pair, for example 3© 3© → 1© 1© and

3© 3© → 1© 4©. When all sixteen vertex configurations are taken into account, these two

example processes have different initial states, but when described in terms of vertex types,

as they are here, the two processes have the same initial state. We will see in Section 2.5

that this must be taken into account when developing a model for dynamics in terms of

vertex populations.

2.4 Vertex dynamics of rotating field protocols

We now use a picture of vertex dynamics and populations to analyse the dynamics of an

ideal array subject to a constant-amplitude rotating applied field. The constant-amplitude

field protocol, studied experimentally by Morgan et al. [67] is a simplification of the one

originally used by Wang et al. [23, 63] in which the field also changes amplitude and under-

goes phase shifts. We will see here that even the simplified protocol leads to rich dynamics,

and that a detailed analysis of these dynamics provides a foundation for understanding

other field protocols and other array geometries.

As mentioned in Section 1.3, the only configurations known to be exactly experi-

mentally reproducible are the polarised configurations, one of which is illustrated in Fig-

ure 1.7. Accordingly, we focus our simulation studies on dynamics that start with a

polarised configuration. Indeed, this is the initial configuration used in experimental

30

studies of not only fixed-amplitude rotating protocols [67] but also more complicated pro-

tocols [23, 40, 48, 56, 63, 65, 66]. The initial configuration used has a net moment in the

+x direction, as shown in Figure 2.6. In all our simulations in this Chapter, we study

20×20 arrays of spins. In this Chapter we will study only open edge arrays, with geometry

shown in Figure 2.6, but in the next Chapter we will return to this protocol and consider

its interplay with edge geometry.

In our simulations (see Section 1.3 for an overview of methods), a continuously rotating

field is approximated by a sequence of fields applied at angles θ, with angular step dθ = 0.01

radians, rotating in a counter-clockwise direction from θ = 0 (the +x axis in Figure 2.6).

Simulations of other angle step sizes (see Section 3.6) indicate that this step size is small

enough to well approximate continuous rotation. (An alternative scheme would be to start

the field at its initial angle, then calculate the field angle required to flip the spin that

is ‘easiest’ to flip and update the field to that angle. Such an algorithm corresponds to

the limit dh/dt→ 0, and is common in studies of systems with avalanche behaviour [104].

However, we find that our simpler algorithm with small dθ steps is adequate to capture

the behaviour of a smoothly rotating field.)

We first make some observations about the energetics of spins in a polarised configura-

tion. The key point is that the external field required to overcome the barrier to flipping

spin i is set by both the intrinsic barrier hc and also the field ~h(i)dip arising from interactions

with other spins. When the array is in a well-defined configuration, such as the polarised

configuration, the interactions are known and as a result so are the external field thresholds

for dynamical processes. We are therefore able to predict several features of the simulated

dynamics. We emphasise, though, that the analysis of this Section only applies to ideal

open edge arrays. The restriction that the system be ideal arises because we assume that

hc is the same for every spin and ~h(i)dip can be determined exactly from a knowledge of the

Ising spin configuration. We will see in Chapter 4 that many of the features of dynamics

that we describe here do not hold when quenched disorder is present in the system, as

it always is in experimental systems. The study presented here is useful as a ‘baseline’

against which to compare the dynamics of a disordered system and uncover the effect that

disorder has.

Our discussion here also assumes that the field starts aligned with the array net moment

(and therefore cannot flip any spins, because it has a positive projection onto all of them),

31

and rotates with small angular step. This second assumption comes into play when we

calculate the field amplitudes required for vertex processes. These are given in terms of

the size of the component of the field antiparallel to the spin. If the field is applied along

the spin axes, then the strength of the parallel component is equal to the strength of the

field. When the field rotates smoothly, it always passes through the directions of the spin

axes. However, if the field were to rotate in steps that did not align it with spin axes, then

the external field strength required for vertex processes would be increased by a factor of

1/ cos(φ), where φ is the smallest angle between the field and the spin axes.

The ‘easiest’ spins to flip are those at the array edges, because edge spins are coupled

to fewer neighbours than bulk spins are. We can calculate the fields acting on spins in the

approximation that spins only interact within vertices (the ‘vertex-only approximation’).

In this approximation, an edge spin has coupling −jN + jL which is −3/2 + 1/√

2 ≈ −0.8

in our reduced units, and a bulk spin has coupling 2jL =√

2 ≈ 1.4. The difference in sign

indicates that an edge spin can lower its energy by flipping, whereas flipping a bulk spin

raises its energy. Accordingly, it is possible to flip edge spins without affecting the bulk,

by applying a field with correct strength.

As the field rotates past θ = π/4, it has a negative projection onto those spins with a

net moment in the −π/4 direction. When this projection is large enough, the edge spins

can flip, creating type 3 vertices, in the process 2© → 3©, as illustrated in Figure 2.7.

Type 3 vertices can propagate on a type 2 background in two processes: 3© 2©→ 1© 3© and

3© 2©→ 2© 3©, which are shown in Figure 2.5. The first corresponds to a zigzag line of spin

flips, and creates a type 1 vertex. The second corresponds to a straight line of spin flips,

and leaves the global vertex populations unchanged, although, as already noted, it does

change vertex configurations and the spatially-dependent vertex populations. The process

3© 2© → 1© 3© requires a smaller external field than the type 3 creation process 2© → 3©,

but the process 3© 2©→ 2© 3© requires a larger field than type 3 creation does.

From these considerations we expect four field regimes. Very low fields have no effect

on the spins because of the intrinsic barriers hc. Very high fields force the magnetisation

to track the rotating field. Between these two regimes we expect two non-trivial regimes.

In the first, which we call the ‘low field’ regime, type 3 vertices can only propagate in the

process 3© 2© → 1© 3©, whereas in the second, which we call the ‘high field’ regime, the

process 3© 2©→ 2© 3© can also occur.

32

өH

x

a) b)

Figure 2.7: How dynamics start at array edges. a) When the field angle passes π/4,it has a negative projection onto spins with moment in the −π/4 direction, as indicatedby the red colouring. b) An example of the type 3 nucleation process 2© → 3©, for thespins indicated by the box in part (a).

The boundaries of the field regimes correspond approximately to the fields required

for these processes in the vertex-only approximation. Appendix B.1 gives the fields for

these and all other vertex processes. In the vertex-only approximation, the low field

regime is hc − 3/2 + 1/√

2 . h . hc, which corresponds to 10.45 < h < 11.25 in the

units we use. The upper boundary of the high field regimes is determined from the field

required for type 3 expulsion in the process 3©→ 2©, since this process must occur if the

magnetisation is to track the applied field, as it does in the very high field regime. This

field is hc − 1/√

2 + 3/2 ≈ 12.05 in the vertex-only approximation.

The existence of field regimes is borne out by simulation results, and boundaries be-

tween the regimes are found at approximately the field strengths predicted, as seen in

Figure 2.8, which gives the final type 1 population as a function of applied field. For

h < 10.25, n1 is zero because no dynamics occur. For 10.25 < h < 10.75, n1 attains a very

large value, close to 0.9. This is the low field regime. The high field regime corresponds

to 10.75 < h < 12.125, with a crossover between the low and high field regimes that we

will discuss below. Finally, for h > 12.125, n1 is zero, and inspection of the magnetisation

shows that it tracks the applied field.

Studying the dynamics in detail further confirms our predictions based on field thresh-

olds for vertex processes. Figure 2.9 shows type 1 and type 3 vertex population evolutions

for fields in the low field regime (h = 10.5), high field regime (h = 12) and crossover (h =

11.25). The array configurations at key times are shown in Figure 2.10 for h = 10.5 and

Figure 2.11 for h = 11.25. The high field dynamics are similar to the crossover dynamics.

Animations of dynamics can be found at http://prl.aps.org/supplemental/PRL/v105/i1/e017201.

In all cases, dynamics start with type 3 nucleation at the array edges in the process

2©→ 3©. This dictates the peak type 3 population fraction, which, as seen in Figures 2.10

33

æ æ

æ

æ æ

æ

æ

ææ æ

æ æ æ

æ

æ æ

10.0 10.5 11.0 11.5 12.0 12.50.0

0.2

0.4

0.6

0.8

1.0

h

Xn1,

fina

l\

Figure 2.8: Final type 1 populations for rotating fields. There are four field regimes:a very low field regime in which no dynamics occur, a low field regime in which n1 attainsa large value, a high field regime in which n1 is not as large and a very high field regime inwhich n1 is zero because the array magnetisation tracks the applied field. The boundariesof these field regimes are indicated by the dashed vertical lines. Error bars are 1 standarddeviation, with statistics performed over 100 independent simulation runs.

Figure 2.9: The evolution of vertex populations shows differences between fieldregimes. Mean population fractions of (a) type 1 and (b) type 3 vertices as a function of‘time’/field angle for an open edge array subject to a rotating field protocol with dθ = 0.01.In both cases, the three field strengths shown are h = 10.5 (black), h = 11.25 (blue) andh = 12 (red). Averages are performed over 10 independent simulation runs, and shadedregions indicate errors (± 1 standard deviation; errors smaller than the line thicknessesare not shown).

34

a) b)

c) d)

Figure 2.10: Vertex dynamics in the low field regime. Snapshots of vertex config-urations at key times for a 20 × 20 island array subject to a rotating field of strengthh = 10.5. At this field strength, dynamics are strongly constrained. The symbols used aregreen circles for type 1 vertices, blue arrows indicating the net moment of type 2 verticesand red crosses representing type 3 vertices. Type 4 vertices do not appear. (a) Type 3vertices are nucleated along array edges at θ ≈ 3π/4. (b) These type 3 vertices propagatein a 3© 2© → 1© 3© process. (c) This process continues as the field rotates. The motion ofone type 3 vertex is indicate by the black path. (d) Eventually, the whole array is coveredin type 1 vertices apart from a domain wall at the centre which is formed when type 3vertices meet and annihilate in the process 3© 3©→ 1© 2©.

35

a) b)

c) d)

Figure 2.11: Vertex dynamics in the crossover/high field regimes. Snapshots ofvertex configurations at key times for a 20 × 20 island array subject to a rotating fieldof strength h = 11.25. At this field strength, type 3 vertices are able to move in alldirections in the array. The symbols used are green circles for type 1 vertices, blue arrowsindicating the net moment of type 2 vertices and red crosses representing type 3 vertices.(a) Type 3 vertices are nucleated along array edges at θ ≈ 3π/4. (b) Type 3 vertices areable to propagate along diagonals in the process 3© 2© → 2© 3©. (c) Type 1 processes aregenerally created in the process 3© 2© → 1© 3©. (d) At longer times, the number of type3 vertices decays to zero because type 3 vertices are removed in annihilation processessuch as 3© 3©→ 1© 2© and once the array edges are covered with type 1 vertices no furthernucleation of type 3 vertices is possible.

36

and 2.11, is equal to the fraction of vertices that are along the array edges. These initial

dynamics are the same for both the low and high field regimes, but the processes that

follow depend on field strength.

In the low field regime, the type 3 population remains constant over many cycles, until

a final sharp decrease. As seen in Figure 2.10 the type 3 vertices propagate towards the

array centre in the process 3© 2© → 1© 3© until the type 3 vertices meet and annihilate.

By this orderly type 1 invasion the system attains a very large final type 1 population

fraction, with only a line of non type 1 vertices remaining at the end (see Figure 2.10).

The reason for the large final type 1 population is that the applied field is only strong

enough to induce the process 3© 2© → 1© 3©, which has a relatively low threshold because

it converts a type 3 vertex to a type 1 vertex and lowers the dipolar energy substantially.

We will see in Chapter 4 that these constrained dynamics are not ‘robust’ against disorder

– if the couplings or island switching barriers are perturbed, the process 3© 2©→ 1© 3© can

be blocked, and other processes allowed.

For stronger applied fields, the process 3© 2© → 2© 3© is possible, and type 3 vertices

can propagate diagonally across the array, as illustrated in Figure 2.11. If spins only

interacted within vertices this propagation would continue all the way to the array edges,

but longer-range interactions prevent this and cause the type 3 vertices to spend more

time in the array bulk. We discuss the role of long range interactions in more detail in

Appendix A.3.

In the crossover and high field regimes, type 1 vertices are generally created near the

array edges, as seen in Figure 2.11. This predominantly occurs in the process 3© 2©→ 1© 3©

but the processes 3© 3©→ 1© 1© and 3© 3©→ 1© 2© also occur. The type 1 vertices cover the

array edges, and cannot be destroyed by the external field. When the spins on the array

edge cannot flip, type 3 nucleation is blocked.

This blocking serves to slow – and eventually stop – dynamics at long times, because

type 3 vertices that are destroyed are not replaced and the number of vertices involved

in dynamical processes decreases to zero. We call this phenomenon ‘trapping’. Trapping

causes the final type 1 populations to be lower in the crossover and high field regime than

in the low field regime because type 1-creating dynamics are inherently self-limiting. We

emphasise that trapping exists when dynamics are required to start at the array edge.

We will see in Chapters 3 and 4 that when disorder is introduced or the field protocol is

37

randomised, dynamics can start in the array bulk and trapping becomes negligible.

2.5 Vertex population dynamics: an analytical model

The model

In the previous Section we saw that the dynamics of open edge arrays under rotating fields

can be described in terms of vertices and their dynamical processes. Single spin flips can

be thought of as vertex pair processes, and the evolution of the vertex populations gives

an indication of which processes occur. Motivated by this, we push the vertex picture

further and develop a model for dynamics in an open edge array, based only on vertex

populations. The goal is to construct a model that is as general as possible, based on ‘first

principles’ rather than being merely a description of the observed dynamics.

In this model, we neglect correlations in positions of vertex types, which is equivalent

to assuming vertices are well-mixed. This assumption does not hold in the low field

regime where dynamics are restricted, but is plausible in the high field regime, to which

we restrict our considerations. We will see the vertex population model captures many of

the essentials of the simulated dynamics of the previous Section. We will also discuss the

points of difference between the population model and simulation results.

We start from the observation that vertex populations are changed by vertex pair

processes in the bulk and single vertex processes at the edges. As with other reaction rate

equations, the rate of a process, the rate of decrease of the population of reactants and the

rate of increase of the population of products are all equal, so that the total population

of (reactants + products) is conserved. For example, the vertex pair process 3© 3©→ 1© 2©

enters the rate equations for population fractions ni as

n1 = +R 3© 3©→ 1© 2©, (2.2a)

n2 = +R 3© 3©→ 1© 2©, (2.2b)

n3 = −2R 3© 3©→ 1© 2©, (2.2c)

where dots indicate time derivatives and R is the rate of a process2. The factor of 2 for n3

2There is a subtlety here: in our simulations, time does not enter, rather, the system relaxes fullyand ‘instantaneously’ under each field application. This is justified because the time scale for islandmagnetisation reversal (nanoseconds [93, 94]) is much faster than the rate of field rotation. However, whenwe write our equations here, processes have rates, as if they take finite time.

38

n1 n3n2

Figure 2.12: Population dynamics without trapping. The three populations in ourpopulation dynamics model, with arrows representing ‘flow’ from one population to an-other via vertex processes. Note that the only steady state solution of this set of popula-tions and processes is n1 = 1, that is, for all population to flow to type 1.

arises because two type 3 vertices are destroyed for every one type 1 and 2 created. Note

that n1 + n2 + n3 = 0, that is, the total population is constant.

In order to write down equations for the vertex populations, then, we need to identify

all possible processes and determine their rates. We make the assumption that ‘high

energy’ processes are not allowed: the process 2© 2© → 3© 3© is forbidden, as are type 1

destruction and type 4 creation. In order to consider finite size effects, we treat array

edges separately by including single-vertex processes, which we discuss below.

All energetically possible one- and two-vertex processes are shown diagrammatically in

Figure 2.12. Without calculating the process rates or solving the corresponding equations,

we can see that for any initial conditions, the system will tend towards n1 = 1. This is

because there are no processes to reduce n1, but type 2 and 3 vertices can both be converted

into type 1 vertices. These conversion processes continue until all vertices are of type 1,

where dynamics must stop.

Clearly, this result does not match our simulation results presented in the previous

Section. This is because we have neglected the process of ‘trapping’, by which type 1

vertices at array edges prevent type 3 nucleation and slow dynamics. In other words,

in simulations the steady state has n1 < 1 because type 1 creation is self-limiting. To

incorporate this effect in the population dynamics equations, we approximate trapping

as a five-vertex process: a type 2 or 3 vertex surrounded by type 1 vertices is trapped

and cannot undergo any other dynamical processes. We introduce two new vertex types,

types 2T and 3T, to represent these inactive trapped vertices. We denote type 2 and 3

vertices that are not trapped by 2F and 3F: the total type 2 and 3 populations are the

39

n1 n3Fn2F

n2T n3T

Trapping Trapping

Figure 2.13: Population dynamics with trapping. The five populations in our pop-ulation dynamics model, with arrows representing ‘flow’ from one population to anothervia vertex processes. The labels 2F, 2T, 3F and 3T refer to type 2 and 3 vertices that arefree (F) and trapped (T).

sums of trapped and free populations. The diagram of all possible processes is now shown

in Figure 2.13.

In fact, in simulations, type 3 vertices are not observed to be trapped, see Section 2.4.

However, in order to incorporate trapping in a way that is consistent with our assumptions

of vertex mixing, we do allow type 3 vertices to be trapped. We will see in Section 2.5

that this has a strong effect on how well the solutions to population dynamics equations

agree with simulation results.

Now that we have listed the vertex populations and processes, all that remains is

to determine the rates of processes. The rate of a vertex pair process i© j© → k© l© is

proportional to the fraction of vertex pairs i© j©, multiplied by the probabilities g and f

that the process is topologically and energetically possible:

R i© j©→ k© l© = ani,jg i© j©→ k© l©f(h, hd) (2.3)

where a is a constant of proportionality with dimensions (1/time) and is the same for all

processes, h is the external field and hd is the contribution from spin-spin interactions

to the energetic barrier for the process. For concreteness, we will calculate explicitly the

rate of the processes 3© 2© → 1© 3© and 3© 3© → 1© 1©. Other vertex pair process rates are

calculated in the same way.

40

In order to take into account array size effects, we consider edges and the bulk sepa-

rately. In an open edge array with N vertices, Ne = 2(√

2N − 1 − 1) are on the edges.

Edge vertices have two neighbouring vertices; bulk vertices have four. (For simplicity,

we treat the four corner vertices as edge vertices, an approximation which becomes more

accurate as the array size increases and the fraction of corner vertices goes to zero.) If the

array has population fraction n2F of free type 2 vertices and n3F of free type 3 vertices

and we neglect correlations in vertex positions, the fraction of pairs 3© 2©, n3F,2F , is

n3F,2F = n3FNe

N× 2n2F + n3F

N −Ne

N× 4n2F (2.4a)

= 4n2Fn3F

(1− N

2

), (2.4b)

where N = Ne/N is the fraction of vertices on the array edges. Using a similar argument,

but taking into account that there are fewer distinguishable combinations of 3© 3© than

3© 2©, the number of 3© 3© pairs is

n3F,3F = 2n23F

(1− N

2

). (2.5)

The topological factor g can be thought of as a way to partially recover the information

lost by using vertex types rather than configurations. g i© j©→ k© l© is simply the number

of ways that the vertex pair i© j© can be arranged so that flipping their shared spin yields

the pair k© l© divided by the number of ways to arrange two vertices3, as illustrated in

Figure 2.14. For the process 3© 2©→ 1© 3©, g 3© 2©→ 1© 3© = 16/128 = 1/8; for 3© 3©→ 1© 1©,

g 3© 3©→ 1© 1© = 4/128 = 1/32.

Whether a process is energetically possible depends on the applied field strength h and

angle relative to the shared spin, and is given by the spin flipping criterion (1.5). The

energetic factor f is given by the fraction of field angles at which a process is possible,

and is given by summing the switching criterion over the four possible shared spin angles

and integrating over applied field angles:

f(h, hd) =1

2π

∫ 2π

0dθ∑α

U(h cos(θ − α)− (hc + hd)

). (2.6)

3As mentioned in Section 2.3, if we used all sixteen vertex configurations for our populations ratherthan the four types, the topological factor would be unecessary because vertices would either be able toundergo a process, or not.

41

4 possible vertices

2

8 possible vertices

x4 pair positions

3

a)

b) 3

2

3

3 3

3

2 2

22

Figure 2.14: How the g factors are calculated, for the example of 3© 2© → 1© 3©.Part (a) shows how the denominator in the ratio is determined: there are eight vertexconfigurations of type 3, four of type 2, and four distinguishable configurations of a pairof distinguishable vertices. This gives 8× 4× 4 = 128 pair arrangements. Part (b) showsthat for a given vertex-type configuration, there are four spin configurations for whichthe shared spin (in red) can be flipped to give a 1© 3© pair. Multiplying this by four(for the four pair arrangements) give a denominator of 16. From these considerations,g 3© 2©→ 1© 3© = 16/128 = 1/8.

U is the Heaviside unit step function, and α represents the spin directions and can take

values π/4, 3π/4, 5π/4, 7π/4.

Combining these factors, the rate of the process 3© 2©→ 1© 3© is

R 3© 2©→ 1© 3©(h) = a×4n2Fn3F

(1−N

2

)× 1

8× 1

2π

∫ 2π

0dθ∑α

U(h cos(θ−α)−(hc−3+

√2))

(2.7)

where −3 +√

2 is the hd value for the process 3© 2© → 1© 3©4. For convenience in what

follows, we write the rates as

R i© j©→ k© l©(h) = ν i© j©→ k© l©(h)ninj . (2.8)

The rates of single-vertex edge processes can be determined in a manner analogous to

the rates of vertex pair processes. The probability of a type i vertex at the array edge is

niN , and the topological and energetic factors are calculated in analogy to pair processes.

There is an important difference though, which is that not all of the nearest neighbours

of the spin that flips are specified by the configuration of the spin’s vertex, as illustrated

in Figure 2.15. We deal with this issue by averaging over the two possibilities, so that

the rate of each process is 12(f(h,+) + f(h,−)), where + and − stand for the coupling

fields when the unspecified spin is in a favourable and an unfavourable alignment with the

4See Appendix B.1 for the interaction energies associated with all vertex processes.

42

2

Figure 2.15: A single edge vertex does not specify all the nearest neighboursof an edge spin. Three of the relevant near neighbours of the yellow indicated spin arespecified by the configuration of its vertex (in this case, of type 2). However, the red spin,which must also be taken into account, is not specified. We deal with this by averagingover the rates of processes for the two possible alignments of the red spin.

flipping spin, respectively. (In principle, one could instead define two-island edge vertices,

but this would add further complexity to the population dynamics equations.)

Trapping is taken to occur whenever a type 2 or 3 vertex is surrounded by type 1

vertices, so the rate of the trapping process is determined by the probability of a type 2 or

3 vertex having four type 1 neighbours: a(1−N )n2F,3Fn41. The factors g and f are both

unity.

Given these process rates, the equations corresponding to the set of processes shown

in Figure 2.13 are

n1 = ν 3© 2©→ 1© 3©n2Fn3F + 2ν 3© 3©→ 1© 1©n23F + ν 3© 3©→ 1© 2©n2

3F + ν 3©→ 1©n3F , (2.9a)

n2F = −ν 3© 2©→ 1© 3©n2Fn3F + ν 3© 3©→ 1© 2©n23F + 2ν 3© 3©→ 2© 2©n2

3F

− ν 2©→ 3©n2F + ν 3©→ 2©n3F − νTn2Fn41,

(2.9b)

n3F = −2(ν 3© 3©→ 1© 1© + ν 3© 3©→ 1© 2© + ν 3© 3©→ 2© 2©

)n2

3F

+ ν 2©→ 3©n2F −(ν 3©→ 2© + ν 3©→ 1©

)n3F − νTn3Fn

41,

(2.9c)

n2T = νTn2Fn41, (2.9d)

n3T = νTn3Fn41. (2.9e)

These nonlinear differential equations can be solved numerically using standard algorithms.

We have used the numerical solver built into Mathematica. The values of the ν are given

43

Process ν

3© 3©→ 1© 1© a(1−N/2)18f(h,−6 +

√2)

3© 3©→ 1© 2© a(1−N/2)14f(h,−3)

2© 3©→ 3© 1© a(1−N/2)12f(h,−3 +

√2)

3© 3©→ 2© 2© a(1−N/2)14f(h,−

√2)

2©→ 3© aN 14f(h, 1/

√2− 3/2)

3©→ 1© aN 116

(f(h, 1/

√2− 9/2) + f(h, 1/

√2− 3/2)

)3©→ 2© aN 1

8

(f(h,−1/

√2− 3/2) + f(h,−1/

√2 + 3/2)

)Trapping a(1−N )

Table 2.1: Rates of vertex processes for Equations (2.9). Rates are calculated usingEquations (2.3), as described in the text.

in Table 2.1. For clarity, the field dependence is not explicitly written.

Results and discussion

Figure 2.16 shows the evolution of n1 and n3 for an applied field equal to the island

switching field hc, for array sizes ranging from 100 to 1, 000, 000 spins. The qualitative

agreement with simulations is good. n1 increases with time before saturating at a steady

state value less than unity, as we expect.

The initial peak in n3 is also reproduced by the population dynamics, at least for small

array sizes. The peak is caused by an initial creation of type 3 vertices out of the type

2 background, and a subsequent decay via processes such as 3© 3© → 1© 2©. The rate of

type 3 creation is linear in N , which decays with array size as N−1/2. Accordingly, for

large arrays the initial type 3 creation is suppressed, and the type 3 population increases

monotonically to a final population that is limited by type 3 decay and trapping.

The array size dependence of the final n1 value is also shown in Figure 2.16. The final

type 1 population decays with array size, but only logarithmically slowly. Recall that

the array size dependence of the steady state solution enters only because of trapping: if

there were no trapping, the system would tend always to n1 = 1. Thus, the details of

how trapping is modeled influence the array size dependence of the final n1. Trapping is

incorporated in Equations (2.9) as a process that occurs when a type 2 or 3 vertex is sur-

rounded by four type 1 vertices, an implementation that is consistent with the assumption

of vertex mixing. However, in simulations, trapping occurs when type 1 vertices cover

the array edges. Therefore, there is no reason to expect that these two types of processes

should give the same array size dependence, since one involves the edges whereas the other

is a bulk process. Checking the actual array size dependence in simulations is a future

44

0 8 Π 16 Π 24 Π 32 Π 40 Π 48 Π

0.0

0.2

0.4

0.6

Θ

n1

Increasing N

0 8 Π 16 Π 24 Π 32 Π

0.0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Θ

n3 Increasing N

a)

b)

!

!

!

!!!!!!

2 3 4 5 60.4

0.5

0.6

log10N

n1,final

Figure 2.16: Evolution of populations according to Equations (2.9). (a) Type 1 and(b) type 3 population fractions for dynamics given by Equations (2.9), for a field h = hcand for arrays of increasing size from 100 to 1, 000, 000 spins. The type 3 population isthe sum of free and trapped type 3 vertex populations. Inset: Final type 1 populationfraction as a function of array size N .

45

a)

b)

Figure 2.17: The effect of type 3 trapping on the evolution of n1 and n3. Bothplots show the evolution of a system of 100 spins. The blue curves indicate the dynamicsgiven by Equations (2.9), and the red curves indicate the dynamics when type 3 verticescannot be trapped.

direction for this work, and would require optimising simulations to be able to study much

larger arrays than the 20× 20 arrays dealt with here.

Another consequence of the choice of how to implement trapping is that the final type 3

population attained is relatively large, compared to simulations where it is approximately

zero for a 20× 20 array, as seen in Figure 2.9. This is because trapped type 3 vertices are

unable to be destroyed, whereas in simulations, all type 3 vertices are eventually destroyed.

To correct for this, we can rewrite Equations (2.9) without a type 3 trapping process, so

that the variables are n1, n2F , n2T and n3. Solving these revised equations gives evolutions

of populations that are closer to those observed in simulations, as seen in Figure 2.17.

However, as noted in the previous Section, neglecting type 3 trapping is inconsistent with

our original assumption of well-mixed vertices, since under that assumption there should

be no reason for type 3 vertices not to be trapped.

Another point of difference with simulations is the failure of Equations (2.9) to ex-

46

æ

æ

ææ

æ æ æ æ æ æ æ æ æ

9.0 9.5 10.0 10.5 11.0 11.5 12.00.0

0.1

0.2

0.3

0.4

0.5

0.6

h

n1

Figure 2.18: The field dependence of steady state type 1 populations. Values arecalculated for an array with 102 vertices, using Equations (2.9). The field regimes seen inFigure 2.8 are not reproduced, because the vertex population dynamics assume verticesare well-mixed, an assumption which breaks down in the low field regime or for very highfields.

hibit field regimes. The rates of processes depend on the applied field strength, but the

existence of field regimes depends on the break down of the vertex mixing assumption –

an assumption the equations are based on. The field dependence of the steady state type

1 population for the population dynamics is as shown in Figure 2.18. Once the field is

above the threshold for the process 2©→ 3© the final type 1 population quickly reaches a

maximum value.

The population equations also lack a mechanism for describing the very high field

regime, for which an array initially in a polarised configuration is always re-polarised by

the external field. This re-polarisation occurs via type 3 vertices nucleating along array

edges and propagating across the array in the process 3© 2© → 2© 3©, in which the initial

and final type 2 vertices have net moment at 90 to each other. However, this process does

not change the population fraction of vertex types, and therefore cannot be described in

the model we use here. Instead, as the field increases, the rates of all processes saturate,

leading to a constant n1 vs h, as seen in Figure 2.18.

Finally, we note that if the initial configuration of the system is random then the vertex

mixing assumption holds in a wider range of cases. We will discuss only briefly the use

of vertex population equations to model a system with random initial configuration, since

such an initial configuration is not experimentally reproducible.

In a random configuration, all four vertex types are present. In order to model the

dynamics, therefore, we need to introduce type 4 vertices and their energetically-allowed

47

processes. These are

4© 4©→ 3© 3©

4© 3©→ 3© 2©

4© 3©→ 3© 1©

4© 2©→ 3© 3©

4© 1©→ 3© 3©

4©→ 3©

Note that type 4 vertices cannot be trapped, since the process 4© 1©→ 3© 3© actually lowers

the energy of the pair. The rates of processes are determined in the same way as other

processes, and the full equations are given in Appendix B.2.

Figure 2.19 shows the field dependence of the final type 1 population for an array

with random initial configuration, for both the population equations and simulations.

The overall agreement is striking. 5 In particular, both simulated and analytical n1 vs h

curves show an increase towards a saturation value. This is because the random initial

configuration does not undergo the same re-polarisation dynamics that occur when the

initial configuration is polarised: the key process in re-polarisation is 3© 2© → 2© 3©, and

this requires a uniform type 2 tiling as a ‘background’ state. This picture is further

confirmed in Section 4.5, where we revisit the population dynamics for a system with

quenched disorder. In that case, the assumption of vertex mixing holds well, but the

initial polarised configuration allows re-polarisation, causing differences in the n1 vs h

curves for simulations and analytical equations.

It would also be interesting to test the array size dependence for random initial config-

urations. Because random configurations contain high energy spins throughout the array,

dynamics start in the bulk as well as the edges. As a result, trapping does not occur

from array edges in the same way it does for an initially polarised system. Accordingly,

it is plausible the array size dependence of the population equations could well match

simulations. This again points to the importance of array edges in perfect systems, but

indicates that randomness can modify the role of edges. We explore this further in Chap-

ter 4, where we discuss randomness in the form of quenched disorder in interactions and

5It is unclear why the population equations give a peak in n1 near h = 7.

48

ææ

æ

æ

æ

æ

æ

æ ææ

æ

æ æ ææ

æ

à à à

à à

à

à

àà

àà

à

à

àà à à à à

4 6 8 10 120.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hXn

1,fi

nal\

Figure 2.19: The population dynamics equations describe well the dynamics ofan array with a random initial configuration. Final type 1 population as a functionof applied field strength for a 20 × 20 array with a random initial configuration. Bluecircles are simulation results, red squares are analytical results. Averages are made over20 independent simulation runs; error bars represent one standard deviation. The dashedhorizontal line at n1 = 1/8 is the expected type 1 population in a random tiling.

switching fields, rather than configurations.

2.6 Conclusion

We have shown that the vertex dynamics driven by rotating fields can be formalised in

terms of equations for vertex population dynamics, at least in cases where the vertices

are well-mixed. It may be possible to improve this treatment by introducing spatially

dependent densities for the populations and estimating effects of correlations on transition

rates, but that is beyond the scope of the present work. We also note that the equations

we have written apply only to the open edge arrays studied in this Chapter, but extension

to other edge types is possible.

49

50

Chapter 3

Selecting vertex processes: field

protocols and edge effects in ideal

systems

3.1 Overview

In the previous Chapter we saw that spin ice dynamics can be cast in terms of vertex

processes. Dynamics starting from a polarised configuration can be thought of as the

motion of type 3 vertices, which occurs in two processes: the low energy 3© 2©→ 1© 3© and

the high energy 3© 2© → 2© 3©. Dynamics are qualitatively different when both processes

can occur (the high field regime of Section 2.4) and when only the low energy process is

allowed (the low field regime).

The ability to ‘select’ only the low energy process exists in the open edge array geometry

because the field threshold for initial type 3 nucleation is intermediate to the thresholds

of each of the two propagation processes. In this Chapter, we show that the couplings

of edge spins in different geometries are markedly different to the coupling in open edge

arrays (Section 3.2). As a result, for magnetisation reversal dynamics, it is possible to

select the low-energy processes only in open edge arrays (Section 3.3). On the other

hand, fields applied at 90 to the initial magnetisation drive only the high energy process

3© 2© → 2© 3©, regardless of edge geometry (Section 3.4). Other intriguing edge effects

are seen for rotating field protocols, and we comment on these briefly. The three edge

geometries give rise to very different dynamics: open edges can select either low energy

51

Open edges Closed edges 4-island edges

Figure 3.1: The three edge geometries studied in this Thesis. The near neighbourinteractions of highlighted spins are categorised using the same scheme as Wang et al. [23]:nearest neighbour (solid lines), next-nearest neighbour L (dotted lines) and next-nearestneighbour T (dashed lines).

type 3 propagation or both propagation types, 4-island edges force ‘cascades’ of high

energy propagation processes, and closed edges always allow both high and low energy

processes (Section 3.5).

The vertex processes that occur are also selected by the sequence of fields applied. We

saw in the previous Chapter that a common feature of rotating protocols is the tendency

towards ‘trapping’, where type 1 vertices near the array edges stop dynamics. In this

Chapter, we consider alternative, disordered, protocols that avoid trapping by starting

type 1 creation in the array bulk rather than its edges (Section 3.6). The interplay of edge

effects and field protocol is a specific example of a more general interplay between field

protocol and inhomogeneity in the artificial spin ice. We emphasise that in this Chapter we

study ideal systems, but in Chapter 4, we study a second type of inhomogeneity, namely

the quenched disorder that is present in experimental systems.

3.2 Array edge geometries

Square ice has a well-defined bulk geometry, but there are several ways that edges can be

configured. Three edge geometries are studied in this Thesis, which we refer to as ‘open’,

‘closed’ and ‘4 island’ edges. They are depicted in Figure 3.1. Edge spins differ from bulk

spins in terms of their near neighbours.

In open edge arrays, islands at the array edges have three nearest neighbours (except

for corner islands, which have two), and one L and one T next-nearest neighbour. The

odd number of nearest neighbours means that the nearest neighbour interactions, which

52

Type 1e Type 2e Type 3e

Figure 3.2: Types of 3-island edge vertices. The division of the 23 = 8 vertex configu-rations into groups is determined by energetics, with E1e < E2e < E3e. Vertices obtainedby rotating those shown here are treated as the same type, because they are energeticallyequivalent.

are strongest, cannot be balanced. In contrast, bulk spins may have equal numbers of

favourable and unfavourable nearest neighbours, resulting in a weak overall coupling to

the local environment.

4-island edge arrays are so called because every vertex of the array has four islands

associated with it. Spins near the edges have different local environments, depending on

whether they are parallel or perpendicular to the edges. However, all spins have an even

number of nearest neighbours, and when these are balanced, the dominant coupling is to

next-nearest neighbours.

Closed edge arrays have the distinguishing feature that their magnetisation configu-

ration cannot be described fully in terms of vertices of four islands: instead, the edge

vertices have three islands. (The ‘2-island’ edge vertices of open edge arrays can be fully

described by the configurations of their neighbouring 4-island vertices.) The 3-island edge

vertices can be divided into three groups, based on energy, in a manner analogous to the

four vertex types of 4-island vertices (see Section 2.2). These are shown in Figure 3.2.

In terms of near-neighbour interactions, islands at the edges of closed edge arrays always

have an even number of nearest neighbours, but like 4-island edge arrays, the numbers of

next-nearest neighbours are reduced for edge islands.

The effects of these differences in local geometry is emphasised in Figure 3.3, which

shows the strength of coupling for each spin of a +x polarised configuration when the full

long-range dipolar interactions are taken into account, for each of the array geometries we

study. We emphasise that the results of this Section are valid for the polarised configura-

tion only. For example, in a random configuration, the distinction between edge and bulk

53

Open edges

Closed edges

4-island edges

Figure 3.3: Projection of the total dipolar field onto each spin. The interactions arelong-range and every spin interacts with every other spin, as described by Equation (1.3).The colour scheme is scaled between the lowest and highest field values for any geometry,allowing comparisons between geometries.

spins is blurred because high and low energy local configurations can occur anywhere.

In all edge geometries, the bulk spins are subject to approximately the same dipolar

field. In contrast, there is a striking difference in couplings at the edges for the open edge

array, compared to the closed and 4-island edge arrays. In open edge arrays, spins along

the upper and lower array edges are in a favourable local configuration, whereas spins along

the two vertical edges are in an unfavourable local configuration. (If the configuration had

a net moment in the ±y direction, the vertical edges would have favourable local fields

and the horizontal edges would be unfavourable.) The magnitude of the difference in

coupling between edge and bulk spins is on the order of the strength of nearest-neighbour

54

interactions, which is given in the literature as being tens of Oe [23, 40, 43]. In closed and

four-island edge arrays, all edges are equivalent and have couplings quite close to those in

the bulk.

3.3 Response to a field at 180 to initial net magnetisation

To see how these differences in edge coupling affect dynamics, we start by studying mag-

netisation reversal, a simple protocol that gives insight that can be applied to understand-

ing more complex field protocols. As mentioned above, in this Chapter we study only

ideal systems, but in the next Chapter we compare these results to those obtained for

systems with quenched disorder. The initial net moment of the array is in the +x direc-

tion, and a field is applied in the −x direction, starting from zero and increasing until the

magnetisation switches to the −x direction; these directions are indicated in Figure 2.6.

The field increases in strength in steps of 0.05 in our reduced units. This step is small

enough to well approximate a continuously increasing field, as we discuss in further detail

in Appendix A.5. While the final configuration of this field protocol is known in advance,

for intermediate field strengths the dynamics are non-trivial and depend strongly on edge

geometry.

Open edge arrays

We first discuss the dynamics of open edge arrays in some detail. In this geometry, the

coupling of edge spins ‘selects’ dynamics in the bulk, with interesting consequences – we

find that small changes in applied field can lead to large changes in magnetisation and

type 1 vertex population n1.

As anticipated in Sections 2.4 and 3.2, dynamics start at the vertical edges of the array

where dipolar coupling reduces the effective barrier to spin flipping. An edge spin can flip

if a field of (hc − 0.98) = 10.27 is applied antiparallel to it. In the case of rotating field

protocols, the external field is rotated anticlockwise from θ = 0 until it has a negative

projection onto the spins with direction −45 and these spins are ‘selected’ to flip first. In

contrast, a field applied in the −x direction has a negative projection on to all spins of a

+x polarised configuration and so any edge spin can flip when the external field surpasses√

2(hc − 0.98) = 14.52. (The factor of√

2 accounts for the field being at 45 to the spin

55

a) b)

c) d)

Figure 3.4: The first cascades of spin flips and resulting type 1 vertex creationin a section of a +x polarised open edge array subject to a −x field. (a) Spinsalong the edge are able to flip, but spins in the bulk are not. The blue/purple highlightingindicates spins that can flip. (b) The first two flips of a cascade (dotted spins). Thezig-zag line of spins continues to flip until it reaches the opposite edge of the array; interms of vertices it can be thought of as the propagation of a type 3 vertex in the process3© 2©→ 1© 3©. The two (purple) edge spins neighbouring the (blue) edge spin that flipped

are now no longer able to flip, because their local environment is energetically favourable.(c) If only blue spins flip, a complete type 1 vertex tiling can be attained. (d) However,if both blue and purple edge spins flip, then incompatible type 1 domains form, and aline of type 2 vertices, which cannot be expelled, remains. This is shown in an exampleconfiguration in which a line of flips starting with a blue spin has propagated all the wayacross the array, and a line of flips starting with a purple spin has partially propagatedfrom the right hand side of the array.

axes.) This is illustrated in Figure 3.4(a). We will see below that the ability of any edge

spin to flip has consequences for the configurations attained during magnetisation reversal.

As seen in Figure 3.5, the magnetisation remains constant until the field passes through

the threshold at h = 14.55. At this point, mx drops sharply. This drop corresponds to

type 3 vertices being created along the array edges and propagating in cascades of spin

flips, given by the process 3© 2©→ 1© 3©, leaving ‘trails’ of type 1 vertices through the array,

as illustrated in Figure 3.4(b). These cascades of flips are an example of avalanche-type

behaviour, because each time a spin flips, its new orientation causes the total field acting

on one of its nearest neighbours to change from being below the threshold for flipping

to above the threshold. The avalanche occurs during a single field application, and this

56

Figure 3.6

Figure 3.7

Figure 3.5: Magnetisation reversal in a perfect system. Normalised net x compo-nent of the magnetisation for open edge arrays (orange squares), closed edge arrays (redtriangles) and 4-island edge arrays (blue circles). The field is applied antiparallel to theinitial net magnetisation and is increased in strength in steps of 0.05 in our reduced units.Averages are made over 10 simulation runs and standard deviations are small enough thatthese curves are well representative of the results for all runs. Arrows indicate the pointson the curves at which the snapshots in Figures 3.6 and 3.7 are taken.

distinguishes it from the chains of type 1 vertices formed over the course of several cycles

of a rotating field (see Section 2.4). Because there is no quenched disorder, there are no

pinned spins to stop the avalanches, and they propagate from edge to edge. At the same

time, edge spins that neighbour an avalanche are blocked from flipping because their local

environment has become favourable.

Because type 3 vertices can only propagate in the process 3© 2©→ 1© 3©, which creates

type 1 vertices, the dynamics produce a state with a very low net magnetisation and a

high type 1 population. For example, the vertex configuration of a simulation at h = 14.8

is shown in the left-hand panel of Figure 3.6. Almost all vertices are of type 1, except for a

‘domain wall’ of type 2 vertices that cannot be driven out, because this would require the

expansion of one of the type 1 vertex domains, which cannot be achieved by an external

field, because type 1 ordering does not couple to a field. (Although the vertices of the wall

do couple to a field, the wall must be moved vertically to enlarge/shrink domains, and the

type 2 vertices have a horizontal net moment, so the wall cannot be driven by this means.)

This domain wall is worth further discussion, because some of the underlying reasons

for its existence will be relevant in later Sections of this Thesis. ‘Domains’ of type 1

ordering are chessboard tilings of type 1 vertices. These can be ordered in two ways,

in analogy to the two orderings of an Ising antiferromagnet on a square lattice. If one

type of ordering has nucleated in one part of an array, and another type of ordering has

57

Figure 3.6: How reversal occurs in an open edge array. The vertex configurationsof an array with open edges that is initially polarised in the +x direction with a field inthe −x direction. The field strengths are 14.8 (left) and 20 (right) in our reduced units.

nucleated elsewhere, then if the domains of ordering grow to meet each other, there must

be a domain wall (of type 2, 3 or 4 vertices) separating them.

We can divide the spins of a polarised configuration into two groups, A and B, such

that flipping all the spins of A, and none of B, leads to one of the ground state orderings

covering the whole array. The other ground state ordering is obtained by flipping all the

spins of B and none of A. In an open edge array with a +x polarised configuration, these

two groups consist of alternating zig-zag horizontal lines of spins, the ends of which are

indicated by the blue and purple edge spins in Figure 3.4(a). As shown in Figure 3.4(c),

if only avalanches of flips that start with blue spins occur, then a complete type 1 tiling

is obtained. Similarly, if all avalanches start with purple spins, the other type 1 tiling is

obtained.

However, because both blue and purple spins can be driven to flip by a field in the −x

direction, the typical scenario, leading to the simulated configuration seen in Figure 3.6, is

for avalanches to start with both blue and purple spins, and incompatible type 1 domains to

form. This is illustrated in Figure 3.4(d). The exact position of the domain wall depends

on the sequence of spin flips, which is determined by a random number generator. In

principle, the wall could be absent if the random number generator gave a sequence of

flips that only caused ‘compatible’ cascades of flips, but the probability of this decays

exponentially with the number of edge spins and is therefore negligible for large arrays.

Indeed, there is also the possibility that more than one wall could be present, especially

58

in larger arrays.

As the field is increased further, the array is re-polarised into the −x direction. The

spins at the domain wall, and those at the top and bottom edges of the array, have higher

energy than the rest, are involved in the next stage of dynamics. At h ≈√

2(hc + 2.55) =

19.3, type 3 vertices can be nucleated and propagated to create lines of type 2 vertices

with −x net moment. This yields the vertex configuration shown in Figure 3.6 at h = 20.

This configuration is stable up to h ≈√

2(hc + 4.15) = 21.6, at which point the field is

strong enough to flip spins that have neighbours in a type 1 configuration, and the entire

array reverses via type 3 nucleation and propagation, in the process 3© 1© → 2© 3©. This

corresponds to the final drop in the mx vs h curve in Figure 3.5. The array is left in a

polarised configuration with n2 = 1, with a net moment in the −x direction.

Closed and 4-island edges

The field strengths required for type 3 nucleation are approximately the same for both

closed and 4-island edge arrays: Figure 3.3 gives values of√

2(11.25 + 0.18) = 16.16 for

closed edge arrays, and√

2(11.25 + 0.25) = 16.26 for 4-island edge arrays. These values

are matched by simulation results, where dynamics start at h = 16.2 and h = 16.3 for

closed and 4-island edge arrays, as seen in Figure 3.5.

The similarity in the field threshold for type 3 nucleation leads to close similarities in

the subsequent dynamics. The initial field is large enough that both type 3 propagation

processes ( 3© 2© → 1© 3© and 3© 2© → 2© 3©) are energetically allowed, and a field applied

at 180 to the initial polarisation can drive both. This leads to complex configurations, as

seen in Figure 3.7. As the field increases, the processes are biased towards increasing the

net −x moment, but there is no single reversal process as there is for open edge arrays.

This is seen in Figure 3.5 as a smoother mx vs h curve.

The striking differences in behaviour between open and closed/4-island edge arrays

makes clear the importance of array edges in selecting dynamical processes in perfect

arrays. This theme will recur throughout this Chapter.

59

Type 1 Type 2 Type 3

Figure 3.7: How reversal occurs in a 4-island edge array. The vertex configurationof an array with 4-island edges that is initially polarised in the +x direction after a fieldin the −x direction has been ramped up in strength to 16.5 in our reduced units. Similarconfigurations are seen in closed edge arrays.

3.4 Response to a field at 90 to initial net magnetisation

We turn now to the dynamics that are induced by a field applied at 90 to the initial

net magnetisation (that is, the y direction in Figure 2.6), and study the dynamics of the

transition from one polarised state to the other. The results are summarised in Figure 3.8,

which shows the net x and y components of the magnetisation as a function of the strength

of the field in the y direction, for the three array edge types. As was the case for fields at

180 to the initial magnetisation, open edge arrays exhibit qualitatively different behaviour

to closed and 4-island edge arrays.

The lower panel of Figure 3.8 shows the y component of the magnetisation, my. This

is equal to 1−mx, because only spins with original net moment in the −45 direction flip

under the applied field, into the direction 135. Each flip subtracts one ‘quantum’ from

the total x magnetisation and adds one quantum to the total y magnetisation, preserving

the relationship mx +my = const.

Open edge arrays

In open edge arrays, the field required to flip spins at the array edges is the same as it was

for the −x field, that is,√

2(hc− 0.98) = 14.52, because the sum of switching barriers and

60

Figure 3.9

Figure 3.10

Figure 3.8: Magnetisation 90 rotation in perfect systems. Normalised net x com-ponent (upper panel) and y component (lower panel) of the magnetisation for open edgearrays (orange squares), closed edge arrays (red triangles) and 4-island edge arrays (bluecircles). The field is applied at 90 to the initial net magnetisation and is increased instrength in steps of 0.05 in our reduced units. Averages are made over 10 simulation runsand standard deviations are small enough that these curves are well representative of theresults for all runs. Arrows indicate the points on the curve where the snapshots of Figures3.9 and 3.10 are taken.

61

Type 1 Type 2 Type 3Type 1 Type 2 Type 3Type 1 Type 2 Type 3

Figure 3.9: How magnetisation rotation occurs in open edge arrays. Vertexconfigurations of an open edge array initially polarised in the +x direction with a fieldapplied in the +y direction. The field strengths are, left to right, h = 15.7, h = 16.05 andh = 16.3.

dipolar interactions is the same, and the field is again applied at 45 to the island axes.

However, in this case, only the edge spins with moment in the −45 direction are driven

to flip by the external field, because the others have a positive projection of the external

field. The x component of the magnetisation decreases for 14.55 ≤ h ≤ 15, when the

edge spins flip, creating a line of type 3 vertices along the array edges. The configuration

attained in this process is shown in the first panel of Figure 3.9.

The next stage of dynamics starts with a sharp drop inmx at h = 16, which corresponds

to the type 3 vertices propagating along array diagonals in the process 3© 2©→ 2© 3©. The

propagation is strongly directed and is the only process that occurs, because the process

3© 2© → 1© 3© requires a component of the field to be in the opposite direction to the

magnetisation of the initial type 2 vertex. The second and third panels of Figure 3.9 show

vertex configurations at two points during the type 3 propagation. As the field is increased

further, the type 3 vertices are expelled, leaving the array in a +y polarised state.

Closed and 4-island edge arrays

For closed and 4-island edge arrays, rotation of the array magnetisation is driven by the

same type 3 propagation process 3© 2© → 2© 3©. The initial spin flips occur at the array

edges at fields of h = 16.2 (closed edges) and h = 16.3 (4-island edges), again matching

the field values required to start dynamics with a −x field. These fields are larger than

the fields required for type 3 propagation, so type 3 nucleation and propagation across the

array occurs in a ‘cascade’ in a single field step. As a result, type 3 vertices are not seen in

the array configuration; see Figure 3.10 which shows vertex configurations of the 4-island

62

Type 1 Type 2 Type 3 Type 1 Type 2 Type 3

Figure 3.10: How magnetisation rotation occurs in 4-island edge arrays. Vertexconfigurations of a 4-island edge array initially polarised in the +x direction with a fieldapplied in the +y direction. The field strengths are h = 16.35 (left) and h = 16.5 (right).

edge array during the reversal. The net magnetisation rotates by 90 very rapidly, as seen

in Figure 3.8.

3.5 Rotating protocols

In Section 2.4, we saw that open edge arrays are able to ‘select’ low-energy vertex processes

when a field of the right strength is applied, and as a result the response to rotating fields

displays distinct field regimes. In this Chapter, we have seen that closed and 4-island edges

lack this ability. Accordingly, we expect that their response to rotating field protocols will

be qualitatively different to that of open edge arrays, despite the fact that the underlying

rules for vertex dynamics are the same. In fact, we will see that not only are closed and

4-island edges different to open edges, but they are also different to each other in their

response to rotating fields. As always, the array has no quenched disorder and is initialised

in a +x polarised configuration, and the field rotates anticlockwise from the +x direction.

Closed edge arrays

Figure 3.11 shows the final type 1 population fraction for closed edge arrays, plotted

against applied field strength. Unlike open edge arrays, closed edge arrays have only one

field regime of type 1-creating dynamics (see Figure 2.8 for comparison).

The lower boundary of the field regime corresponds to the field required for type 3

nucleation. As expected from our study of single direction protocols, this field is large

enough that once type 3 vertices have been nucleated, which happens when the field

has rotated to approximately the +y direction, the process 3© 2© → 2© 3© can also occur.

63

æ ææ

æ

æ

æ

æ

æ

æ

æ

æ

æ æ æ æ

11.5 12.0 12.5 13.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

h

Xn1,

fina

l\

Figure 3.11: Final type 1 population as a function of applied field strength fora closed edge array subject to a rotating field. There is only one non-trivial fieldregime in addition to the two trivial very low and very high field regimes. In the very lowfield regime (h ≤ 11.5) no dynamics occur. In the very high field (h ≥ 12.6125) regimen1 is zero because the magnetisation tracks the applied field. The non-trivial regime(11.5 < h < 12.6125) is described in the main text. Error bars are 1 standard deviation,with statistics performed over 100 independent simulation runs.

This drives the type 3 vertices across the array and they are expelled. Indeed, the array

configuration at early times (shown in Figure 3.12(a)) resembles the configurations seen

when a +y field is applied to an array in a +x polarised configuration. Because the

field required to create type 3 vertices is large enough that the high-energy propagation

process 3© 2© → 2© 3© must occur, there is no ‘low field’ regime, unlike open edge arrays

(see Section 2.4).

As the field continues to rotate past the +y direction, new dynamics can occur in the

form of the edge vertex process 2e© 2e© → 1e© 1e©. Figure 3.12(b) shows an example of this.

Type 1e vertices are stable against applied fields and block type 3 vertices from leaving

the array. The type 3 vertices are instead ‘reflected’ at the array edges and continue to

move around in the bulk of the array, creating type 1 vertices, primarily in the process

3© 2© → 1© 3©. Figure 3.12 (c) and (d) shows these dynamics; note the similarity with

Figure 2.11.

4-island edge arrays

Whereas 4-island edge arrays respond to a single direction field protocol in a very similar

manner to closed edge arrays (see Sections 3.3 and 3.4), when it comes to a rotating field,

the two geometries exhibit quite different behaviour. This is because 4-island edge arrays

64

a) b)

c) d)

Figure 3.12: Response of a closed edge array to a rotating field. Snapshots ofvertex configurations at key times for a closed edge array of 512 islands subject to arotating field of strength h = 12.5. (a) At early times diagonal lines of spins flip, resultingin lines of rotated type 2 vertices. (b) The type 2e vertices are able to convert to type 1e.(c, d) Type 3 vertices move through the array and are ‘reflected’ off the type 1e vertices,creating type 1 vertices near the edges.

65

Figure 3.13: Typical normalised net array magnetisation m as a function of fieldangle θ for h = 11.75 (left) and 12.25 (right). The rotating field is applied to a 4-islandedge array of 480 islands. The net magnetic moment is reduced by the field protocol butthis occurs in a way that leaves vertex population fractions unchanged.

lack any mechanism to block type 3 vertex expulsion. The result of this is that type 3

vertices always propagate across the array and are expelled in similar cascades to those

seen when a field is applied at 90 to the initial magnetisation, and no processes that

create type 1 vertices can occur.

As a result, vertex population fractions are not useful for describing the state of the

system, because n2 = 1 always. However, examination of the net magnetisation (Fig-

ure 3.13) reveals three field regimes: very low and very high regimes as for the other edge

geometries, and an intermediate regime 11.25 < h < 12.5 where the action of the rotating

external field is to reduce the net magnetisation.

Vertex configurations at key times for a field in the intermediate regime are shown in

Figure 3.14. Although all vertices are of type 2 always, the moments of the type 2 vertices

are rearranged into a short-range vortex ordering reminiscent of the type 1 ordering for

individual islands. This reduces the dipolar energy of the array considerably, from −307.6

(−0.64 per spin) for the initial polarised configuration to −406.7 (−0.85 per spin) for the

final configuration for h = 12.25. This is shown in Figure 3.15.

3.6 Random and large ∆θ protocols

Motivation

Until now, we have studied field protocols for which the sequence of applied fields is

ordered. We have seen that such protocols can lead to orderly dynamics, especially in

open edge arrays where the edge geometry makes it possible to select only dynamical

66

Type 1 Type 2 Type 3 Type 1 Type 2 Type 3 Type 1 Type 2 Type 3

Figure 3.14: Vertex configurations of a 4-island edge array subject to a rotat-ing field of strength h = 11.75. Snapshots are taken at θ = 3π/4, 5π/4, 7π/4. Thearray lowers its energy without introducing type 1 vertices by arranging the type 2 vertexmoments into short range vortex orderings.

æ æ

æ

æ æ æ

æ

æ æ æ æ æ æ

11.0 11.5 12.0 12.5 13.0 13.5 14.0

-400

-380

-360

-340

-320

h

XEdi

p,fi

nal\

Figure 3.15: Final dipolar energy Edip vs field strength h for a 4-island edge array,subject to a rotating applied field. Although vertex populations are not changed bythe field-induced dynamics, there is a drop in Edip corresponding to the rearrangement oftype 2 vertices shown in Figure 3.14. Averages are taken over 3 simulation runs, and errorbars represent 1 standard deviation.

67

processes that create type 1 vertices. In general, however, the response to a sequence of

applied field involves a mixing of vertices, taking the system from a ordered polarised state

to a lower-energy but less ordered state.

These ideas of mixing are explored by Nisoli et al. [65, 66], who model ac demagneti-

sation in terms of entropy maximisation in order to calculate an effective temperature.

While we do not employ this formalism, we do have the aim in this Section to find field

protocols that randomise and mix the vertex configurations. Intuitively, mixing dynamics

should be more ‘robust’, since they are less dependent on selecting an exact sequence of

vertex processes and instead have the ‘goal’ of driving as many processes as possible. This

possibility of robust energy-lowering dynamics motivates studying in more depth dynamics

that involve randomisation. While such dynamics do occur under regular field protocols,

it is plausible that a randomised field protocol will be more effective at inducing them.

We will see in this Section that this is the case. In Chapter 5, we will interpret this in

terms of complex networks.

Here we focus on the effects of protocol rather than array edge type, and so we limit

our study to open edge arrays. In all our simulations, a sequence of fields of constant

magnitude h is applied. For random θ protocols, each field angle θ is taken from a uniform

distribution between 0 and 2π. Different simulation runs use different field sequences, so

the average results we obtain are made over a sample of all possible sequences. We also

study rotating field protocols with large ∆θ, in which h is constant and θ increases in

steps that are large and incommensurate with 2π.

Experimentally, random field protocols can be implemented by rotating a sample

through a random angle, stopping the rotation, and applying a field pulse. Large ∆θ

protocols can be implemented by pulsing the field at regular times. To our knowledge,

there are no published results of such experiments in the literature. We also note that

there is a subtle difference between any experimental realisation of a random or large ∆θ

protocol and our simulations presented here. We simulate the evolution under a field that

is instantaneously turned on with amplitude h at each field application. In experiments,

the field must be ramped up from zero at each field application and then ramped down.

The configurations obtained by pulses with finite ramp rate are not necessarily the same

as those obtained by pulses with infinite ramp. The differences can be substantial, as

we discuss in Appendix A.5, but we focus only on infinite ramps, partly for simplicity of

68

1 2 3 54

Figure 3.16: Random θ protocol field regimes. Final type 1 population of an openedge array as a function of applied field strength for a random field protocol (blue circles)with the results for a rotating field protocol (see Section 2.4) shown for comparison (redtriangles). The dashed vertical lines indicate the boundaries of the field regimes discussedin the text, which are indicated by number. The shaded area indicates the field strengthsfor which n1 is always fluctuating and the ‘final’ populations are taken at an arbitrarycutoff time of 2000 simulation steps.

simulation, and partly because infinite ramp rates provide an idealisation that makes the

effects of randomness in driving apparent.

Random θ simulation results

As is the case for the rotating protocols, the behaviour for random field protocols can be

divided into several field regimes that are seen in n1 vs h plots and can be explained in

terms of thresholds for vertex processes.

Figure 3.16 shows n1 vs h averaged over 10 independent random field protocols. There

are five non-trivial field regimes. In the trivial very low field regime, no dynamics occur

because the applied field cannot overcome island switching barriers. In the trivial very

high field regime, which is not shown, each field application re-polarises the array and no

type 1 vertices are created. We discuss now the five intermediate field regimes.

Regime 1—

In the first, 10.45 < h < 11.25, the dynamics are the same as those of the low field regime

for the rotating field. This is because the field is only able to drive dynamical processes

when its projection onto island axes is sufficiently large (c.f. Equation (1.5)). Application

of the field at other angles has no effect. This is also the case for the small dθ rotating

protocol, where dynamics occur only in discrete steps at θ ≈ 3π/4 and θ ≈ 5π/4 (see

69

Figure 2.9). The importance of the sequence of field angles will be a theme in this Section

and later in the Thesis, where we discuss these sequences in terms of paths on complex

networks.

Regime 2—

The second regime, 11.25 < h < 12.04 has the same boundaries as the high field regime for

rotating fields, but the final n1 values attained are significantly higher: approximately 0.6

rather than 0.3. This reflects a difference in the dynamics. Under the rotating protocol,

type 3 vertices always reach an array edge. In contrast, under the random θ protocol, it

is possible for type 3 vertices to propagate part way across the array and remain in the

bulk, if the field is applied at an angle such that its projection onto the island axes is not

large enough to drive complete propagation.

The type 3 vertices in the bulk can create type 1 vertices in the process 3© 2©→ 1© 3©.

Examples of this dynamic are seen in Figure 3.17. These type 1 vertices serve as nucleation

sites for type 1 domains: type 1 vertices cannot be destroyed at these field strengths, so

when type 3 vertices propagating across the array encounter type 1 vertices in the bulk,

they cannot propagate further and remain next to the type 1 vertices until a field is applied

at an angle that allows them to be ‘reflected’ in the process 3© 2© → 1© 3©. This adds to

the domain, in a sort of ‘deposition’ dynamics.

These dynamics of bulk nucleation and growth of type 1 domains allow for larger final

type 1 populations than the type 1 invasion dynamics that occur for rotating fields. This

is because the array edges remain largely ‘clear’ of type 1 vertices, which avoids trapping

and allows continued type 3 nucleation at the edges. We will see in the next Chapter that

disorder is similarly able to reduce the effect of trapping by allowing dynamics to start in

the array bulk.

Regime 3—

In the third regime, 12.04 < h < 13.54, even larger type 1 populations are reached as the

field strength is increased. Type 1 domain nucleation and growth occurs as it does in the

second regime, but for these fields an additional process 1© 3© → 3© 2© can occur, that is,

type 1 vertices can be destroyed. This enables a more efficient exploration of phase space,

because type 1 domains may shrink as well as grow until a favourable configuration is

attained. Indeed, n1 fluctuates significantly over time.

Regime 4—

70

a) b) c)

d) e)Type 1

Type 2

Type 3

Figure 3.17: Nucleation and growth of a type 1 domain in the array bulk, causedby a random θ protocol. The type 3 vertex circled in part (a) propagates part wayacross the array, as shown in (b), and is then deflected in the process 3© 2©→ 1© 3© beforepropagating slightly further, as shown in part (c). As shown in (d), the type 3 vertexpropagates until it reaches a type 1 vertex, where it stops because it cannot pass throughthe type 1 vertex. It is later deflected in the process 3© 2© → 1© 3©, as shown in part (e).Field directions during propagation events are indicated in the top right of each frame;intermediate field applications in which no relevant dynamics occur have been omitted.The field strength is h = 11.75.

71

The increase in n1 with h ‘saturates’ for 12.84 < h < 13.54 where the final type 1 pop-

ulation attained is n1 = 1. In other words, in this regime, the random field protocol

represents a reproducible means of obtaining the ground state of the system. The ground

state search is driven by large fluctuations, and as a result the system’s history is not

important. Instead, n1 fluctuates over the range [0, 1) until eventually a sequence of fields

is applied to take the system to the ground state. The mean number of field applications

before the ground state is reached from the initial polarised state is ∼ 2000, but com-

parison of simulation runs shows that the value of n1 is uncorrelated between different

simulation runs at 500 steps prior to reaching the ground state – history prior to this

point is irrelevant.

Regime 5—

The fifth regime is characterised by continual large fluctuations, with no final state reached.

This is because the field, h > 13.54, is strong enough that Zeeman energy can overcome

the dipolar coupling of even the ground state, and no configuration exists that cannot be

changed by an applied field. The type 1 populations shown in Figure 3.16 are taken at

the arbitrary time of 2000 field applications. The large error bars indicate that the states

of different simulation runs at this time are uncorrelated. We discuss these continually

fluctuating dynamics using complex networks tools in Chapter 5.

Conclusion—

In summary, for the right field strengths, the random θ protocol is efficient at driving the

system to a low energy state, and in certain cases even the ground state. It does this by

forcing the system to explore a large region of phase space, until a state is attained that

has sufficiently low dipolar energy to ‘trap’ the dynamics. For the ideal arrays studied in

this Chapter, it is possible to balance the Zeeman energy of the external field with the

dipolar energy so that the only trapping state is the ground state, such as in the fourth

non-trivial field regime, 12.84 < h < 13.54. In Chapter 4, we see that quenched disorder

prevents this balance, by causing some spins to be ‘loose’ enough to always track the

external field.

Large ∆θ simulation results

We now show that a regular field protocol with fixed h and a field step ∆θ large and

incommensurate with 2π can also create large populations of type 1 vertices, in a similar

72

a) b)

Figure 3.18: Initial field angle is important for large ∆θ protocols. Configurationsof open edge arrays after 250 field applications with field angle θ increasing in steps of∆θ = π/2.5 ≈ 1.26. In (a) the first field angle is θ1 = 0, in (b) it is θ1 = π/4.

manner to random θ protocols.

One subtlety in the implementation of large-∆θ protocols is that the exact sequence of

field angles is important, and sequences of field angles that are ‘phase shifted’ relative to

one another can drive the system along very different dynamical pathways. An example of

this is illustrated in Figure 3.18, which shows the final configurations of open edge arrays

after application of two different field protocols with step size π/2.5 1. In the first field

protocol, the first field angle of the sequence is θ1 = 0, in the second it is θ1 = π/4. The

final configurations are very different: the θ1 = π/4 protocol yields large type 1 domains

whereas the θ1 = 0 protocol has a significant population of type 2 and 3 vertices. Although

the choice of initial field angle clearly plays a role in determining the dynamics, we limit

our study to field protocols starting at θ1 = 0, rather than attempting to explore the entire

space of possibilities.

Figure 3.19 shows the type 1 population attained by an array after 2000 field appli-

cations with step size ∆θ between 0 and π. We have sampled field strengths between

h = 10.5 and h = 13.375 in steps of 0.125. The ∆θ values are sampled in steps of 0.05. As

with the random θ protocol, these field protocols are able to drive the system to very low-

energy, high-n1 states. In particular, for (h = 13.25, 13.375, ∆θ = 2.35) and (h = 13.125,

∆θ = 1.6) the type 1 population fraction after 2000 steps is exactly 1. We point out that

at these field strengths, the random θ protocol also obtains n1 = 1. We see below this is

1We note that in this case, the system had short range interactions, with jN = 1.5, jL = 1/√

2 andjT = 1/(2

√2).

73

Figure 3.19: Effectiveness of the large ∆θ protocol at type 1 vertex creation.Type 1 population fraction attained after 2000 steps of a field h rotating from θ1 = 0 withfield angle step ∆θ. For (h = 13.25, 13.375, ∆θ = 2.35) and (h = 13.125, ∆θ = 1.6) thetype 1 population attained is 1. The legend indicates the colour scale used to representn1 values.

attained via similar processes to those that occur under a random field protocol.

For low fields, the dynamics proceed as for the (small dθ) rotating protocol, for any

∆θ. This is the same behaviour as the random θ protocol and it occurs for the same

reason: the field projection is only large enough to induce dynamics when θ ≈ 3π/4, 5π/4;

other field angles have no effect.

On the other hand, the value of ∆θ is important for stronger fields that can induce

dynamics for a range of field angles. Figure 3.20 shows the ∆θ dependence of n1 for h =

13.125, a field strength in the third non-trivial field regime for random θ protocols. When

∆θ is small, no type 1 vertices are created because the magnetistation tracks the applied

field, as it does for the rotating field protocols of Section 2.4. However, for ∆θ > 0.9,

n1 almost always reaches a very large value. The exceptions to this occur when ∆θ is

approximately a rational fraction of 2π, as indicated by the drop in n1 seen for ∆θ ≈ 2π/3

and the large error bar for ∆θ ≈ π/2.

The similarities in dynamics between random and large ∆θ protocols – and their

differences with the small dθ rotating protocols – can be understood by considering the

island switching criterion (1.5), which depends on the component of the total field parallel

74

0 Π

4Π

23 Π

4Π0 Π

32 Π

3Π

0.0

0.2

0.4

0.6

0.8

1.0

DΘ

Xn1,

fina

l\

h=13.125

Figure 3.20: ∆θ dependence of type 1 creation for fixed h. Type 1 populationfraction attained after 2000 steps of a field h = 13.125 rotating from θ1 = 0 in steps of∆θ. Averages are made over 10 trials, error bars represent 1 standard deviation.

to the island axes. For a small dθ protocol, the projection reaches approximately the

same peak value every cycle. For strong driving fields, this periodicity induces a periodic

response and the magnetisation tracks the large applied field. In contrast, the sequences

of projections for large ∆θ protocols are more complex, and in some cycles, the projection

is large enough to induce a response that falls short of complete reorientation of the

magnetisation. Under such fields, dynamics similar to those of the random θ protocol can

occur, with type 1 domains nucleating in the array bulk (see Section 3.6).

3.7 Conclusion

We draw two conclusions from this Chapter. First, that in an ideal system, array edges are

able to ‘select’ different vertex processes in the bulk, meaning that the same field protocol

can lead to very different outcomes for arrays with different edge geometries. Second, that

both random and large ∆θ field protocols can be more effective than rotating fields at

driving a perfect finite spin ice to a low-energy, large n1 configuration. This is because

such protocols apply complex sequences of field projections, which can select different

vertex processes than periodic sequences can. At the right field strengths, the selected

vertex processes allow type 1 domain nucleation in the array bulk. These dynamics avoid

trapping and thereby continue for longer times, reaching larger final type 1 populations.

75

76

Chapter 4

Disorder effects

4.1 Overview

We have already seen that inhomogeneities in artificial spin ice open new pathways for

dynamics. As seen in Chapters 2 and 3, in ideal systems, these inhomogeneities are asso-

ciated with array edges, which we have seen control dynamics by ‘selecting’ the dynamical

processes that can occur in the bulk. However, even with inhomogeneities in the array,

the driving field is still global and deterministic – rather than local and randomised –

and dynamics are strongly constrained. Many processes are not possible (for example,

2© 2©→ 3© 3©), and many configurational states are not accessible.

On the other hand, as noted in the Introduction, in real experimental systems ran-

domness does enter, in the form of quenched disorder. Disorder may lower the energy of

randomly-chosen states and energy barriers, making new states accessible, but at the same

time it raises other energies to make states inaccessible. In this Chapter, we explore the

effects of these changes in accessibility of states, via numerical simulations and analytical

arguments.

We first show that the strength of disorder is more important than its origin or the

precise details of how it affects the island interactions or switching. We show in Section 4.2

that there are two disorder regimes, weak and strong. In subsequent work, we focus on

switching field disorder because it is the simplest to understand, and the strong disorder

regime because, as we will see in Section 4.4, it is experimentally relevant.

In the strong disorder regime, spin flipping dynamics in a polarised configuration can

start in the array bulk, and not just the edges. This causes the dynamics to differ signifi-

77

cantly from those seen for perfect arrays. In particular, type 1 creation can occur over a

wider field range and edge effects are negligible. Similar disorder effects are seen for both

single direction field protocols (Section 4.3), and rotating field protocols (Section 4.4). We

also comment in Section 4.5 on how the effects of disorder on the response of a system to

rotating field protocols modify the vertex population dynamics equations of Chapter 2.

We then turn to more general considerations about how disorder affects field driven

dynamics. In Section 4.6, we present simulation results for random θ and large ∆θ field

protocols. We show in Section 4.7 that disorder breaks symmetry between clockwise

and anticlockwise rotating fields, an effect which random θ and large ∆θ protocols ‘take

advantage’ of better than rotating field protocols do. However, although disorder increases

the number of accessible states, it does not always do so in a way that promotes access to

the global ground state, as we prove in Section 4.8.

4.2 Disorder types and regimes

Experimental artificial spin ices are typically fabricated via a process in which magnetic

material is deposited through a mask [23, 24, 42, 47, 48]. The accuracy with which one

can grow islands of identical thickness and with a shape that perfectly matches the mask

is limited, and real islands are subject to roughness, and may also have positions and

orientations that are imperfect. Micromagnetic studies [60] of artificial spin ice show

that this disorder is important, as evidenced by the differences between simulations where

scanning electron micrographs of real islands were used to define islands and those where

ideal islands were used.

Different types of disorder can be categorised by whether they affect interaction ener-

gies (we call this ‘energetic disorder’), the switching fields (‘switching disorder’), or both.

For example, disorder in island positions affects the energy of any pair of islands, but does

not affect their switching fields, which are an inherent property of each island. On the

other hand, island edge roughness affects both energetics and switching characteristics,

in a way that is difficult to disentangle. The effect of edge roughness on the switching

characteristics of sub-micron elements has been studied extensively in both simulation and

experiment [86, 87, 105–108]. For example, in Permalloy wires with mean width ∼ 0.6 µm,

the wire coercivity was shown to increase with peak-to-peak edge roughness [107] (although

78

it should be noted that those structures are significantly larger than the single-domain is-

lands of artificial spin ice, so the results should not be directly comparable). At the same

time, roughness affects magnetisation configurations of elements at remanence [109–111],

which in turn affects interactions in strongly coupled arrays such as artificial spin ice.

In this Thesis, however, we take advantage of the fact that numerical simulations offer

full control over disorder, and study the effects of different types of disorder separately.

This allows us to see that their effects are similar. In terms of energetic disorder, we study

disorder in island positions, disorder in island orientations, and a random perturbation to

pairwise interactions, which is intended as a simplified ‘representation’ of energetic disor-

der, because it does not lead to correlations in island pair energies, unlike the positional

and orientational disorder. We also study disorder in island switching fields. Switching

field disorder has the advantage that its effects are simplest to understand, since it affects

each spin separately in an uncorrelated way. Although we have not studied disorder that

affects both energies and switching fields, it seems reasonable from our results to assume

such disorder will also have similar effects to the types we have studied.

An important consideration is whether disorder changes the ground state of the system.

For example, in the random field Ising model [112], in which disorder is introduced as a

random uncorrelated field acting on each Ising spin, if the dimensionality is below three, the

ground state is always paramagnetic, but its exact configuration depends on the disorder

realisation (see Reference [113] for a review). In ideal artificial spin ice, the ground state

is a uniform tiling of type 1 vertices, and energetic disorder can change this by making

type 1 vertices locally unfavourable, though whether this occurs depends on the disorder

distribution and the system size. We will see below that very strong energetic disorder is

required for there to be a significant probability of a type 1 vertex having higher energy

than a type 2 vertex. For the 20 × 20 spin arrays we study in this Thesis, it is safe to

assume that the ground state is unchanged by disorder, but in the limit of infinite system

size, if there is not a cutoff in the distribution of energetic disorder, there must exist at

least one vertex whose lowest energy configuration is not of type 1 and the ground state

must be changed. On the other hand, switching field disorder does not alter the nature

of the ground state of the system, since it changes only the energies of barriers between

states and not the energies of the states themselves. Therefore, with regards this aspect

of disorder, energetic and switching disorders are not equivalent.

79

Pairwise interaction disorder

Figure 4.1: Determination of the strength of energetic disorder. The mean energyplus/minus one standard deviation of a (top-to-bottom) type 4, 3, 2, 1 vertex, subject topositional, orientational or pairwise interaction disorder. The point where the type 2 andtype 3 bands cross is taken to be the transition between the weak and strong disorderregimes. The distributions are calculated over 200 independent disorder realisations.

In order to measure the strength of disorder that affects energies, we consider the

configuration of a single vertex. In the absence of disorder, the four vertex types have

distinct single-valued energies with E1 < E2 < E3 < E4, but when energetic disorder is

present the energy of a vertex configuration depends on the disorder realisation. When

many disorder realisations are considered, we get a distribution of energies. In Figure 4.1

we characterise these distributions by the mean value plus/minus one standard deviation,

giving energy bands for each vertex type. These bands become wider as the disorder

strength increases, and eventually overlap occurs, which signifies that the probability of

a randomly-selected disorder realisation giving energies ‘out of order’ is significant. We

take the weak disorder regime to be that where there is no overlap, and the threshold of

the strong disorder regime being the point where two bands first cross (in practice, this is

always the type 2 and type 3 bands, since they are closest initially). The transition occurs

at σ(ri) ≈ 0.05 for island positional disorder, σ(φ) ≈ 0.3 for island orientational disorder1

and σ(Epair) ≈ 0.225 for pairwise interaction disorder. (Note that ri = rx,y refers to each

component of an island’s position.)

Unlike energetic disorder, switching disorder leaves the energies of configurational

states unchanged and cannot be characterised in this way. However, an analogous mea-

sure can be constructed by considering the applied field required for a 2© → 3© process

and a 3© → 2© process. When all switching fields are equal, the applied fields required

are hc + 1/√

2 and hc − 1/√

2, respectively. In analogy with the approach taken above

for energetic disorder, we take the boundary between weak and strong disorder to be the

1As an aside, we note that orientational disorder reduces the strength of the dipolar coupling betweennearest neighbours (see Appendix C.2), thereby raising the energy of a type 1 vertex and lowering theenergy of a type 4 vertex.

80

0.2 0.4 0.6 0.8 1.0ΣHhcL

10.0

10.5

11.0

11.5

12.0

12.5

13.0hswitch

Figure 4.2: Determination of the strength of switching disorder. The meanplus/minus one standard deviation of the external field required for the processes 2©→ 3©(upper band) and 3©→ 2© (lower band). The bands cross at σ = 1/

√2 ≈ 0.7.

point where the plus/minus one standard deviation bands meet. In this case, the standard

deviation σ(hc) is the standard deviation in the field required, so the crossing point can be

easily determined to be at σ = 1/√

2 ≈ 0.7, as seen in Figure 4.2. We emphasise that this

measure of disorder strength is relative to the inter-island coupling, not the mean switch-

ing field, since changing hc does not affect the disorder strength at which the bands meet.

This is in contrast to Daunheimer et al. [61], who quote switching field disorder values

as fractions of mean switching fields. However, by using the nearest-neighbour coupling

as the measure for disorder strength, both energetic and switching disorder are measured

relative to the same quantity and can be compared meaningfully.

To see that the strength of disorder is more important than its origin, we compare in

Figures 4.3 and 4.4 the final type 1 population attained by open edge arrays, subject to

rotating field protocols, for a perfect system and systems subject to different types of weak

disorder (Figure 4.3) and strong disorder (Figure 4.4). In each disorder regime, the four

types of disorder all have similar effects on the n1 vs h curves. In particular, switching

disorder has similar effects to energetic disorder. This reflects the fact that dynamics are

determined through the switching criterion −(~h(i)dip+~h)·Mi > h

(i)c , which is affected equally

by changes to the energy landscape (through ~hdip) and changes to the switching field hc.

On the other hand, even though different disorder types have similar effects, the differ-

ence between the weak and strong disorder regimes is striking. As expected, in the weak

disorder regime disorder acts as a small perturbation on the perfect system, but in the

strong disorder regime the n1 vs h curve is strongly altered from the perfect case. We

explore why this is so in the rest of this Chapter, via simulations of dynamics of systems

in the strong disorder regime.

81

9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.00.0

0.2

0.4

0.6

0.8

1.0

h

n1,final

OrientationPositionPairwiseSwitchingNo disorder

Figure 4.3: The weak disorder regime. Final Type 1 population vs applied field h fora perfect 20 × 20 system (black open circles) and four different types of weak disorder:island position disorder with standard deviation in each component of the positions of0.01 (green filled triangles); island orientation disorder with standard deviation 0.06 (redfilled circles); island pairwise interactions perturbed with standard deviation 0.045 (bluefilled squares); and switching field disorder with standard deviation 0.14 (orange emptytriangles). In all cases, the disorder strength σ = 0.2σ∗, where σ∗ is the transition fromweak to strong disorder.

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Orientation

Position

Pairwise

Switching

No disorder

Figure 4.4: The strong disorder regime. Final type 1 population vs applied field hfor a perfect 20× 20 system with open edges (black open circles) and four different typesof strong disorder: island position disorder with standard deviation in each componentof the positions of 0.06 (green filled triangles); island orientation disorder with standarddeviation 0.36 (red filled circles); island pairwise interactions perturbed with standarddeviation 0.27 (blue filled squares); and switching field disorder with standard deviation0.84 (orange empty triangles). In all cases, the disorder strength σ = 1.2σ∗, where σ∗ isthe transition from weak to strong disorder.

82

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0.0 0.5 1.0 1.5 2.00.2

0.4

0.6

0.8

1.0

ΣΣ

*

n 1n 1

*

Figure 4.5: Final n1 at h = 10.5, vs disorder strength. Blue circles represent switch-ing field disorder, red squares represent disorder in island positions. The array has 20×20spins. The n1 values have been scaled to n∗1, their value in an ideal system, and thedisorder strengths have been scaled to σ∗, the transition point from weak to strong dis-order. Averages are made over ten simulation runs and error bars represent one standarddeviation.

To examine the transition from weak to strong disorder, we plot the final n1 value

obtained by a rotating field protocol with h = 10.5 against disorder strength in Figure 4.5.

At this field strength, an ideal system is in the low-field regime, and as disorder strength

is increased, the final type 1 population is reduced. In order to compare switching field

disorder with disorder in island positions, we scale the n1 values by n∗1 ≈ 0.91, the n1

value obtained in an ideal system, and we scale the strength of each disorder by σ∗, the

transition point from weak to strong disorder. In agreement with the results already seen

in Figures 4.3 and 4.4, the two curves coincide within error bars almost everywhere. While

the transition from weak to strong disorder is not sharp, for σ . 0.2σ∗, the n1 values are

close to those of the ideal system, and for σ & 1.5σ∗ there is a clear regime in which

increasing disorder strength does not affect n1.

In the rest of this Thesis, we will focus on switching field disorder, an approach taken

by other authors [31, 33, 45, 52, 61]. Unless otherwise stated, in our simulations we use

a Gaussian distribution of island switching fields, with mean hc = 11.25 and standard

deviation σ = 1.875 ≈ 2.7σ∗, a value well into the strong disorder regime. These values

are based on experimental data obtained by Shawn Pollard and Yimei Zhu at Brookhaven

National Laboratory [114], and we will see in Section 4.4 that they give results compatible

also with experiments performed at the University of Leeds.

83

4.3 Single direction field protocols

We first study the effects of disorder on the dynamics of an array initially in the +x

polarised configuration, subject to a field applied in the −x direction. In Section 3.3 we

studied such protocols acting on perfect systems. Recall that in perfect systems the array

edges are important for determining the response of the system to an applied field, with

open edge arrays exhibiting different behaviour to closed and 4-island edge arrays. Also,

in perfect systems reversal occurs via large jumps in, e.g., mx, as type 3 vertices move

across the array in long avalanches.

Disorder has three key effects on dynamics. The first is that the differences between

the edge types disappear. Figure 4.6 shows the net x component of magnetisation as the

field strength is increased, for the three edge types. The difference in mx curves between

the three edge types is insignificant. The second effect of disorder is that dynamics occur

over a wider range of field strengths. In a perfect system, the transition between +x and

−x polarised states occurs between h ≈ 14.5 and h ≈ 21.45 for open edge arrays, and over

a narrower field range for closed and 4-island edge arrays (see Section 3.3). In contrast, in

a strongly disordered system, magnetisation reversal occurs between h ≈ 10 and h ≈ 25

for all three edge types. The third effect is that the reversal occurs smoothly and there

are no large avalanches. We note that disorder has similar effects on dynamics when the

field is applied at 90 to the initial magnetisation. Details are given in Appendix C.1.

We can understand these three effects by examining the configurations taken by the

array during the reversal. The vertex configurations of an open edge array at three field

values (h = 12, h = 16 and h = 22) are shown in Figure 4.7. Similar configurations

are seen for closed and 4-island edge arrays. Because edge effects are suppressed, in the

following discussion we focus on how disorder affects open edge systems.

The most obvious difference in the dynamics is that type 3 vertex creation can occur

in the array bulk as well as at the edges. The type 3 creation occurs in the process

2© 2© → 3© 3©. This can be observed in the configuration at h = 12 (Figure 4.7). In

a perfect system the edge spins are all easier to switch than the bulk spins, but when

disorder is present the easiest spins to flip are distributed throughout the array. In other

words, edge effects are caused by differences in coupling on the order of nearest-neighbour

interactions, and are therefore suppressed by strong disorder that is also on the order of

84

No disorder

Strong disorder

Figure 4.6: Magnetisation reversal in a disordered system. Normalised net xcomponent of the magnetisation for open edge arrays (orange), closed edge arrays (red) and4-island edge arrays (blue). The field is applied antiparallel to the initial net magnetisationand is increased in strength in steps of 0.05 in our reduced units. In the upper panel,shown for comparison, there is no disorder (see Figure 3.5) whereas in the lower panelthe standard deviation in island switching fields is σ = 1.875. Averages are made over 10simulation runs, with each run having a different disorder realisation, and shaded regionsindicate one standard deviation in mx. The dashed vertical lines indicate the field valuesat which the configuration snapshots in Figure 4.7 are taken.

Type 1 Type 2 Type 3Type 1 Type 2 Type 3Type 1 Type 2 Type 3

Figure 4.7: How reversal occurs in a disordered system. Vertex configurations ath = 12, h = 16 and h = 22 for an open edge array of 20 × 20 spins subject to a fieldapplied at 180 to the original (+x) magnetisation.

85

nearest-neighbour coupling.

The dominant type 3 propagation process is still 3© 2© → 1© 3©, as evidenced by the

large number of type 1 vertices in the configuration at h = 16 in Figure 4.7. However, the

process 3© 2©→ 2© 3© also occurs, which was not possible in the ideal system. This process

is the source of the type 2 vertices with net moment in the y direction seen at h = 16.

As the field is increased further, the array re-polarises into the −x direction. Because

of the large switching fields of some islands, this is a gradual process, as seen by the

configuration at h = 22, in which most vertices are of type 2 with a −x net moment, but

some are still of type 1 or 3. This, and the ability of some spins to switch at very low fields,

leads to a smooth magnetisation reversal that occurs over a wide range of field values.

These results agree with both experiments and other simulation work. Bulk nucleation

of chains of spin flips and smooth magnetisation reversal are seen in experiments of both

square [43, 60] and kagome [45, 52, 61] ices (although the authors of Reference [52] report

that thinner films show ‘sharper’ magnetisation reversal [53]). Similarly, simulations of

hysteresis loops in colloidal ‘spin’ ices with disorder in the heights of barriers between

colloid positions also show smooth hysteresis loops [31].

In ideal open edge arrays, we saw that type 3 vertices can propagate from one edge of

the array to the other in avalanche processes. When disorder is present, avalanches can

still occur, but their length is limited because when they encounter spins with very high

switching fields, they are stopped. We do not make a quantitative analysis of avalanche

dynamics, but, as mentioned in Chapter 1, avalanches that appear during the magnetisa-

tion reversal of connected and disconnected kagome lattices have been studied by other

authors [45, 62], and their findings suggest that such an analysis performed on square ice

could be interesting, because those two studies find different distributions of avalanche

lengths.

To conclude this Section, we note that we have seen that disorder allows processes

to occur that are forbidden in perfect systems, thereby opening new ‘pathways’ for the

dynamics. However, disorder also closes pathways. For example, for the θ = 180 protocol,

the open edge array does not attain the same low-energy, high-n1 state that it does in a

perfect system (see Section 3.3, in particular Figure 3.6), because bulk nucleation of type

3 vertices blocks the orderly invasion of type 3 vertices via avalanches along well-defined

paths. In other words, disorder has blocked a dynamical pathway. This idea recurs in this

86

Chapter, and in Chapter 5, we make the concept of dynamical pathways more rigorous in

a network picture of dynamics.

4.4 Rotating field protocols

In order to further explore how disorder affects dynamics, we simulate rotating field proto-

cols, acting on arrays with strong disorder, and compare the simulation results with those

obtained experimentally.

Simulation results

We saw in the previous Section that strong disorder allows the process 2© 2© → 3© 3© to

occur in the bulk, at sites where hc is small. At the same time, the edge islands of the

array are not always able to flip at low applied fields: some have large hc values and are

pinned. We expect these considerations to also be important when the system is subject to

a rotating applied field. This allows us to make some general predictions about dynamics

under rotating fields in the presence of strong disorder.

The first is that the array edge geometry will be unimportant in determining the

response of the system to an applied field. We have already seen that this is the case for

single direction field protocols, and there is no reason to expect differently when the field

protocol is changed.

The second is that for open edge arrays, type 1 vertex creation in the low field regime

will be suppressed, because – as was the case for +x polarised arrays subject to a field in

the −x direction – the orderly propagation of type 3 vertices in the process 3© 2©→ 1© 3©

is blocked because type 3 vertices can be nucleated in the bulk. On the other hand, in

the high field regime for open edge arrays, we expect that the blocking that occurs in

ideal systems when the edges are covered with type 1 vertices will not occur. As was the

case for the random θ protocol, when the dynamics start in the bulk domains of type 1

vertices can grow outwards, avoiding this blocking. This should lead to larger final type 1

populations for those field strengths.

The suppression of type 1 creation for low fields and enhancement of type 1 creation

for higher fields leads us to expect that for sufficiently strong disorder the two field regimes

of the perfect system should be replaced by a single peak in n1 vs h. Our expectations are

87

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8 9 10 11 12 13 14 150.0

0.1

0.2

0.3

0.4

0.5

h

Xn1,

fina

l\

Figure 4.8: Strong disorder eliminates edge effects, including field regimes. Meanfinal type 1 vertex population n1 vs applied field strength h for systems subject to disor-der in hc (standard deviation 1.875), for open edge arrays (orange open squares), closededge arrays (red triangles) and 4-island edges (blue circles). Averages are made over tenrealisations of disorder, error bars represent one standard deviation.

borne out by simulation results. Figure 4.8 shows the final net type 1 populations plotted

against field strength for arrays of the three different edge types subject to switching field

disorder with σ = 1.875. The curves for the three different edge types are the same, as

anticipated, all featuring a single peak near h = 11.75.

Comparison with experimental results

The results presented above agree well with results from experiments performed at the

University of Leeds, as reported in Reference [67]. In those experiments, five nominally

identical open edge arrays and five closed edge arrays, of the same number of islands as

studied in this Thesis, with a thin film structure of Cr(2nm)/Permalloy(30nm)/Al(2nm),

were patterned on a single Si chip, using methods described in Reference [42]. The arrays

consisted of islands that were nominally 85 nm by 280 nm on a lattice of 400 nm constant,

and nearest neighbour interactions in such a system are of the order of tens of Oersteds.

After being prepared in a polarised state using a large in-plane field at 45 to the island

axes, the system was subject to hundreds of rotations of a rotating in-plane ‘hold’ field

with fixed amplitude Hh, to attain a steady state. The external field was then rapidly

ramped down to zero – the field ramp rate was ∼ 10, 000 Oe/s (to be compared to a

rotation period of ∼ 30 ms), so that the field range in which non-trivial dynamics occur

was crossed within a single rotation of the sample. Hold field amplitudes between 411

Oe and 606 Oe were studied, in 22 Oe steps, which is enough to resolve well the n1 vs h

88

a) b)

Figure 4.9: Vertex populations vs hold field for (a) experiment and (b) the-ory. The symbols represent vertices of type 1 (circles), 2 (triangles), 3 (diamonds) and 4(squares), with open (closed) symbols for open (closed) edge arrays. Each data point isthe average over several runs; error bars represent the standard error.

curve for this system. Final configurations at zero field were imaged using magnetic force

microscopy.

The plot of final vertex populations vs hold field is shown in Figure 4.9(a). The agree-

ment with the theoretical results shown in Figure 4.8 is quite good, with the same overall

shape of n1 vs h and the same suppression of edge effects. To attain better quantitative

agreement, we also simulate field protocols in which, after rotation, the field is ramped

down over half a cycle to h = 8, a field strength too low to induce dynamics. These results

are shown in Figure 4.9(b), and the agreement with experimental results is striking. In

simulations, the disorder is always switching field disorder, with σ = 1.875. We do not

aim to exactly quantify disorder strength, because (a) uncertainty in the precise field ramp

rates in experiments introduces an extra fitting parameter; (b) disorder in our simulations

is an effective switching barrier distribution that models disorder in both switching char-

acteristics and interactions, and does not give direct information about island properties;

and (c) we are in the strong disorder regime where n1 is not sensitive to disorder strength

(see Figure 4.5). Even without an exact quantification of disorder, it is clear that the

magnitude of disorder effects is of the same order as nearest-neighbour interactions.

4.5 A note on vertex population dynamics

The vertex population dynamics equations of Section 2.5 can be readily extended to de-

scribe a system with switching field disorder. Because strong disorder allows type 3 vertices

89

to be nucleated in the array bulk, the assumption of vertex mixing is reasonable. However,

as we see in this Section, the vertex population dynamics are still unable to capture the

field-dependence of the dynamics, for the same reason as in the ideal system: the array

re-polarisation processes of the very high field regime are not accounted for, and instead,

in the vertex population equations the rates of type 1 vertex-creating processes increase

with field strength.

We modify the equations of Section 2.5 in two ways to include disorder effects. First, we

introduce the bulk type 3 nucleation process 2© 2©→ 3© 3©. Second, we change the energetic

factor f in the vertex process rates from its step function form in Equation (2.6). The step

function of Equation (2.6) described the fact that the probability a process is energetically

allowed depends on the switching criterion (1.5). In an ideal system, all switching fields

are equal to hc and the probability a process is allowed is simply 0 when the total field

is smaller in amplitude than hc and 1 when the total field is larger. When disorder is

present in the form of a distribution of switching fields, the probability that a process is

energetically allowed can be calculated from the cumulative distribution function of the

switching field distribution. For a Gaussian distribution, the energetic factor f is

f(h, hd, µ, σ) =1

2π

∫ 2π

0dθ∑α

(1 + erf(

h cos(θ − α)− hd − µ√2σ

)

)/2 (4.1)

where h is the external field, hd is the contribution from spin-spin interactions to the ener-

getic barrier for the process, µ and σ are the mean and standard deviation of the switching

fields, respectively, α represents the spin directions and can take values π/4, 3π/4, 5π/4, 7π/4,

and the error function erf(x) = 2√π

∫ x0 dy exp(−y2).

The dynamics are described by Equations (2.9), with the addition of a term +2ν 2© 2©→ 3© 3©n22F

to n3F and subtraction of the same term from n2F . The rate ν 2© 2©→ 3© 3© is a(1 −

N/2)14f(h,

√2, µ, σ). The other rates are unchanged from those shown in Table 2.1, apart

from the change in definition of f described above.

The field dependence of the steady-state n1, for an array with 102 vertices, a mean

switching field of µ = 11.25 and a switching field standard deviation of σ = 1.875 is shown

in Figure 4.10. As already mentioned, because the population dynamics equations do not

account for re-polarisation, the n1 vs h curve does not match the simulated n1 vs h of

Figure 4.8. Instead, it displays the same saturation as the n1 vs h curve for the vertex

90

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0 5 10 15 200.00.10.20.30.40.50.60.7

hn 1

,fin

al

Figure 4.10: The field dependence of steady state type 1 populations, whendisorder is present. Values are calculated using the approach described in the text, foran array with 102 vertices.

population dynamics of an ideal system (see Figure 2.18), albeit with a slower growth

in n1 as would be expected from gradual, rather than step-like, increase in f with field

strength.

4.6 Random θ and large ∆θ protocols

We have already seen in Section 3.6 that in an ideal system, random θ protocols and large

∆θ protocols are able to avoid trapping by nucleating type 1 domains in the array bulk.

Although disorder allows dynamics to start in the array bulk for a uniform rotating field

protocol, it turns out that random θ and large ∆θ field protocols are still more effective

at generating low-energy, high-n1 configurations.

This is demonstrated in Figure 4.11, which compares the type 1 population after 2000

random field applications to the steady-state type 1 population for a rotating field protocol.

For all field strengths, the random θ protocol attains a larger n1. Similar n1 vs h curves

are obtained by large ∆θ protocols, as seen in Figure 4.12, which shows n1 after 2000 field

applications, as a function of h and ∆θ. In all cases, the n1 vs h curve has a single peak,

which for the large ∆θ protocols shows a dependence on ∆θ.

We make two observations. First, the random field protocol is more effective at creating

type 1 vertices than the rotating field protocol, at all field strengths, and the large ∆θ

protocol is typically at least as effective as the rotating protocols, with its effectiveness

increasing as ∆θ becomes larger than ∼ π/4. Second, unlike in a perfect system, the

91

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8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

h

Xn1,

fina

l\

Figure 4.11: Random θ protocols are more effective at type 1 creation thanrotating protocols. Mean final type 1 vertex population n1 vs applied field strength hfor systems subject to disorder in hc (standard deviation 1.875), under a rotating appliedfield (red triangles) and a random θ protocol (blue circles). Averages are made over tenrealisations of disorder, error bars represent one standard deviation.

Figure 4.12: Mean type 1 population attained by large ∆θ protocols. n1 valuesare measured after 2000 steps of a field h rotating from θ1 = 0 with field angle step ∆θ.Averages are made over 10 runs, each with a distinct realisation of disorder. The legendindicates the colour scheme used to represent n1 values.

92

non-periodic field protocols are not able to take the system to a complete type 1 vertex

tiling. We discuss these observations in the next Sections.

4.7 Symmetry breaking by disorder

We saw in the previous Section that a random field protocol is more effective than a

rotating field protocol at generating low-energy, high-n1 states in a system with strong

quenched disorder, even though neither protocol is subject to blocking by type 1 vertices

at the array edges. Evidently, then, blocking is not the only factor that determines the

effectiveness of a field protocol in finding low energy states. In this Section, we return

to rotating field protocols, and show that in disordered systems, the sense of rotation

matters.

In the studies of rotating applied fields presented in Chapters 2 and 3, the field was

always rotated in an anticlockwise direction. It is easy to see that a clockwise rotating field

would yield the same results in terms of vertex populations. This is because the horizontal

centre of the array is an axis of reflection symmetry for a +x polarised configuration and

the field is initially applied along this direction. Any sequence of spin flips caused by an

anticlockwise rotating field can be mapped onto a reflected sequence of spin flips caused by

a clockwise rotating field. The resulting configurations are different, but the populations

of vertex types are the same. 2

On the other hand, in an ice with quenched disorder, two spins related by reflection

are not interchangeable. For example, if the disorder is in island switching fields, two spins

related by reflection will, in general, have different hc values. We note that symmetry is

restored upon averaging over disorder realisations: in the case of switching field disorder,

this is because every set of switching fields can be mapped onto a reflected set of switching

fields. However, in this Section we will show that symmetry breaking can still be impor-

tant, even when dealing with averages. We discuss the specific example of switching field

disorder because it is simplest, but note that these arguments can be made for any type

of quenched disorder.

As an example that demonstrates the principles of symmetry breaking by disorder, we

2We note a subtlety, which is that in simulations, dynamics are partly determined by a random numbergenerator, even in the ideal system. As a result, two independent simulations of the same field protocolcan lead to different outcomes. This is, however, a computational point, not a physical one, and averagingover independent simulation runs prevents this from being a problem.

93

H H HH H

H H HH H

HHHH H

H

H

H

HHHH H H

No disorder

Disorde

r

Key to colours: hc=10 hc=11 hc=12 hc=13

Figure 4.13: Disorder in switching fields can allow type 1 vertex creation for onesense of rotation and block it for the other. The top two rows show a vertex withall switching fields equal to 10. Under an applied field h = 11, the configuration oscillatesbetween different type 2 vertices, with no other configurations allowed. Both senses ofrotation give this oscillation, though the order of configurations accessed is different forclockwise (top row) and anticlockwise (second row) rotation. On the other hand, when theislands have switching fields of 10, 12, 11, 13 (starting at the top left island in the vertexand working left-to-right, top-to-bottom), clockwise rotation (third row) causes oscillationbetween type 2 and type 3, whereas anticlockwise rotation (bottom row) causes a stabletype 1 vertex to be created.

94

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Figure 4.14: Accessing both senses of rotation avoids blocking in a disorderedsystem. Mean final type 1 vertex fraction n1 as a function of applied field strength h.For an undisordered system, a clockwise rotating field (orange diamonds) an anticlockwiserotating field (blue squares) and a field protocol that alternates sense of rotation every 2cycles (red circles) all give the same final n1. When disorder is included (as a switchingfield distribution with σ = 1.875), the clockwise and anticlockwise rotating fields (lightblue circles and green inverted triangles, respectively) give same results as each otheron average, but the alternating field protocol (purple triangles) leads to a larger final n1.Averages are made over 100 simulation runs, and error bars represent 1 standard deviation.

show that for a suitable distribution of hc values, the vertex process 2© → 3© → 1© can

be allowed under one sense of rotation but not the other, as illustrated in Figure 4.13.

In the absence of disorder, the two senses of rotation give equivalent results: for a the

chosen external field strength, the net moment of the vertex simply tracks the rotating

field. However, when disorder is present the situation is different. Under an anticlockwise

rotating field, it is possible to create a type 1 vertex that is stable under continued rotation

of the field (even if the sense of rotation of the field is reversed). On the other hand, if

the applied field rotates clockwise, the first step of type 1 creation, 2© → 3© can occur

but the process 3© → 1© is blocked because the spin that needs to flip is pinned by its

large hc value. Instead, the type 3 vertex will subsequently convert back to type 2 and

an oscillation between type 2 and 3 will be set up. Although this example is of a single

vertex with island switching fields selected to illustrate the point, the same phenomena

can be observed in simulation results, which we discuss below.

On average, the number of vertices for which type 1 creation is blocked under clockwise

and anticlockwise fields is equal, and symmetry is maintained, as seen in Figure 4.14, where

the average type 1 vertex populations generated by clockwise and anticlockwise rotating

fields are the same, both for ideal and disordered systems. However, the key point is this:

95

a field protocol that samples both senses of rotation can, in the disordered system, access

processes that are allowed for one sense of rotation but not the other. As an example, we

simulate a protocol where the field rotates anticlockwise for two cycles, then clockwise for

two cycles, repeated for ten cycles total. As seen in Figure 4.14, this protocol gives the

same results as clockwise and anticlockwise rotating fields for the perfect system, because

the two senses of rotation are equivalent. However, in a disordered system the alternating

protocol creates a significantly larger final type 1 population than the protocols with a

single sense of rotation.

These results also shed light on the effectiveness of the random field protocol for at-

taining low energy states in disordered systems. The alternating protocol is able to get

around blocking by breaking the sequence of field angles and sampling a different sequence.

Similarly, the random θ protocol can be considered to sample all possible sequences of field

angles, thereby avoiding blocking even more effectively. In the next Chapter, we revisit

this idea in the context of complex networks.

4.8 Inaccessibility of the ground state for disordered sys-

tems

We have seen that in the presence of disorder, spin ice does not attain a complete type

1 tiling from an initial polarised state, even after application of a random field protocol,

which is able to obtain the ground state in an ideal system. That result was from simulation

studies and is in agreement with experimental work, where typically only short range

ground state order is observed after demagnetisation [23, 40, 48, 63]. The open question

is: to what extent this is inherent to the nonequilibrium driven dynamics of a frustrated

system, and to what is the role played by disorder? Simulation studies of a nanopatterned

superconductor ‘spin’ ice suggests that disorder is at least partly responsible [33]. In

those studies, ac ‘demagnetisation’ was able to attain the ground state of an ideal system,

but when a distribution of heights of barriers between vortex pinning sites is introduced,

higher-energy vertices persist.

We now prove analytically that the transition from a polarised state to the ground

state is impossible for any field protocol with constant field amplitude, for any sufficiently

large system subject to sufficiently strong disorder. We give quantitative lower bounds on

96

the system size and disorder strength for which this limitation holds. Our argument is a

specific example of a broader problem of how disorder affects access to a set of degenerate

states, and should also be applicable to, e.g, antiferromagnetically coupled magnetic dots

with a distribution of switching barriers, such as those proposed for logic gates and data

storage [115–117].

A related problem has been studied in the 1-dimensional random field Ising model. In

that system, the ground state depends on the disorder realisation, and actually becomes

more accessible when disorder strength is increased [118, 119]. The model is of an Ising

ferromagnet with nearest neighbour coupling of strength J , with the additional energy

term of a random field hi acting on each spin. The hi are taken from a Gaussian dis-

tribution with standard deviation R. The dynamics of the system are such that spins

always align with the direction of the net field acting on them (up or down). When the

external field is zero, spins with |hi| > 2J align with the random field always, because

even if aligning opposite to the random field gains coupling energy, the loss of random

field energy is larger. Such spins are therefore ‘frozen’. In the limit R →∞, the fraction

of frozen spins goes to unity because the portion of the Gaussian distribution in the finite

range [−2J, 2J ] becomes vanishingly small. As a result, the ground state is trivial: in

the ground state, every spin aligns with its random field. The ground state can always

be reached from a saturated state, simply by turning off the external field and allowing

spins to align with their local field. We will see that in our system, which is subject to

switching field disorder that does not change the nature of the ground state (as discussed

in Section 4.2), the effect of disorder is very different.

One reason we limit ourselves to cases where the ground state is not changed by

disorder is because a key ingredient of our argument is the two-fold degeneracy of the

ground state, and hence the possibility of separate domains of ground state order forming.

Our argument considers two mechanisms by which disorder can force nucleation of multiple

ground state domains. We calculate an upper bound on the probability P (not blocked)

that neither mechanism operates; P (blocked) = 1 − P (not blocked) is a lower bound on

the probability the ground state is blocked. The first mechanism involves spins pinned

into their initial polarised configuration. The second involves spins that are so ‘loose’ they

always align with the external magnetic field.

The first mechanism depends on the initial state being polarised. As was previously

97

B

A

Figure 4.15: The relationship between polarised and ground state configurations.The left hand figure shows a +x polarised state, with dark and light arrows indicatingthe two ground state orderings. The right hand figure shows one of those ground stateorderings, obtained by reversing all spins belonging to the other ordering. If, for example,spins A and B in the left hand figure are both unable to flip, it is impossible for the wholearray to take a ground state configuration.

discussed in Section 3.3, in both a polarised state and the ground state, one can divide the

array into chains of spins aligned north-to-south (in an open edge array, these chains are

zig-zag lines of spins). In the polarised state all chains have aligned net moments, whereas

in the ground state neighbouring chains have oppositely aligned net moment, as illustrated

in Figure 4.15. The chains of the polarised state can be divided into two groups based on

their ground state alignment. To transform a polarised configuration into a ground state

configuration, all the spins of one group must flip. Suppose one spin from each of the two

groups – e.g, spin A and B in Figure 4.15 – is pinned and remains always in its initial

state. Then two ground state domains must form. This gives a necessary condition for

the ground state to be attainable: all the spins of at least one group must be able to flip.

The probability that none of the spins in one group are pinned is (1− Ppin)n/2, where

n is the total number of spins in the array and Ppin is the probability of any given spin

being pinned. We describe below how this probability is calculated, but note here that it

depends on switching barrier distribution and the applied field strength, but we treat it as

configuration-independent. The probability that at least one of the two groups contains

no pinned spins is 2(1− Ppin)n/2 − (1− Ppin)n, where (1− Ppin)n has been subtracted to

avoid double-counting cases where none of the spins in the array are pinned.

The second mechanism by which the ground state may be blocked is related to the

first, but is slightly more complex and does not rely on an initial polarised state. In the

case of pinned spins, disorder forces spins to retain the relative alignment they held in the

polarised state, which may be opposite to the relative alignment they require for ground

98

A

BC

D

Figure 4.16: The four groups of spins: A, B, C and D, based on their direction in the GS.

state ordering. On the other hand, if two spins that are antiparallel in the ground state

are ‘loose’ and always align with the external field, then the ground state is blocked. It

should be emphasised that two loose spins are required for the ground state to be blocked,

because if only one spin is loose, then the rest of the array could be prepared in a ground

state and then the external field used to correctly align the loose spin. We divide the spins

of a ground state configuration into four groups, A, B, C and D, based on their alignment.

This is illustrated in Fig. 4.16. It is not possible for an A spin and a C spin to both be

aligned to an external field as well as their relative ground state alignment, and the same

is true for B and D spins.

As a result, a necessary condition for the ground state to be attainable is that if

group A contains loose spins then group C does not, and if group B contains loose spins

then group D does not. Table 4.1 lists all allowable combinations of groups containing

loose spins, and the probabilities of each of these. The cases are all independent, so

we sum over them to obtain the probability that the GS is not blocked by loose spins:

4(1− Ploose)n/2(1− (1− Ploose)

n/4)2 + 4(1− Ploose)3n/4(1− (1− Ploose)

n/4) + (1− Ploose)n

where Ploose is the probability of a single spin being loose (given below).

Combining the results of the above two arguments, the probability that the ground

state is unattainable from an initial polarised state is 1− P (not blocked), or

P (blocked) = 1−[2(1− Ppin)n/2 − (1− Ppin)n

]×[4(1−Ploose)

n/2(1−(1−Ploose)n/4)2+4(1−Ploose)

3n/4(1−(1−Ploose)n/4)+(1−Ploose)

n],

(4.2)

where we have made the assumption that the two blocking mechanisms are independent

99

Groups Probability

None (1− Ploose)n

A only

(1− (1− Ploose)n/4)(1− Ploose)

3n/4B onlyC onlyD only

A and B only

(1− (1− Ploose)n/4)2(1− Ploose)

n/2A and D onlyB and C onlyC and D only

Table 4.1: Combinations of groups of spins (see Fig. 4.16) that can contain loose spinswithout blocking the ground state, and the probabilities with which they occur. In allother cases, the ground state is blocked.

so the probability of neither of them occurring is simply the product of the probabilities

of each of them not occurring.

We now give expressions for the probabilities Ppin and Ploose, which depend on field

strength h and switching barrier distribution. We calculate lower bounds on Ppin and Ploose

under the assumption of a Gaussian distribution of switching fields and using maximally

‘strict’ definitions of ‘pinned’ and ‘loose’.

We define a ‘pinned’ spin as one whose switching barrier is so high that it cannot flip

even if the local environment is maximally unfavourable. Recall that spins flip according

to the condition

−(~h(i)dip + ~h) · Mi > h(i)

c . (4.3)

Rearranging, under the assumption that the external field is applied antiparallel to the

spin, the probability that a spin is pinned is simply

Ppin = P (hc > |h| − hdip, unfav), (4.4)

where hdip, unfav < 0 is the parallel component of the dipolar field acting on a spin in a

maximally unfavorable local environment. We take hdip, unfav to be −6+√

2, the field that

must be overcome for the process33© 3©→ 1© 1©, calculated under the approximation that

spins only interact within vertices (effectively averaging over longer-range configurations;

see Table B.1 for a list of all vertex processes). Then for a Gaussian distribution of hc

3hdip is more negative for a 4© 4© pair, but such a configuration is not seen in our simulations.

100

values,

Ppin(h) =

∫ ∞h−(−6+

√2)dhc

1√2πσ

e−(hc−µ)2

2σ2

=1

2

(1 + erf

(µ− h− 6 +√

2√2σ

)),

(4.5)

where µ = 11.25 is the mean switching field and σ is the standard deviation, and the error

function erf(x) = 2√π

∫ x0 dy exp(−y2).

In a similar manner, we define a ‘loose’ spin as one that aligns with an external field

even when its neighbours are ground state ordered. The probability that a spin is loose is

Ploose = P (hc < |h| − hdip, fav) (4.6)

where hdip, fav > 0 is the parallel component of the dipolar field acting on a spin in a

maximally favorable local environment. We take hdip, fav to be 6−√

2, the field that must

be overcome for the process 1© 1©→ 3© 3©. Then, for a Gaussian distribution of hc values,

Ploose(h) =

∫ h−(6−√

2)

−∞dhc

1√2πσ

e−(hc−µ)2

2σ2

=1

2

(1 + erf

(−µ+ h− 6 +√

2√2σ

)).

(4.7)

Using these expressions for Ppin and Ploose, we can determine the probability of blocking

from (4.2) for a given system size and spread in hc values. As seen in Figure 4.17, for

σ = 1.875, the probability of blocking is always greater than 65% for the 20 × 20 spin

system we study. Figure 4.18 shows that the minimum of P (blocked) grows rapidly with

both disorder strength and array size. We note that in the limit of an infinite system,

finite probabilities of pinned and loose spins lead to finite populations of spins in both GS

alignments, and the GS is necessarily blocked.

We emphasize that these results give a lower bound on the probability the ground

state is unattainable, since we have both been conservative in our estimates of Ppin and

Ploose, and neglected other blocking mechanisms such as the jamming seen in simulations

of systems without quenched disorder (see Section 2.4). While large P (blocked) indicates

the ground state is inaccessible, small P (blocked) does not mean it can be reached. The

results apply to any field protocol where the field amplitude is fixed, such as the random

θ and large ∆θ protocols studied in this Thesis. An open problem is whether protocols

101

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Figure 4.17: Ground state blocking probability vs applied field. The probabilityof the ground state being unattainable from an initial polarised configuration for a systemwith 20× 20 spins and a spread in hc values of σ = 1.875.

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Figure 4.18: Ground state blocking probability grows with array size and dis-order strength. Left: Minimum value of P (blocked) vs number of spins in the array,for switching field standard deviations between 1 (blue circles) and 2 (green triangles) insteps of 1/3. Right: Minimum value of P (blocked) vs switching field standard deviationσ(hc), for array sizes ranging from 20×20 (blue circles) to 100×100 spins (blue triangles).

102

with varying field amplitude, such as ac demagnetisation protocols, face similar blocking.

4.9 Conclusion

To conclude the Chapter, we comment on the importance of a careful study of disorder

effects. On the one hand, disorder can remove blocking in the system and allow dynamics

that are not possible in perfect arrays. This allows the system to attain low energy, high-

n1 configurations, especially if the field protocol is not constrained to a regular sequence

of fields. However, as we saw in Section 4.8, it is not always the case that disorder makes

the global ground state of the system more accessible. In the remainder of this Thesis, we

approach the study of disorder effects using a complex network representation of dynamics

to allow quantitative comparison of perfect and disordered systems.

103

104

Chapter 5

Network representation of

dynamics

5.1 Introduction and overview

Up to now, our discussion has centred on the dynamics of spins in the array, cast in terms

of vertex dynamics. We have seen that the effects of array edge geometry, field protocol,

and quenched disorder can be described in terms of the vertex processes they allow or

forbid. We have also seen that the vertex picture can be formalised via vertex population

dynamics equations. In this Chapter, we shift our focus to an alternative model for

dynamics, in which the external field ‘transports’ the system through its configurational

phase space. Because the set of configurations that can be taken by a finite number of

Ising spins can be enumerated exactly, we can map this process of transportation onto

a directed network in which nodes are spin configurations and links represent dynamical

transitions.

The network representation of dynamics provides a natural framework in which to

explore effects of field protocols and disorder. We will see that the network picture allows

us to immediately see the effects of varying disorder strength, in a way that is independent

of field protocol. At the same time, it provides an elegant description of the differences

between rotating and random field protocols and the interplay between field protocols

and disorder. It also enables us to calculate in a straightforward way the effects of, for

example, being able to individually control a single spin in the array. One of the goals

of this work is to take advantage of the network tools that have been developed in many

105

other contexts [120–123], such as technological and social networks, and apply them to

understand the dynamics of artificial spin ice.

a b

Figure 5.1: Network representation of the driven dynamics of an athermal Isingsystem. In (a), the two spins are identical with switching thresholds of hc = 10 anda ferromagnetic coupling of strength J = 1. In (b) the left-hand spin is subject to anadditional field of strength h′ = 3 in the up direction (indicated by the red arrow). Thismodifies the possible transitions between states, as shown by the networks of possiblefield-driven transitions for an external field of strength h = 11.5 that can point either upor down.

The essentials of the approach taken, and its use in understanding disorder effects,

can be understood by constructing explicitly a network for the field-driven dynamics of a

pair of athermal, ferromagnetically-coupled Ising spins subject to external field h, which

is allowed to point either up or down. The two spins can take four configurations: up-up,

up-down, down-up and down-down. These are the four network nodes. The hamiltonian

of the system is

H = −Jσ1σ2 − h(σ1 + σ2

), (5.1)

where the σ can take values ±1. For concreteness, we set the coupling strength to be

J = 1, and the spin switching threshold (following the criterion (1.5)) to be hc = 10.

Then an external field of strength h = 11.5 can induce the transitions between states

shown in Figure 5.1(a). For example, the transition from up-up to down-down is allowed,

but it is not possible to pass from the up-up to the up-down state, because the global

nature of the driving field and the ferromagnetic coupling prevents the spins from flipping

independently.

Suppose now that one spin is subject to an additional field of strength h′ = 3, in the

up direction, such as might occur in a random field Ising model [112]. In other words, the

hamiltonian is now

H = −Jσ1σ2 − h(σ1 + σ2

)− h′σ1. (5.2)

106

We again construct the network for an external field of h = 11.5, as illustrated in Fig-

ure 5.1(b). This network is different to that of the ideal system. For example, the random

field acting on the first spin pins it so that the transition from up-down to down-down

is no longer allowed. Thus we see the fundamental idea of the network picture: changes

to the allowed transitions between states are reflected by changes to the network. These

can be measured and analysed using network theoretic tools that have been developed in

other contexts.

The outline of the rest of this Chapter is as follows. In Section 5.2 we comment on

related studies of network representations of dynamical systems, before turning in Sec-

tion 5.3 to describe in more detail how the spin ice dynamics networks are constructed,

and some of their basic properties. We then present our results, exploring how different

network properties relate to the energetics and dynamics of artificial spin ice. In Sec-

tion 5.4, we discuss the global network topology, as revealed by the distributions of the

numbers of links into and out of nodes. These links are determined by the energy of spin

configurations and the barriers between them. In Section 5.5 we study the number of

states that can be reached from a polarised configuration, and the reversibility of dynam-

ical transitions between those states. The network representation of dynamics provides

an elegant description of the differences between rotating and random field protocols and

the interplay between field protocols and disorder, which we discuss in Section 5.6. Fi-

nally, in Section 5.7, we study the structure of the spin ice dynamics networks and discuss

how small changes to the properties of the artificial spin ice can lead to large changes in

dynamics.

5.2 Related work

In this Section, we comment on previous studies that have used networks to describe the

phase space and dynamics of various systems, in order to explore that context of our

network studies. This type of analysis has been applied in a wide variety of contexts,

from the six-vertex model to protein folding, as we will see below. This Section is not

intended as a complete review of network theory; interested readers are suggested to refer

to References [120–123].

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Exact enumeration of phase space

Many systems with interesting physics have a discrete phase space that can be enumerated

exactly. For example, a system of N Ising spins has 2N configurations. Another example is

that of lattice polymers, which can be described by the positions of monomers on a discrete

lattice. Such phase spaces can readily be mapped onto networks, by identifying network

nodes with configurations, and using links to represent transitions that are allowed under

the dynamics of the system. Here we describe some examples of such mappings in more

detail.

Scala et al. [124] use complex networks to give insight into the structure of the phase

space of 2D lattice polymer chains. In those studies, the configurations of a chain with

fixed ends are mapped onto network nodes. A pair of nodes is linked if one conformation

can evolve to another via a single elementary Monte Carlo move. The authors construct

an unweighted network where all moves with non-zero probability are represented by a

link. This is effectively the high-temperature limit, and stands in contrast with simula-

tions where Monte Carlo moves are weighted according to temperature and interactions.

Their studies produced evidence of a small-world structure [125], in which the network

distance between two configurations grows logarithmically with the network size but the

local structure is lattice-like. Although many small-world networks are scale-free, that

is, the distribution of the number of links at each node follows a power law, the lattice

polymer phase space networks are found to have a binomial degree distribution, so that

the probability of a node having k links decays rapidly with k and there are no ‘hubs’

linking many nodes.

Protein conformations have also been studied by Ravasz et al. [126], who construct

network representations of the conformation space of ball-chain model proteins, a model

in which balls are connected by thin rods, and the angles between neighbouring rods can

take a discrete set of values. Thus the configurational phase space of this model is discrete,

and each configuration can be defined by the set of bond angles. Like Scala et al., Ravasz

et al. find the network has small-world characteristics, and a degree distribution that is

inconsistent with a scale-free network.

Interestingly, studies of protein conformations based on sampling molecular dynamics

(MD) simulations – in this case, by taking snapshots of conformations at regular time

intervals and defining nodes as conformations and links as transitions between them – find

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the networks to be scale free [127]. This appears to be in contradiction with the results

for discrete models, but Ravasz et al. resolve this by pointing out that when energetics

are taken into account, as they are in MD simulations, the network studied is not the

complete conformation network, but a network generated from it by allowing only links

that are associated with a maximum gradient in energy. It has been shown that these

‘gradient networks’ can be scale-free, even if they are generated from a non-scale-free

network [128–130].

Another example of biology-related networks comes from the work of Li et al. [131]

who use a network to describe the evolution of a cell towards a steady state. Network

nodes represent states, defined by the on/off characteristics of 11 key proteins. Links

represent allowed evolutions. The network provides an easy way to examine the stability

of the evolution towards the state of steady cell growth, a fixed point which attracts 86%

of other states.

Networks have also been used to study certain classical physics problems. Han et

al. [132–134] describe the state spaces of triangular Ising antiferromagnets, the six vertex

model and 1D and 2D diffusive lattice gas models in terms of undirected networks. Network

nodes represent system states, which are discrete, and pairs of nodes are linked if evolution

between the two states via an elementary ‘move’ (for example, a single spin flip) is allowed.

Dynamics are reversible in those systems, so the links are undirected. The networks have

a Gaussian degree distribution and the small-world property, and in certain systems –

triangular antiferromagnets on their ground state manifold, the six-vertex model, and the

1D lattice gas model – the networks have global self-similarity, that is, they are fractal.

This is shown by defining the distance between a pair of nodes as the number of links in

the shortest path on the network between them, and then using a box covering method.

In that method, boxes that can contain subnetworks of maximum pair-distance lB are

placed over the network until all nodes are in boxes: if the number of boxes NB follows a

power law NB ∼ l−dBB with the dimension dB fractional, then the network is fractal.

The studies of Han et al. also deal with ergodicity, in other words, whether one can

expect time-averaging to be equivalent to averaging over an ensemble. Networks describing

the six-vertex model with free boundary conditions are fully connected, indicating ergodic

dynamics since all states are accessible, but the six-vertex model with periodic boundary

conditions is not ergodic and has a disconnected phase space network [133], meaning that

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not all states are mutually accessible. We touch on related issues of accessibility of states

in the context of artificial spin ice networks in this Thesis.

Sampling phase space

Of course, many systems do not have a discrete phase space; instead, their states are

described by continuous variables. Such systems are still amenable to study via discrete

networks if an appropriate discretisation scheme is applied.

One example where networks have been used to study a continuous system is that

of dynamical maps. These take the form xn+1 = f(xn), and describe how a continuous

variable (in this case, x) evolves in discrete time (which is indicated by the subscript).

The approach taken by Borges et al. [135] and Kyriakopoulos and Thurner [136] is to

subdivide the phase space into discrete ‘boxes’, each of which is represented by a node in

the network. A directed link exists between nodes i and j if it is possible for the system

to evolve from a point in phase space in box i to a point in box j in a single time step.

As would be expected, network properties depend on how the discretisation is performed.

Studies [135] of the quadratic map, xn+1 = 1−ax2n, whose behaviour depends on the value

of a ∈ [0, 2], reveal that as the chaotic regime of the dynamical system is approached, the

network can exhibit strong dependence on box size.

Several authors have also studied network representations of energy landscapes of

glasses [137–141], clusters of atoms [142, 143], and proteins [127, 144–147]. Rather than

attempt to deal with networks in which nodes represent every possible microstate of the

system, these authors have instead constructed networks in which nodes represent states

that have been sampled in some way – typically (free) energy minima, but snapshots from

MD simulations at regular time intervals have also been used – and links represent allowed

transitions. The use of sampling allows for much more complex systems to be studied than

would be possible if every microstate was enumerated, and as mentioned above, raises the

possibility of performing similar studies on large artificial spin ice arrays.

Generic properties of phase space networks

To conclude this Section, we discuss briefly some studies of the generic properties of phase

space networks. The authors of these studies do not construct network representations of

particular realisations of systems, instead they describe from first principles what structure

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networks should have, or how dynamics should occur on a network.

Bertotti et al. [148–150] describe the random field Ising model [112] in terms of a

network of ‘basins’ of mutually-connected configuration states. Two states A and B are

in the same basin if there exists a field sequence that can drive the system from A to B,

and vice-versa. These basins form a directed network, and the authors study the structure

of this type of network. For example, all paths end at the basin containing the saturated

states, because one can always apply a very large field to saturate the system. We will

see this system bears some resemblance to ours because the athermal, driven dynamics

are similarly constrained and strongly directed. However, the question of connectedness

of states is more complex for artificial spin ice, because there are two spin axes, long-range

interactions and high energetic barriers to switching.

Baronchelli et al. [151] build on earlier work describing random walks on networks to

construct a model for glassy dynamics. Rather than study a particular network represen-

tation of a glass, they describe general results that should hold for any network of energy

minima connected by saddle points. By assuming that the time to escape from an energy

minimum is exponential in the depth of the minimum, they are able to model quanti-

ties such as the mean first passage time for a random walk on the network, a quantity

which is related to the global relaxation time for the glass. These results for a time-

dependent, thermally-driven system are not directly applicable to our studies of artificial

spin ice, but it is possible that they might be applied to studies of thermal artificial spin

ice [42, 58, 152, 153].

5.3 Network construction and properties

In this Section, we describe the methods we use to construct the artificial spin ice phase

space networks, and some of the basic properties those networks have. Much of what we

do can be readily understood in terms of the simple networks we constructed in Section 5.1

for the two Ising spins, so most of this Section will be structured around a comparison

between those networks and the artificial spin ice networks.

In order to compare different systems, we construct networks for each of them. For

example, when we study disorder effects, we construct a separate network for each disorder

realisation. However, in all cases, the network nodes are the same – as already described,

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they represent spin configurations, which form a discrete state space. The number of

configurations is exponential in the system size, and this dramatically limits the systems

we can study directly using network techniques. We construct networks for a 4× 4 square

spin ice array with open edges, whose 16 Ising spins can take 216 = 65, 536 configurations.

Increasing the system size to even 5 × 5 increases the number of configurations – or

equivalently, network nodes – to approximately 33, 000, 000.

In Appendix D.1, we show simulation results for a 4 × 4 array, and these show that

this small system that is feasible to analyse exhibits dynamics that are sufficiently similar

to those of larger arrays that the results presented here can be applied more broadly. Of

course, the smaller system does have differences with the larger array. For example, we

have seen already that 20× 20 arrays can exhibit domains of ground state ordering (see,

for example, Section 3.3), but such features are not supported by the 4 × 4 arrays. One

direction for future work is to study larger systems. For example, we note that simulations

of dynamics of larger arrays are effectively a sampling of their phase space networks, but

we have not yet rigorously interpreted or applied this observation.

The second ingredient of a network is its set of links. In the simple networks con-

structed in Section 5.1, links represented the possible evolutions of the system under a

single application of an external field, which had fixed amplitude and could take one of

two directions (up or down). So, for example, the link from the down-down state to the

up-up state represented two spin flips which occur as a cascade. There are two key points

here. The first is that links represent cascades of flips, rather than, say, single spin flips.

The second is that the links of the network depend on the applied fields used to construct

the network. To illustrate this, Figure 5.2 shows the network for the ideal two-spin system

that has been constructed with only up-pointing external fields.

We choose to study networks in which links represent cascades of spin flips because

networks defined in this way provide a description of dynamics with a useful level of detail.

If the network links represented single spin flips (as allowed under the criterion (1.5)), the

networks would be relatively dense, since most spin configurations have several spins that

can flip. Alternatively, network links might represent the system’s evolution under a field

protocol, so that links would point from an initial configuration to the final configurations

it might evolve to after, say, a full cycle of a rotating external field. However, such networks

would provide relatively little information about dynamics. We find that the definition we

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Figure 5.2: Network for two ferromagnetically-coupled Ising spins, if the exter-nal field can only point up. The two spins are identical with switching thresholds ofhc = 10 and a ferromagnetic coupling of strength j = 1. The external field has strengthh = 11.5.

use, of links representing cascades of spin flips, gives rise to networks that are feasible to

analyse and which are a rich source of information about dynamics. In fact, the networks

in which links represent cascades of spin flips can be derived from single spin flip networks,

and we take advantage of this in our network construction algorithm, which is described

at the end of this Section.

We emphasise that because links are defined in terms of dynamics, they are directed.

In other words, the existence of a link i→ f does not necessarily imply the existence of a

link f → i. The reason for this is that dynamical transitions are not, in general, reversible:

the athermal field-driven dynamics must involve transitions that lower the sum of dipolar

and Zeeman energies, due to the nature of the energy barriers in the system as described

by the switching criterion (1.5). In the literature, it is quite common for systems whose

links are naturally directed to be approximated by undirected networks. For example, in

the studies of dynamical systems by Borges et al. [135], which we describe more fully in

Section 5.2, nodes represent regions of phase space and links represent the trajectory of

a dynamical map, and are therefore naturally directed. However, the authors disregard

the directions of links and are concerned only about the existence of a link between two

regions of phase space. Such an approximation makes for simpler analysis – for example,

undirected graphs have symmetric adjacency matrices. However, in this work we retain

the directed nature of links, because of the importance of the irreversibility of dynamics,

which we have already seen in simulations.

The set of fields used to construct the networks is also important, and indeed, one of

the goals of this Chapter is to understand what networks can tell us about the different

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field protocols studied in previous Chapters. In line with the work presented in previous

Chapters, and in the interests of simplicity, we construct separate networks for each field

amplitude h we study. As we saw in our studies of large ∆θ field protocols (see Sections

3.6 and 4.4), dynamics depend on the set of applied field angles. We generate links for field

angles θ = 0, π/128, 2π/128, . . . We show in Appendix D.2 that this choice of field angles

yields networks with properties that approach the expected limit of a continuously varying

field angle, but that are still computationally tractable. Although we only study networks

for fixed field amplitude h, our work can readily be extended to networks describing

transitions for more than one field amplitude, simply by ‘adding’ the links for a set of field

amplitudes h1, h2, h3, . . .

In effect, the network links are labelled by the range of field angles for which they are

‘active’, θmin < θ < θmax. A field protocol, that is, a sequence of field angles, corresponds

to the paths on the network that are generated by following links that are active for each

angle in the sequence in turn. The existence of a network path from a node i to a node

j indicates that there exists one or more sequences of field angles that can be applied to

a system prepared in state i to drive it into state j. A network constructed for a field

amplitude h contains information about all possible field protocols with fixed amplitude

h. However, in many cases we only wish to know whether a field protocol exists that can

drive the system from one state to another, rather than the details of the protocol. In

that case, the labels on links are unimportant.

When we ignore the range of field angles for which a link is active, we essentially treat

it as unweighted. In other words, if a node has kout links pointing out of it, a random

walk on the network would follow any of those links with probability 1/kout, without

regard for whether that link is active for a wide or narrow range of field angles. An

alternative approach would be to weight the network. Weighted networks, in which links

have a measure such as cost or capacity associated with them, are the more complete

means of describing many real-world systems. For example, in shipping networks, links

can be weighted by the number of ships travelling between two ports [154] or quantity

of cargo transported on a route each year [155]. Alternatively, in social networks links

may be weighted by the time two individuals spend in contact [156]. In the spin ice

dynamics networks, a natural choice of weighting w is based on how often a field with a

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randomly-chosen direction would activate a link:

wi→f (h) =1

2π

∫ 2π

0dθ

ni→f (h, θ)∑j ni→j(h, θ)

, (5.3)

where ni→j(θ) is the number of cascades of spin flips that take the system from configu-

ration i to configuration j, when a field of strength h and angle θ is applied. However,

for simplicity of analysis in this work we neglect these weights and are concerned only

with the existence of links. A study of weighted spin ice dynamics networks is an obvious

future direction this work could take.

Our method for constructing the networks is as follows. In order to enumerate all

network links, we first determine all energetically-allowed single spin flips for each of the

216 configurations, for all fields (h, θ). This is simple to do, using code already written

for the numerical simulations of previous Chapters. The sets of allowed single spin flips

can be stored as transition matrices T (h, θ), where Tij(h, θ) = 1 if configuration i can

be transformed into configuration j by a single spin flip that is allowed under a field

(h, θ) according to criterion (1.5), and Tij(h, θ) = 0 otherwise. Note that T (h, θ) is not

symmetric, because only flips that lower the system’s total (dipolar and Zeeman) energy

are allowed.

We then take advantage of the fact that our network links, which are dynamical cas-

cades, are made of sequences of single spin flips. The non-zero entries of T 2 give permitted

transitions involving two spin flips. Similarly, three spin flips are described by T 3, and

so on. The non-zero entries of T i are not all equal to 1, so we apply the sign function

(sign(x) = 1 if x > 0 and 0 otherwise) to each element of T i to ‘normalise’ the matrix,

and keep it unweighted. Because there are 16 spins in the system and spins can only flip

once in a dynamical cascade, the maximum length of a sequence of spin flips is 16, and

we must have sign(T i) = sign(T 16) for all i > 16. We denote sign(T 16) as A(h, θ), and

Aij(h, θ) = 1 if configuration i can evolve by a cascade of spin flips into configuration

j, under the field (h, θ), and Aij(h, θ) = 0 otherwise. We define the network adjacency

matrix A(h) by A(h) = sign(∑

θ A(h, θ)), where the sum is over all θ between 0 and 2π.

A describes the network representation of all possible dynamics under fields (h, θ), and

it is the properties of A for various field strengths and disorder realisations that will con-

cern us in this work. We emphasise that this method of network construction is an exact

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enumeration over all possible transitions between configuration states allowed for a given

set of external fields.

Finally, a note on disorder realisations. We study disorder effects by implementing

switching field disorder, as described in Chapter 4. However, because of the small number

of islands, the actual mean of the hc values drawn from the Gaussian distribution can

deviate significantly from 11.25, and is 11.25 + ∆. We subtract ∆ from each hc value

to fix the mean at 11.25, because the network properties are quite sensitive to the mean

switching field relative to the applied field, as we will see in Section 5.5. Different disorder

realisations lead to different networks, but as we will see, disorder realisations with similar

switching field standard deviation give rise to networks with similar properties.

5.4 Node degrees and energetics

We now present results from our network analysis of dynamics. In this Section we describe

what the distribution of node degrees shows about the global topology of the network;

however, we will see in subsequent Sections that the degree distributions are insufficient

to completely describe the network topology. We also discuss how a purely local property

of network nodes, namely their degree, can be related to spin ice physics.

The in- (out-) degree of a network node is the number of links pointing into (out of)

it. In an undirected network, the two quantities are the same, but in a directed network

they are different. The degree distribution N (kin(out)) is the distribution of the number of

nodes with degree kin(out). The in degree of node v is given by

kin =∑i 6=v

Aiv, (5.4)

and the out degree is given by

kout =∑j 6=v

Avj , (5.5)

where A is the adjacency matrix, defined in Section 5.3. In this definition, we do not count

self-links.

One reason why node degrees are important is because they can give information about

the global network topology, via the degree distribution N (k). For example, Amaral et

al. [157] study the degree distributions of a variety of real-world networks, such as power

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grids and the neural network of C. elegans, and find three characteristic behaviours: scale-

free, where the degree distribution is a power-law without a sharp cutoff; ‘broad-scale’,

that is, a power-law with a sharp cutoff; and ‘single-scale’, where the degree distribution

has a rapidly decaying tail. Another example of networks with rapidly decaying degree dis-

tributions comes in the form of Erdos-Renyi random graphs [158], for which an undirected

link between any pair of nodes is present with probability p and absent with probability

1− p. Such networks have Poissonian degree distributions.

Directed networks can be described by three distributions: the joint in- and out-degree

distribution, which gives the probability that a randomly selected node has in-degree kin

and out-degree kout, and the two separate degree distributions, which are obtained by

integrating the joint distribution. In Figure 5.3, we plot these three distributions for the

network describing an undisordered spin ice at h = 11, and in Figure 5.4 we plot the

separate in- and out-degree distributions at h = 7.

The first feature to notice is that the shape of the kout distribution depends on field

strength. At h = 11, the out-degree distribution has a peak near kout = 10, but at

h = 7, it decreases monotonically. The reason for this change is clear when one considers

that physically, a node with high out-degree represents a configuration that can ‘decay’

into many others when fields are applied, while a node with low out-degree represents a

‘stable’ configuration. ‘Stability’ here refers to how a configuration can be modified by

an external field, as measured by, for example, the probability that a field with direction

chosen uniformly at random is able to drive the system to a new configuration. A node

with kout = 0 corresponds to a spin configuration that does not respond to an external

field. When the field is weak, few configurations are affected by the field and most nodes

have kout = 0. As the field strength is increased, fewer and fewer configurations are stable

against the field, and N (kout = 0) goes to zero.

The other key feature of the distributions is that they are different to those seen

for other frustrated systems. Han et al. [132–134] show that networks describing the

phase space of other frustrated spin models and lattice gas models have Gaussian degree

distributions, which the distributions shown in Figures 5.3 and 5.4 are clearly not. We

attribute this to a difference in dynamics between our systems and those studied by Han

et al. For example, in their model of dynamics for a triangular antiferromagnet, network

nodes represent single spin flips, and only spin flips that conserve energy are allowed. Such

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a b

c

Figure 5.3: Degree distributions for a perfect system, subject to a field h = 11.(a) The joint in- and out-degree distribution, for kin < 50 and kout < 30. (b) The separateout-degree distribution. (c) The separate in-degree distribution. Disordered systems havesimilar degree distributions.

a b

Figure 5.4: Degree distributions for a perfect system, subject to a field h = 7.(a) The out-degree distribution. (b) The in-degree distribution.

zero-energy flips must preserve the number of satisfied and unsatisfied nearest-neighbour

bonds. Since each spin has six neighbours, only spins with 3 satisfied and 3 unsatisfied

bonds can flip, and the number of such spins in a configuration determines the degree

of its corresponding network node. There are more configurations with an intermediate

number of flippable spins than configurations in which most spins can be flipped (high-k

nodes) or in which few can be flipped (zero-k nodes), so a symmetric degree distribution

is expected. We see below that the situation is more complicated for artificial spin ice

networks.

In order to further interpret the degree distributions of artificial spin ice, we take a more

‘local’ approach and explore how the in- and out-degree of a node relates to its location in

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a b

Figure 5.5: How node degree depends on configuration energy. The (a) out degreeand (b) in degree of 1000 randomly selected network nodes (spin configurations), plottedagainst their dipolar energy. Blue (dark grey) circles represent the perfect system, orange(light grey) squares represent the disordered system with σ = 2.05.

the energy landscape. Figure 5.5 shows the in- and out-degrees of 1000 randomly selected

nodes, vs the energy of the spin configurations they represent, for networks describing a

perfect and a disordered system (with σ = 2.05) subject to a field of amplitude h = 11.5.

For both perfect and disordered systems, high-energy configurations correspond to nodes

with low in-degree and high out-degree, and low-energy configurations correspond to nodes

with high in-degree and low out-degree.

The correlation between energy and node degree has been studied in other cases where

energy landscapes are described by networks, such as studies of atom clusters or spin

glasses. In most cases, configurations or energy basins that have low energy also have

high degree and correspond to network hubs [139–143].1 Doye and Massen [143] offer

an explanation for low-energy basins of atom clusters corresponding to network hubs,

proposing that the energy landscape has a hierarchical structure with smaller basins of

attraction surrounding larger ones.

Unlike studies of undirected networks, we have two k vs energy trends to consider, one

for kout and one for kin. As noted above, a node’s out-degree is related to its stability

against external fields. Although the stability of a configuration is not completely corre-

lated with its energy and the out-degree vs energy data displays some spread, Figure 5.5(a)

shows a clear relationship between energy and out-degree. This can be understood by con-

sidering the extremes of high and low energy: At the field strength studied here (h = 11.5),

1One exception to the rule of high-degree nodes having low energy is seen in the studies of ball-chainmodel proteins by Ravasz et al. [126]. They find that if interactions along the chain are attractive, thelow energy configurations are tightly-packed. Such configurations correspond to nodes with low degree,because the tightly-packed chains have less space in which to move and rearrange, unlike higher-energyopen configurations.

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all spins in a type 4 tiling, in which all nearest-neighbour interactions are maximised, can

be flipped by an external field, so there are many pathways out of that configuration. On

the other hand, the ground state of the system is stable against application of external

fields. Disorder does not change this picture significantly.

The energy dependence of the in-degree has a wider spread, especially when in-degree

is low. This is because the barriers to ‘entering’ a state are topological as well as energetic.

A low in-degree may correspond to a state with high energy or it may correspond to a low

energy state with antiferromagnetic ordering that is ‘hard’ to access with a global external

field, which tends to create ferromagnetic ordering. This idea was discussed in relation to

the accessibility of the ground state of the system in Section 4.8. Nevertheless, it is the

case that only low-energy configurations have high in-degree, as seen in Figure 5.5(b).

To conclude this Section, we comment on how disorder in the artificial spin ice affects

node degrees. The switching field disorder that we study does not affect energetics, but it

does affect network connectivity by changing the barriers between states. As we will see

in Section 5.5, the effect of disorder is typically to add links to the network. This must

affect the degree distributions, since the sum of all kin(out) is simply equal to the number of

links in the network. However, as seen in Figure 5.6, nodes that have low (high) degree in

the perfect system typically also have low (high) degree in a disordered system. In other

words, disorder perturbs the system but does not fundamentally change the stability of

configurations against external fields. Because links are added to the network, the degree

is increased more often than it is decreased.

5.5 Accessibility of states and reversibility of dynamics

Suppose one wished to make a data storage device out of an artificial spin ice system,

by using external magnetic fields to ‘write’ a state by setting the system’s magnetisation

configuration, which could be ‘read’ later. The usefulness of such a scheme depends partly

on the answer to two questions: how many distinct configurational states are available,

and can those states be accessed reliably? In principle a system of N Ising spins has 2N

configurations and can store N bits of information, but the effective information capacity

is lower if only some fraction of those configurations can be realised. In this Section,

we see that network tools are ideal for studying these questions, and for uncovering ef-

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Figure 5.6: Correlation of in- and out-degrees for the perfect and disorderednetworks. The plots are constructed by selecting 2000 configurations at random anddetermining their in- and out-degree in the networks for an ideal system and a disorderedsystem (σ = 2.02). The red line of slope 1 separates the nodes whose degree is increasedby disorder from those whose degree is decreased by disorder.

fects of disorder and field strength on the accessibility of states. In particular, we focus

on the accessibility of states from the polarised configurations, which will allow further

interpretation of our simulation results of previous Chapters.

In line with the rest of the work in this Thesis, we focus on states that can be accessed

using a sequence of fields with fixed amplitude. As mentioned in Section 5.3, such a

sequence of fields corresponds to a ‘walk’ on the network, in which at each step only

a link that is active at that field angle is followed. There may be more than one walk

corresponding to a sequence of fields, since there may be more than one link active for a

given field angle. A random field protocol, in which the field angle is selected uniformly at

random from [0, 2π) at each step of the sequence, is ‘approximately’ a random walk on the

network. ‘Approximately’ here refers to the fact that in simulations, the walk is weighted

according to Equation (5.3), but a random walk on the networks we study is unweighted.

The total set of nodes that can be reached from an initial node v, following network

paths of any length, is given by the fixed point of repeatedly multiplying the unit vector ~v

(all entries zero, except for non-zero entry v) by the adjacency matrix A. Each state can

be accessed via one or more field protocols, but a given field protocol may not be able to

access all of them. We return to this question of ergodicity below.

Figure 5.7 shows how the fraction of configurations that can be reached from the

+x polarised configuration depends on field strength, for both the perfect system and

121

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Figure 5.7: How the number of accessible configurations depends on fieldstrength. The fraction of configurations that can be reached from an initial polarised con-figuration, vs applied field amplitude, for an ideal system and two disorder realisations. Inall cases, the polarised configuration is counted in the number of accessible configurations.

two disorder realisations (σ = 2.05 and σ = 2.22). For weak fields, the +x polarised

configuration is stable against applied field and no other states are accessible from it.

However, for suitable field values, approximately 10% of all configurations can be reached

from the +x polarised configuration. In the very high field limit (not shown), applied

fields always polarise the system, so from the +x polarised configuration there are 3 other

states accessible, namely, the other polarised states.

Although the curves for the perfect and disordered systems in Figure 5.7 have similar

form, for applied field strengths near the mean island switching field (hc = 11.25) the

difference between perfect and disordered systems is large. For example, when h = 11.5,

in the perfect system the subnetwork that contains the +x polarised configuration has

only 5 nodes. When strong disorder (σ = 2.02) is present, the subnetwork contains 1814

nodes, an increase of almost three orders of magnitude. The striking difference between

the two systems is illustrated in Figure 5.8, which shows the two subnetworks.

A more quantitative view is given in Figure 5.9(a), which shows how the number of

states accessible from the +x polarised configuration depends on disorder strength, for

h = 11.5. There is a jump in the number of accessible states when disorder is turned

on. This is because two of the configurations that can be reached from the +x polarised

configuration have spins that, in the perfect system, require an external field of 11.74

to switch. A small disorder-induced decrease in the switching barrier for these spins

allows them to flip at h = 11.5, opening new dynamical pathways. This interpretation is

confirmed by the network for h = 11.75. Its dependence of the number of accessible states

on disorder strength is shown in Figure 5.9(b). For this field strength, the fraction of states

122

a b

Figure 5.8: The size of the subnetwork of states accessible from the +x polarisedconfiguration is dramatically increased by disorder. (a) In a perfect system, thesubnetwork contains 5 nodes. (b) When strong disorder (σ = 2.02) is present, the sub-network contains 1814 nodes. The applied field strength in both cases is h = 11.5. Thelarge purple node in each network represents the +x polarised configuration, and the othernodes are colour-coded so that darker nodes are closer (in terms of network path) to the+x polarised node. Node positions are not significant.

a b

Figure 5.9: How the number of accessible configurations depends on disorder.The fraction of configurations that can be reached from an initial polarised configuration,vs disorder strength, for an applied field of strength h = 11.5 (a) and h = 11.75 (b). Forh = 11.5, there is a jump in the number of accessible states when disorder is turned on, butfor h = 11.75, the number of accessible states increases smoothly. Each point representsa single disorder realisation. In both figures, the initial configuration is counted in thenumber of accessible states.

123

accessible from the +x polarised configuration in the undisordered system is ∼ 10−3, and

increases smoothly with disorder strength. The large impact of small disorder-induced

changes to the system will be discussed further in Section 5.7.

At this point, it is worth commenting on a similarity with ‘order by disorder’ phenom-

ena. As pointed out by Villain et al. [159], a frustrated system in thermal equilibrium can

have one (or a small number) of its degenerate states selected as a ground state by ther-

mal fluctuations. This occurs because different states that are degenerate in energy can

support different low-energy excitations, and the number and quantity of these determine

the size of the free energy basin surrounding each state. The ground state in thermal equi-

librium is determined by these basins, and not simply the energy landscape. For example,

a system of Ising spins connected by ‘springs’ to form a deformable triangular lattice has

a ground state in which frustrated bonds form straight stripes, even though the energy of

zig-zag stripes is equal, because of the differences in vibrational modes between different

configurations [160]. In our athermal nonequilibrium system, the degeneracy of states is

already lifted by the nature of interactions, and disorder does not make the ground state

accessible. However, it does add links between states and allow the system to explore its

energy landscape to find low-energy states that are inaccessible in a perfect system, as we

saw in Chapter 4.

The number of states that can be reached from an initial polarised state is a starting

point for describing dynamics, but this quantity does not give a complete picture. One

question it gives little information about is that of ergodicity – once a transition has been

made from the polarised state to another state, what further transitions can be made?

Is it still possible to access all of the other configurations that are accessible from the

polarised configuration? Can transitions be reversed? This is important for information

storage applications, where it is important that the state of the system can be easily and

reliably ‘re-written’.

A useful network theoretic concept here is that of strongly connected components

(SCCs). An SCC of a directed network is a set of nodes for which paths exist between

every pair of nodes, taking the directions of the links into account. For example, in the

network shown in Figure 5.10(a), the nodes A, B, C form an SCC, but D is not a member

of the component because there is no path from D to the other nodes. We determine the

SCCs of a network using the algorithm in Reference [161], as implemented by the software

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Figure 5.10: Small example networks. (a) An example directed network containing 4nodes. The nodes A, B, C form a strongly connected component, but D is not a memberof the component because there is no path from D to the other nodes. (b) An exampleof how a node may be accessible, but not reproducibly so. The highlighted node can bereached by applying a field of angle θ to node A, or a field of angle α to node B, but bothof these field angles also activate links to other nodes, so it is not possible to construct afield sequence that is guaranteed to pass through the highlighted node.

a b

Figure 5.11: Tuning field strength and/or increasing disorder increases the frac-tion of nodes in large strongly connected components (SCCs). (a) Fraction of allnodes that are in the largest SCC and the SCC that contains the +x polarised configu-ration, as a function of field strength. For the system without disorder, the +x polarisedconfiguration is always in the largest SCC. (b) Fraction of all nodes that are in the largestSCC, vs disorder strength (standard deviation in island switching fields). The appliedfield strength is h = 11.5. Each point represents a single disorder realisation.

package Mathematica.

Dynamics within an SCC are reversible, provided the correct field protocol is applied.

In terms of networks, it is possible to travel along network links from any node of the SCC

to any other node of the SCC; in terms of artificial spin ice dynamics this means that for

any configuration in the SCC, there exists a sequence of fields to drive the system from

that configuration to any other configuration in the SCC, and back again.

The sizes of two SCCs in particular are of interest: the SCC that contains the +x

polarised configuration (or one of the other three polarised configurations: this choice is

arbitrary), and the size of the largest SCC in the network. Figure 5.11(a) shows these

sizes, plotted against field strength, for the system without disorder and the two disordered

systems. For a perfect system, the +x polarised configuration is always in the largest SCC.

Below a threshold field, all SCCs have size 1, that is, no dynamical transitions can

125

be reversed. Above the threshold, the size of the largest SCC grows by three orders

of magnitude to take in approximately ten percent of all nodes, before decreasing again

for large field strengths. In the limit of very strong fields the largest SCC has size 4,

and consists of the four polarised configurations. This limit holds for both perfect and

disordered systems because strong external fields overcome disorder. When disorder is

present, the +x polarised configuration is only in the largest SCC when the external field

is sufficiently strong.

Comparison of Figures 5.7 and 5.11 indicates the strong correlation between the num-

ber of accessible states and SCC size. In general, the number of states accessible from a

node must be greater than or equal to the size of its SCC. For example, in the network

shown in Figure 5.10(a), the number of nodes reachable from A is 4, while it is in an SCC

of size 3. In fact, in an ideal artificial spin ice, for h ≥ 12, all four polarised configurations

are in the largest SCC and for h ≥ 14 all states that can be reached from the +x polarised

configuration are in the same SCC. Similar results hold for the disordered systems.

The existence of large SCCs that include the polarised configurations implies that

for correctly-tuned fields, the information storage capacity of the artificial spin ice is

maximised, with several thousand configurations accessible from one another, making it

possible to ‘write’ a configuration and then ‘rewrite’ a new configuration by applying a

suitable sequence of fields. However, one should be careful for two reasons. First, if the

number of states accessible from a given starting state (e.g, a polarised configuration) is

larger than the SCC size, there will be dynamical ‘dead ends’, that is, states that can be

entered but not exited. Second, the existence of a path into a node does not guarantee that

it is possible to reliably access that configuration. This is illustrated in Figure 5.10(b),

where the highlighted node can be accessed, but it is not possible to construct a field

sequence that is guaranteed to pass through it, because from both ‘precursor nodes’ A

and B the field angle required to activate the link into the highlighted node also activates

a link into another node. A more detailed study of these two points is a topic for future

work.

We close this section by commenting on the effect of disorder. While the general

trends for SCC sizes as a function of field strength are the same for disordered and perfect

systems, for applied fields close to the mean island switching field of 11.25, the difference

in SCC size between perfect and disordered systems is substantial: around two orders of

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magnitude for h = 11. Figure 5.11(b) shows the size of the largest SCC for a range of

disorder realisations, at an applied field strength of h = 11.5. The size of the largest SCC

increases continuously with disorder strength, although, as might be expected, the spread

in values for different realisations of strong disorder is substantial.

5.6 A note on field protocols

At this juncture, it is worth discussing briefly what the network picture can tell us about

field protocols. The network picture provides a natural framework in which to understand

field protocols, as walks on the networks. As already hinted in the previous Section,

disorder in the spin ice opens new paths in the network, but the correct field protocol is

required to follow these paths.

Recall that links are ‘active’ only for certain field angles. A uniform rotating field

protocol, or any other regular sequence of field angles, corresponds to a walk on the

network that at each node can only follow links corresponding to the ‘next’ field angle of

the sequence. In other words, even if a node has high out degree, a regular field protocol

may only be able to follow one of the links. As a result, regular field protocols are highly

restricted walks on the network, as illustrated in the left-hand sketch in Figure 5.12. Even

though links exist to exit the cycle, the links are ‘active’ only for angles outside the regular

sequence and cannot be accessed.

In some cases, these restricted walks are very effective at reaching low energy states.

An example of this is seen in the low field regime of the rotating field protocol acting

on a perfect array with open edges, described in Section 2.4. There is only one path the

dynamics can follow from a polarised configuration, which takes the system to a high n1,

low energy state. However, these dynamics are not robust against disorder, as seen in

Section 4.4. In terms of networks, disorder disrupts the path by removing some of its

links.

By way of contrast, random field protocols act approximately as a random walk on the

network. At each step of the walk, all links out of a node are accessible, because the field

angle is selected randomly. In terms of the dynamics of the underlying spin ice system,

such a random walk is more likely to avoid being trapped by higher-energy local minima,

since if a pathway exists out of an energy basin the random walk will ‘find’ it and follow

127

Figure 5.12: The difference between rotating and random field protocols. Illus-trated on the left is the restricted walk on the network that a rotating protocol performs,with green arrows representing links followed and black arrows representing links that can-not be followed because they are active for field angles ‘out of sequence’ with the rotatingfield. On the right, a random protocol can follow any link, and hence explore more of thenetwork.

it. This is illustrated in the right-hand sketch of Figure 5.12.

Large ∆θ protocols have properties intermediate to uniform rotating and random pro-

tocols. Like uniform rotating protocols, large ∆θ protocols cannot follow every out link

of a node. However, if they enter a cycle in the network, they will eventually be able to

exit, because the sequence of field angles is incommensurate with 2π so the sequence will

eventually give a field angle that corresponds to an exit link. As a result, large ∆θ proto-

cols have behaviour that is closer to random protocols than uniformly rotating protocols,

as seen in Section 3.6.

5.7 Network structure and ‘rewiring’

We saw in Section 5.4 that the energetics of artificial spin ice plays a key role in determin-

ing network properties on a local, i.e: node, level. Similarly, in Section 5.5, we have seen

how the global properties of the network relate to dynamics, which can be observed in

simulation and are experimentally testable. In this Section, we comment on the relation-

ship between the local and global scales of the network, showing that purely local network

properties are insufficient to predict the global structure and that instead, correlations

exist in the network that can be related to the dynamics of artificial spin ice.

We first demonstrate the existence of correlations in the artificial spin ice dynamics

networks. We do this by comparing the spin ice networks with networks that retain some

properties of the spin ice dynamics networks but are otherwise randomised. Comparison

with randomised networks is frequently used to reveal the structure of real-world net-

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works [162–167]. For example, Kitsak et al. [167] compare several real-life social networks,

such as the network of contacts between inpatients in Swedish hospitals, with randomised

networks that are generated from the real-life networks in a way that preserves the degree

of each node, and find that real-life networks contain hubs (nodes with high degree) in

the network periphery, whereas in the randomised networks the hubs are all near the net-

work core, where the core and periphery are defined using a k-shell decomposition [168].

(This feature of the real-life network structure has implications for epidemic spreading,

because the authors also find that the efficiency of the spread of an epidemic from a node

is controlled more by how close to the network core the node is, rather than its degree.)

We study two types of randomised networks. The first type of randomised network

is an Erdos-Renyi random graph [158] generated from the artificial spin ice network by

preserving the total number of links but not the degree distributions. In other words, any

link i → f exists with equal probability, p = n(links)/232 where n(links) is the number

of links2 and 232 is the total number of possible links between 216 nodes. The links are

directed, with random direction. The uncorrelated random networks represent a dynamics

in which the possibility to pass from any state to any other state does not depend on energy

at all. In particular, the relationship between the energy of a configuration and the degree

of its network node is lost.

A more sophisticated approach to constructing random networks is to preserve the

relationship between configuration energy and node degree, by creating ‘maximally ran-

dom’ networks consistent with a given joint in- and out-degree distribution. Under such

a scheme, high-energy, unstable configurations have many links out, and low-energy, sta-

ble configurations have few links out, for example. This gives a network that, at least

locally, represents a dynamics much more closely approximating the actual artificial spin

ice dynamics.

As discussed by Maslov et al. [166], more than one algorithm exists to generate such

a randomised network. The ‘local rewiring algorithm’ used by Maslov et al. strictly

conserves the node degrees, but for our purposes the simpler ‘stub reconnection algorithm’

of Newman et al. [169] is sufficient. In essence, the links of the network are cut, so that

each node has kin ‘in-stubs’ and kout ‘out-stubs’. A random out-stub is then chosen and

2A note on definitions: If nodes A and B are linked in both directions, that is, (A→ B) and (B → A),we count each directed link separately, to give two links. Unless otherwise specified, we do not countself-links, that is links pointing from a node to itself. Under this definition, the number of links is givenby

∑i 6=j Aij .

129

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connected to a random in-stub, to make a directed link. This process is repeated using

the remaining unconnected in- and out-stubs, until the network has been fully re-wired.

This algorithm does not prevent multiple links from being created between a pair of nodes,

but because our networks have relatively few links, this has only a negligible effect on the

resulting rewired networks. We simply replace multiple links from node i to node j by a

single link.

The comparison between these networks and the spin ice dynamics networks is illus-

trated in Figure 5.13, where we plot the fraction of nodes in the largest SCC (strongly

connected component; see Section 5.5) against the number of links in the network. (For

the spin ice dynamics networks, the size of the largest SCC is not a single-valued function

of the number of links. However, as seen in Figure 5.11(a), it is a single-valued function

of field strength, which parameterises the curve, and the ‘doubling back’ of the SCC size

vs the number of links occurs because the number of links has a peak near h = 17.)

The randomised networks have very different global properties to the spin ice dynamics

networks, although, surprisingly, they are quite similar to each other. While all networks

show an increase in the size of the largest SCC with number of network links, the spin

ice dynamics networks have a much reduced tendency towards large SCCs than the ran-

domised networks do. This is because the connections in the spin ice dynamics networks

are not simply dictated by the degree distribution. This in turn is because the dynamics

are not simply dictated by the energies of states but also by the barriers between them.

In the study of growing random networks, the emergence of a giant SCC as the number

130

of (directed) links is increased is treated as a percolation transition [169–174]. A percola-

tion transition on a (regular) lattice corresponds to the emergence of a path that connects

opposite edges of the lattice when links are added between sites; in a similar way, the

emergence of an SCC that contains a finite fraction of network nodes corresponds to that

fraction of nodes being connected so that a random walker can visit all of them. In many

random networks, and the randomised networks studied in this Section, the largest SCC

grows with the number of network links until it contains 100% of nodes. In contrast, the

maximum SCC size in artificial spin ice networks is limited. Studies of a model for highly

clustered random networks [175] suggest that this may be because of clustering. However,

this point warrants further study for two reasons. First, it is not known how the size of the

largest SCC in the artificial spin ice networks scales with the number of network nodes,

because we have only studied networks with 216 nodes. Second, the model used by New-

man [175] for highly clustered random networks assumes that each node belongs to one or

more groups, and nodes can only be connected within groups. This is a plausible model

for, e.g, social networks, where people really do belong to groups such as workplaces or

football clubs, but it is not clear that the same method should apply directly to artificial

spin ice dynamics.

An intriguing feature of Figure 5.13 is that even though the SCCs of artificial spin ice

networks do not grow as rapidly as those of randomised networks, there is still a dramatic

increase in the size of their largest SCCs, caused by a relatively small increase in the

number of network links. For example, at fixed field strength h = 11.5, increasing disorder

strength from σ = 0 to σ = 3.3 has a relatively small impact on the number of links –

which increases from 736, 720 to 1, 060, 814 – but the size of the largest strongly connected

component size grows from 3 nodes (∼ 10−5 of all nodes) to 19% of all nodes.

The fact that a relatively small change in the links of the network can alter the con-

nectedness of nodes so dramatically suggests that for fields close to the mean switching

field of 11.25 the perfect system is ‘almost’ well connected. This notion is supported by the

jump in the number of states accessible from a polarised configuration at h = 11.5 when

disorder is turned on, shown in Figure 5.9(a), which we have already seen is caused by

disorder reducing the switching barriers of spins that in the perfect system have barriers

slightly higher than the applied field.

131

5.8 Conclusion

In this Chapter, we have shown that a network model for the dynamics of artificial spin ice

allows us to quantify how applied field strength and quenched disorder affect the system’s

behaviour. Increasing disorder strength and tuning the applied field increase the number

of states accessible to field-driven dynamics and the reversibility of dynamical transitions,

via a ‘re-wiring’ of the network that involves relatively few links. This suggests that the

highly restricted dynamics of a perfect system subject to a sub-optimal field are caused

by a small number of dynamical pathways being blocked. We have also shown that the

degree of a network node, a local property of the network, can also be related to the

physics of the system, via the relationship between the degree of a node and the energy of

the configuration it represents.

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Chapter 6

Summary and outlook

6.1 Summary of this Thesis

The overarching theme of this Thesis was an investigation of how the dynamics of artificial

spin ice are affected by driving field protocols and differences between islands, which may

be caused by finite array sizes or quenched disorder. Two frameworks were used to describe

dynamics: interacting vertices, and phase space networks. Here we summarise the main

findings of this work.

In Chapter 2, we saw that the treatment of array vertices as interacting objects aids

the interpretation of spin dynamics. By considering which vertex processes are allowed for

different external driving amplitudes, we were able to understand the origin of field regimes

for rotating field protocols acting on open edge arrays: array edges act as nucleation sites

for dynamics when the initial configuration is polarised, and the strength of the external

field required to induce dynamics at array edges, relative to the fields required for other

vertex processes, determine the subsequent dynamics in the bulk.

The picture of interacting vertices was quantified by modelling the dynamics in terms of

vertex population dynamics in a ‘mean field’ approximation, that is, under the assumption

that vertices are well-mixed and the spatial dependence of their population densities is

unimportant. A comparison of the results of this analytical approach with simulation

results showed that these assumptions are reasonable for certain field strengths, but when

the external field is weak, vertices are not well mixed, and when the external field is strong,

the dominant process is re-polarisation, which cannot be described in a model that does

not account for spatial dependences of vertex populations. We also saw that simulations

133

of an array with random initial configuration – for which vertex mixing and spatially-

independent populations are valid assumptions – agreed well with results calculated from

vertex population dynamics.

In Chapter 3, we extended the results of Chapter 2 by comparing dynamics for

arrays with three different edge geometries: open edges, closed edges and 4-island edges.

We found that not only array edge geometry but also the sequence of applied fields control

vertex processes in an ideal system. In certain cases, such as an open edge array subject to

a rotating field with amplitude in the low field regime, edges can select particular vertex

processes that drive the system in an orderly way to a low-energy, high n1 configuration.

However, we saw that another effective means to generate low-energy states is via random

or large ∆θ field protocols, which have the goal of randomising the configuration and

driving as many vertex processes as possible. Because these protocols do not depend on an

exact sequence of spin flips occurring, we proposed – and in subsequent Chapters, proved

– that their ability to generate low-energy states is more robust than that of rotating and

single-direction field protocols.

Building on these results for ideal systems, in Chapter 4 we studied the effects of

quenched disorder. We showed that in our model for spin interactions and switching,

different types of disorder have similar effects on dynamics and the most important char-

acteristic of disorder is its strength relative to interactions, not its origin. We found that

while weak disorder acts to perturb the system from its ideal dynamics only slightly, strong

disorder allows new vertex processes. This strong disorder regime was the focus of the rest

of the Chapter, since it was found that experimental systems exhibit dynamics consistent

with having strong disorder.

We showed that strong disorder is able to break the symmetry between clockwise and

anticlockwise rotating fields, and that this asymmetry could be exploited in the design

of field protocols intended to drive the system to low-energy states. However, although

disorder opens connections to new states, we showed that strong disorder is also able to

block dynamical pathways from polarised states to the two-fold degenerate ground state,

for any constant-amplitude field protocol. Finally, we also commented on the implications

of disorder for vertex population dynamics, and showed that although disorder promotes

vertex mixing, it does not prevent re-polarisation, and therefore vertex population dynam-

ics are subject to similar shortcomings when applied to disordered systems as they are for

134

ideal systems.

Finally, in Chapter 5, we presented an alternative to the vertex picture of dynamics,

which was obtained by mapping the system’s dynamics onto a directed network. In this

picture, network nodes represent spin configurations, and links represent allowed evolutions

via cascades of single spin flips. This allowed us to quantitatively describe how the external

field and quenched disorder affect the connectedness of states and the reversibility of

dynamics. In particular, we showed that for optimal field strengths, a substantial fraction

of all states can be accessed using external driving fields, and this fraction is increased

by disorder. However, to access these states, an appropriate field protocol is required,

and the network picture provides a natural framework to discuss this. We were able to

intuitively understand the ability of random and large ∆θ field protocols to generate low-

energy states in terms of the pathways available on the network. We also showed how the

degree of a network node is related to the energy of the configuration it represents. We

thereby demonstrated how correlations in the network result from properties of artificial

spin ice dynamics.

6.2 Future directions

The scope of the work presented in this Thesis is necessarily limited, and a number of obvi-

ous extensions present themselves. We describe here three directions in which preliminary

work has been performed.

Alternative field protocols

One avenue of research that promises a rich variety of phenomena is that of field protocols.

In this Thesis, we have focussed exclusively on field protocols in which either the field angle

or direction is held constant. Even within such constraints, there are many interesting

possibilities. One example is that of a ‘random reversal’ field protocol, in which the field is

rotated anticlockwise from θ0 to θ1, then clockwise from θ1 to θ2, then again anticlockwise

from θ2 to θ3, and so on, with the θi selected uniformly at random from the interval [0, 2π).

Such a protocol is ‘intermediate’ to the field protocol studied in Section 4.7 in which the

field switches sense of rotation every 4π radians and the random θ protocols where the

field is simply applied at random angles and not rotated. In terms of our complex network

135

picture of field protocols (see Section 5.6), the random reversal protocol should be better

able to escape closed loops of states than a uniformly rotating protocol is, because it can

access links that are not active for field angles in the regular sequence.

We have performed some preliminary simulations of such a protocol on a 20× 20 open

edge array with strong switching field disorder (σ = 1.875), and find that after 50, 000

field steps (with step size dθ = 0.01 radians), the type 1 populations attained are at least

as high as those obtained from a field protocol in which the sense of rotation changes

every 4π radians, although never higher than those of the random θ protocol. An obvious

topic for enquiry is whether the random reversal field protocol can be optimised to be

significantly better at generating low-energy states than a protocol with regular reversals,

like the random θ protocols are.

A secondary motivation for the study of the random reversal field protocol is that it is

a more ‘realistic’ random field protocol than the random θ protocols studied in this Thesis,

which, as mentioned in Section 3.6, suffer from the drawback that to be experimentally

implemented the external field must be ramped up and down for each field application,

and this can have an effect on results, as shown in Appendix A.5. In contrast, the random

reversal protocol can be studied experimentally, simply by rotating a sample in a magnetic

field and switching the sense of rotation at random times. Alternatively, future work could

also involve a more detailed study of field ramp effects.

Of course, another extension to the work of this Thesis could be made by applying

the techniques used here to the study of field protocols in which the field varies both its

amplitude and direction, such as ac demagnetisation. In the field protocol used by Wang

et al. [23, 40, 63], the field is rotated with constant amplitude for ∼ 100 cycles. While

the sample continues to rotate, the field direction is switched in the laboratory frame and

its amplitude is reduced. Note that the sense of rotation of the field in the frame of the

sample does not change. The field direction switching occurs very fast, at a rate of ∼ 104

Oe/s, relative to a sample rotation rate of 1000 rpm. Thus, as a first approximation we

can treat it as an instantaneous phase shift of the rotating field by 180. Our preliminary

simulations of such a field protocol reveal that these phase shifts are important, as shown

in Figure 6.2.

We discussed in previous Chapters two ways in which the work presented in this Thesis

could be extended and applied to understanding ac demagnetisation protocols. First, the

136

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Figure 6.1: Preliminary results for a field protocol in which the sense of rotationchanges at random times, acting on an array with strong disorder. The red circlesrepresent a field protocol in which the field rotates with step dθ = 0.01 radians for 50, 000steps, but reverses sense of rotation at random times, as described in the text. The othercurves shown represent other fixed-amplitude field protocols which have been studied inthis Thesis: the uniform rotating protocol (black triangles; see Section 4.4), a field protocolwith regular reversals of sense of rotation (blue inverted triangles; see Section 4.7) and therandom θ protocol (green squares; see Section 4.4). In all cases, disorder is implementedas a switching field distribution with σ = 1.875.

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Xn1\

Figure 6.2: Preliminary results from simulations of ac demagnetisation in adisordered system. A 20 × 20 open edge array with strong switching field disorder(σ = 1.875) is simulated. The two data sets both represent the time evolution of n1 asthe system is subject to a field that rotates with a fixed amplitude for 10π radians beforestepping down in amplitude by ∆h = 0.1. In the portion of the evolution shown, h isbetween 12.8 and 12.5. The red triangles represent a field protocol where the field stepsdown but does not change direction; the blue circles represent a field protocol where thefield switches direction by 180 when it steps down in amplitude, the effects of which canbe seen in the spikes in n1.

137

Figure 6.3: MFM image of an as-grown square artificial spin ice, showing ex-tensive ground state ordering. Red and blue colouring indicate magnetic poles, anddomain walls and small excitations are clearly visible on the ground state background. Theimage has been false-coloured using the software package WSxM [176]. Image courtesy ofJason Morgan.

argument presented in Section 4.8 about disorder blocking pathways from polarised states

to the ground state was valid only for constant-amplitude field protocols, and it would

be interesting to either extend it to cover field protocols with varying field amplitude

or to prove that such an extension is not possible. Second, in Chapter 5 we studied

networks representing the dynamics possible for fixed field amplitudes, but as mentioned

in Section 5.3 it is straightforward to also construct networks for varying-amplitude field

protocols. Doing this should give a more fundamental insight into the nature of dynamics

that occur during ac demagnetisation.

Response of the ground state to external magnetic fields

Although the polarised configurations are the only configurations known to be exactly

reproducible in experiments, dynamics starting from other ordered configurations may still

be amenable to experimental study. For example, Morgan et al. [42] have shown that as-

grown samples of square artificial spin ice can display extensive ground state ordering. An

example magnetic force microscopy image from those experiments is shown in Figure 6.3.

The image has been coloured so that the domains of ground state ordering, which have no

magnetic charge on a vertex level, form a relatively uniform background on which domain

walls and small excitations above the ground state (that is, clusters of typically up to 8

spins which are aligned against the local ground state order) are clearly visible.

Although the configurations cannot be exactly reproduced, the large areas of type 1

138

H H

a) b) c)

Figure 6.4: How a ground state configuration responds to strong external fields.a) The ground state configuration. b) When a field is applied parallel to one of theisland axes, those spins align with the field, generating an ordered type 3 vertex tiling. c)Applying the field at 45 to the island axes generates a polarised type 2 tiling. Flippedspins are indicated by dotted arrows.

ordering provide a background on which spin flip dynamics can occur, just as polarised

configurations provide a uniform background for type 3 vertex propagation. The response

of a type 1 tiling to a strong external field is relatively straightforward to predict, as illus-

trated in Figure 6.4. If the field is applied at 45 to the spin axes, a polarised configuration

is obtained. Alternatively, if only one sublattice of spins is aligned to an external field

then the final configuration is an ordered tiling of type 3 vertices. Such a configuration

would provide yet another possible background for other dynamics. It might also allow

connections to be made with kagome ices, where ‘charge-ordered’ states can readily be

generated and studied [45, 64, 177].

The dynamics at intermediate field strengths are also of interest. For example, the

dynamics when the applied field is off-axis should involve avalanches as zig-zag lines of

spins flip to generate a polarised state, as seen in Figure 6.4(c). Indeed, such dynamics in

an ideal open edge array were already discussed in the studies of magnetisation reversal

in Section 3.3. On the other hand, as seen in Figure 6.4(b), if the field is perfectly on-axis

then no nearest- or next-nearest neighbour spins flip, so any avalanches would have to

be driven by longer-range coupling which is typically negligible in strength. Varying the

field direction allows us to tune the amplitude of the external drive on different spins.

The situation is also interesting when domain walls or small excitations are present, as

these may respond themselves to the field. For example, our preliminary results shown in

Figure 6.5 indicate that a domain wall can change its total magnetic moment in response

to the external field.

139

Figure 6.5: How two ground state domains respond to a field applied along oneof the spin axes. The 20 × 20 array is initialised in the vertex configuration shown forh = 9. A field is applied along the direction indicated by the black arrow, ramping upfrom h = 0 in steps of 0.05. Key configurations during the ramp-up are shown. The finalconfiguration retains a domain wall, whose net moment has rotated in order to have apositive projection onto the field direction to minimise Zeeman energy. Strong disorder ispresent in the form of switching field distribution with σ = 1.875. Green circles representtype 1 vertices, blue arrows represent type 2 vertices, and red crosses represent type 3vertices.

Modifying and controlling the behaviour of artificial spin ice

We saw in our network studies of Section 5.7 that small changes to the system can have

large effects on the accessibility of states in artificial spin ice. This result may find ap-

plication in the control of artificial spin ice. Such control has already been studied from

an experimental perspective. For example, in experimental studies of field-driven reversal

in artificial kagome spin ice, islands were deliberately modified in order to serve as ‘start’

and ‘stop’ sites for avalanches of spin flips [45], and simulations of a colloidal model for

artificial spin ice reveal that using different barrier heights for different sublattices leads

to a rich array of stable states that are different to those seen when all barriers are the

same [32].

As an alternative, one might imagine a system where a small number of macrospins are

controlled directly via, for example, current-driving switching. In a network picture, such

modifications of the spin ice system are equivalent to deliberately creating and removing

certain links. As seen already in Section 5.7, such re-wiring can have a dramatic effect on

network connectedness.

The effect of having control over a single spin is illustrated by the networks shown

in Figure 6.6. Network (a) is the network of states accessible from the +x polarised

configuration (the large red node) for a perfect system, at a field strength of h = 11.5.

There are two other configurations accessible via a single field application, and a further

140

a

b

Figure 6.6: The ability to control one spin independently opens new paths inphase space. The network of states accessible from the +x polarised configuration (thelarge red node), at a field strength of h = 11.5, for (a) a perfect system, and (b) a systemwhere the lower-left corner island can be flipped independently of the others.

two accessible if a second field is applied, making a total of 5 nodes. (This is the same

network that was shown in Figure 5.8(a).) In contrast, network (b) is the network of

configurations accessible from the same initial configuration, for the same field strength,

but in a system where the lower-left corner island can be flipped independently of the

others. This network contains 128 nodes. In other words, the ability to separately control

a single spin yields a twenty-fold increase in the number of states accessible from the +x

polarised configuration.

These preliminary results demonstrate the value of the network picture of artificial

spin ice dynamics for studying these problems. Future work might take advantage of

other network properties. For example, in studies of how epidemics spread on networks,

tools have been developed to determine which nodes are most important in determining the

properties of transport on the network [167]. Since field-induced dynamics are essentially

the same as network transport in this picture, applying these tools to networks describing

spin ices may offer a way to determine how to modify the spin ice to control dynamics as

desired.

141

142

Bibliography

[1] J. D. Bernal and R. H. Fowler, A Theory of Water and Ionic Solution, with ParticularReference to Hydrogen and Hydroxyl Ions, The Journal of Chemical Physics 1, 515(1933).

[2] W. F. Giauque and M. F. Ashley, Molecular Rotation in Ice at 10K. Free Energyof Formation and Entropy of Water, Physical Review 43, 81 (1933).

[3] W. F. Giauque and J. W. Stout, The entropy of water and the third law of ther-modynamics. The heat capacity of ice from 15 to 273K, Journal of the AmericanChemical Society 58, 1144 (1936).

[4] L. Pauling, The Structure and Entropy of Ice and of Other Crystals with SomeRandomness of Atomic Arrangement, Journal of the American Chemical Society57, 2680 (1935).

[5] M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske, and K. W. Godfrey, Ge-ometrical Frustration in the Ferromagnetic Pyrochlore Ho2Ti2O7, Physical ReviewLetters 79, 2554 (1997).

[6] S. T. Bramwell et al., Spin Correlations in Ho2Ti2O7: A Dipolar Spin Ice System,Physical Review Letters 87, 047205 (2001).

[7] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S. Shastry, Zero-pointentropy in ‘spin ice’, Nature 399, 333 (1999).

[8] H. Kadowaki, Y. Ishii, K. Matsuhira, and Y. Hinatsu, Neutron scattering study ofdipolar spin ice Ho2Sn2O7: Frustrated pyrochlore magnet, Physical Review B 65,144421 (2002).

[9] R. Siddharthan et al., Ising Pyrochlore Magnets: Low-Temperature Properties, “IceRules” and Beyond, Physical Review Letters 83, 1854 (1999).

[10] R. G. Melko, B. C. den Hertog, and M. J. P. Gingras, Long-Range Order at LowTemperatures in Dipolar Spin Ice, Physical Review Letters 87, 067203 (2001).

[11] J. Snyder, J. S. Slusky, R. J. Cava, and P. Schiffer, How ‘spin ice’ freezes, Nature413, 48 (2001).

[12] K. Matsuhira, Y. Hinatsu, and T. Sakakibara, Novel dynamical magnetic propertiesin the spin ice compound Dy2Ti2O7, Journal of Physics: Condensed Matter 13,L737 (2001).

[13] B. C. den Hertog and M. J. P. Gingras, Dipolar Interactions and Origin of Spin Icein Ising Pyrochlore Magnets, Physical Review Letters 84, 3430 (2000).

[14] S. V. Isakov, R. Moessner, and S. L. Sondhi, Why Spin Ice Obeys the Ice Rules,Physical Review Letters 95, 217201 (2005).

143

[15] M. J. P. Gingras, Spin Ice, in Introduction to Frustrated Magnetism: Materials,Experiments, Theory, edited by C. Lacroix, P. Mendels, and M. Frederic, chap. 12,pp. 293–330, Springer, 2011.

[16] C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice,Nature 451, 42 (2008).

[17] L. D. C. Jaubert and P. C. W. Holdsworth, Signature of magnetic monopole andDirac string dynamics in spin ice, Nature Physics 5, 258 (2009).

[18] D. J. P. Morris et al., Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7,Science 326, 411 (2009).

[19] H. Kadowaki et al., Observation of Magnetic Monopoles in Spin Ice, Journal of thePhysical Society of Japan 78, 103706 (2009).

[20] S. T. Bramwell et al., Measurement of the charge and current of magnetic monopolesin spin ice, Nature 461, 956 (2009).

[21] S. R. Dunsiger et al., Spin Ice : Magnetic Excitations without Monopole SignaturesUsing Muon Spin Rotation, Physical Review Letters 107, 207207 (2011).

[22] L. R. Yaraskavitch et al., Spin dynamics in the frozen state of the dipolar spin icematerial Dy2Ti2O7, Physical Review B 85, 020410(R) (2012).

[23] R. F. Wang et al., Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscaleferromagnetic islands, Nature 439, 303 (2006).

[24] E. Saitoh, M. Tanaka, H. Miyajima, and T. Yamaoka, Domain-wall trapping in aferromagnetic nanowire network, Journal of Applied Physics 93, 7444 (2003).

[25] M. Tanaka, E. Saitoh, H. Miyajima, and T. Yamaoka, Magnetization process andmagnetic structure in a NiFe honeycomb nano-network, Journal of Magnetism andMagnetic Materials 282, 22 (2004).

[26] T. Yamaoka et al., Applications of High-Resolution MFM System With Low-MomentProbe in a Vacuum, IEEE Transactions on Magnetics 41, 3733 (2005).

[27] M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and Y. Iye, Domain structuresand magnetic ice-order in NiFe nano-network with honeycomb structure, Journal ofApplied Physics 97, 10J710 (2005).

[28] M. Tanaka, E. Saitoh, H. Miyajima, and T. Yamaoka, Asymmetric field varia-tion of magnetoresistance in Permalloy honeycomb nanonetwork, Journal of AppliedPhysics 99, 08G314 (2006).

[29] M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and Y. Iye, Magnetic interactionsin a ferromagnetic honeycomb nanoscale network, Physical Review B 73, 052411(2006).

[30] A. Libal, C. Reichhardt, and C. J. Olson Reichhardt, Realizing Colloidal ArtificialIce on Arrays of Optical Traps, Physical Review Letters 97, 228302 (2006).

[31] A. Libal, C. Reichhardt, and C. J. Olson Reichhardt, Hysteresis and Return PointMemory in Artificial Spin Ice Systems, Arxiv preprint arXiv:1108.3584 (2011).

[32] C. J. Olson Reichhardt, A. Libal, and C. Reichhardt, Multi-Step Ordering in Kagomeand Square Artificial Spin Ice, New Journal of Physics 14, 025006 (2012).

144

[33] A. Libal, C. J. Olson Reichhardt, and C. Reichhardt, Creating Artificial Ice StatesUsing Vortices in Nanostructured Superconductors, Physical Review Letters 102,237004 (2009).

[34] Z. Zheng, F. Wang, and Y. Han, Glass Transitions in Quasi-Two-Dimensional Sus-pensions of Colloidal Ellipsoids, Physical Review Letters 107, 065702 (2011).

[35] D. Davidovic et al., Correlations and Disorder in Arrays of Magnetically CoupledSuperconducting Rings, Physical Review Letters 76, 815 (1996).

[36] D. Davidovic et al., Magnetic correlations, geometrical frustration, and tunabledisorder in arrays of superconducting rings, Physical Review B 55, 6518 (1997).

[37] S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler, and A. J. Heinrich, Bistability inAtomic-Scale Antiferromagnets, Science 335, 196 (2012).

[38] K. Yamamoto, C. R. Hogg, S. Yamamuro, T. Hirayama, and S. A. Majetich, Dipolarferromagnetic phase transition in Fe3O4 nanoparticle arrays observed by Lorentzmicroscopy and electron holography, Applied Physics Letters 98, 072509 (2011).

[39] J. M. Luttinger and L. Tisza, Theory of Dipole Interaction in Crystals, PhysicalReview 70, 954 (1946).

[40] X. Ke et al., Energy Minimization and ac Demagnetization in a Nanomagnet Array,Physical Review Letters 101, 037205 (2008).

[41] A. Remhof et al., Magnetostatic interactions on a square lattice, Physical ReviewB 77, 134409 (2008).

[42] J. P. Morgan, A. Stein, S. Langridge, and C. H. Marrows, Thermal ground-stateordering and elementary excitations in artificial magnetic square ice, Nature Physics7, 75 (2011).

[43] J. P. Morgan, A. Stein, S. Langridge, and C. H. Marrows, Magnetic reversal of anartificial square ice: dipolar correlation and charge ordering, New Journal of Physics13, 105002 (2011).

[44] E. Mengotti et al., Building blocks of an artificial kagome spin ice: Photoemissionelectron microscopy of arrays of ferromagnetic islands, Physical Review B 78, 144402(2008).

[45] E. Mengotti et al., Real-space observation of emergent magnetic monopoles andassociated Dirac strings in artificial kagome spin ice, Nature Physics 7, 68 (2011).

[46] N. Rougemaille et al., Artificial Kagome Arrays of Nanomagnets: A Frozen DipolarSpin Ice, Physical Review Letters 106, 057209 (2011).

[47] Y. Qi, T. Brintlinger, and J. Cumings, Direct observation of the ice rule in anartificial kagome spin ice, Physical Review B 77, 094418 (2008).

[48] C. Phatak, A. Petford-Long, O. Heinonen, M. Tanase, and M. De Graef, Nanoscalestructure of the magnetic induction at monopole defects in artificial spin-ice lattices,Physical Review B 83, 174431 (2011).

[49] G. Moller and R. Moessner, Artificial Square Ice and Related Dipolar Nanoarrays,Physical Review Letters 96, 237202 (2006).

145

[50] G. Moller and R. Moessner, Magnetic multipole analysis of kagome and artificial icedipolar arrays, Physical Review B 80, 140409(R) (2009).

[51] G.-W. Chern, P. Mellado, and O. Tchernyshyov, Two-Stage Ordering of Spinsin Dipolar Spin Ice on the Kagome Lattice, Physical Review Letters 106, 207202(2011).

[52] S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, Directobservation of magnetic monopole defects in an artificial spin-ice system, NaturePhysics 6, 359 (2010).

[53] S. Ladak, D. Read, T. Tyliszczak, W. R. Branford, and L. F. Cohen, Monopoledefects and magnetic Coulomb blockade, New Journal of Physics 13, 023023 (2011).

[54] S. Ladak, D. E. Read, W. R. Branford, and L. F. Cohen, Direct observation andcontrol of magnetic monopole defects in an artificial spin-ice material, New Journalof Physics 13, 063032 (2011).

[55] R. V. Hugli et al., Emergent magnetic monopoles, disorder, and avalanches inartificial kagome spin ice (invited), Journal of Applied Physics 111, 07E103 (2012).

[56] J. Li et al., Comparing artificial frustrated magnets by tuning the symmetry ofnanoscale permalloy arrays, Physical Review B 81, 92406 (2010).

[57] L. A. S. Mol, A. R. Pereira, and W. A. Moura-Melo, Extending spin ice con-cepts to another geometry: the triangular artificial magnetic ice, Arxiv preprintarXiv:1112.0515 (2011).

[58] Z. Budrikis et al., Domain dynamics and fluctuations in artificial square ice at finitetemperatures, New Journal of Physics 14, 035014 (2012).

[59] C. Nisoli, On thermalization of magnetic nano-arrays at fabrication, New Journalof Physics 14, 035017 (2012).

[60] K. Kohli et al., Magneto-optical Kerr effect studies of square artificial spin ice,Physical Review B 84, 180412(R) (2011).

[61] S. Daunheimer, O. Petrova, O. Tchernyshyov, and J. Cumings, Reducing Disorderin Artificial Kagome Ice, Physical Review Letters 107, 167201 (2011).

[62] Y. Shen et al., Dynamics of artificial spin ice: continuous honeycomb network, Arxivpreprint arXiv:1112.1857 (2011).

[63] R. F. Wang et al., Demagnetization protocols for frustrated interacting nanomagnetarrays, Journal of Applied Physics 101, 09J104 (2007).

[64] A. Schumann, B. Sothmann, P. Szary, and H. Zabel, Charge ordering of magneticdipoles in artificial honeycomb patterns, Applied Physics Letters 97, 022509 (2010).

[65] C. Nisoli et al., Ground State Lost but Degeneracy Found: The Effective Thermo-dynamics of Artificial Spin Ice, Physical Review Letters 98, 217203 (2007).

[66] C. Nisoli et al., Effective Temperature in an Interacting Vertex System: Theory andExperiment on Artificial Spin Ice, Physical Review Letters 105, 047205 (2010).

[67] Z. Budrikis et al., Disorder strength and field-driven ground state domain for-mation in artificial spin ice: experiment, simulation and theory, Arxiv preprintarXiv:1111.6491 (2011).

146

[68] A. Mehta and S. F. Edwards, Statistical mechanics of powder mixtures, Physica A157, 1091 (1989).

[69] H. A. Makse and J. Kurchan, Testing the thermodynamic approach to granularmatter with a numerical model of a decisive experiment, Nature 415, 614 (2002).

[70] C. Song, P. Wang, and H. A. Makse, Experimental measurement of an effectivetemperature for jammed granular materials, Proceedings of the National Academyof Sciences of the United States of America 102, 2299 (2005).

[71] P. E. Lammert et al., Direct entropy determination and application to artificial spinice, Nature Physics 6, 786 (2010).

[72] J. C. Slater, Theory of the Transition in KH2PO4, The Journal of Chemical Physics9, 16 (1941).

[73] E. H. Lieb, Exact Solution of the Problem of the Entropy of Two-Dimensional Ice,Physical Review Letters 18, 692 (1967).

[74] E. H. Lieb, Residual Entropy of Square Ice, Physical Review 162, 162 (1967).

[75] E. Lieb, Exact Solution of the F Model of An Antiferroelectric, Physical ReviewLetters 18, 1046 (1967).

[76] E. Lieb, Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric,Physical Review Letters 19, 108 (1967).

[77] F. Wu, Exactly Soluble Model of the Ferroelectric Phase Transition in Two Dimen-sions, Physical Review Letters 18, 605 (1967).

[78] R. J. Baxter, Exactly solved models in statistical mechanics (Academic Press, Lon-don, 1982).

[79] D. Levis and L. F. Cugliandolo, Out-of-equilibrium dynamics in the bidimensionalspin-ice model, Europhysics Letters 97, 30002 (2012).

[80] M. J. Donahue and D. G. Porter, National Institute of Standards and TechnologyReport No. NISTIR 6376, 1999 (unpublished).

[81] T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, A Systematic Ap-proach to Multiphysics Extensions of Finite-Element-Based Micromagnetic Simula-tions: Nmag, IEEE Transactions on Magnetics 43, 2896 (2007).

[82] W. Scholz et al., Scalable parallel micromagnetic solvers for magnetic nanostruc-tures, Computational Materials Science 28, 366 (2003).

[83] L. D. Landau and L. M. Lifshitz, On the theory of the dispersion of magneticpermeability in ferromagnetic bodies, Physik. Zeits. Sowjetunion 8, 153 (1935).

[84] T. Gilbert, A Phenomenological Theory of Damping in Ferromagnetic Materials,IEEE Transactions on Magnetics 40, 3443 (2004).

[85] A. Aharoni, Micromagnetics: past, present and future, Physica B 306, 1 (2001).

[86] J. Gadbois and J.-G. Zhu, Effect of edge roughness in nano-scale magnetic barswitching, IEEE Transactions on Magnetics 31, 3802 (1995).

[87] Y. Nakatani, A. Thiaville, and J. Miltat, Faster magnetic walls in rough wires,Nature Materials 2, 521 (2003).

147

[88] P. Politi and M. G. Pini, Dipolar interaction between two-dimensional magneticparticles, Physical Review B 66, 214414 (2002).

[89] G. Moller and R. Moessner, Artificial Square Ice and Related Dipolar Nanoarrays,Physical Review Letters 96, 237202 (2006).

[90] E. C. Stoner and E. P. Wohlfarth, A Mechanism of Magnetic Hysteresis in Hetero-geneous Alloys, Philosophical Transactions of the Royal Society of London. SeriesA, Mathematical and Physical Sciences 240, 599 (1948).

[91] C. Tannous and J. Gieraltowski, The Stoner-Wohlfarth model of ferromagnetism,European Journal of Physics 29, 475 (2008).

[92] A. F. Voter, Introduction to the kinetic Monte Carlo method, in Radiation Effectsin Solids, edited by K. E. Sickafus, E. A. Kotomin, and B. P. Uberuaga, chap. 1,pp. 1–23, Springer, 2007.

[93] D. Chumakov et al., Nanosecond time-scale switching of permalloy thin film elementsstudied by wide-field time-resolved Kerr microscopy, Physical Review B 71, 014410(2005).

[94] X. Han et al., Magnetic Instability Regions in Patterned Structures: Influence ofElement Shape on Magnetization Reversal Dynamics, Physical Review Letters 98,147202 (2007).

[95] H. G. Evertz, G. Lana, and M. Marcu, Cluster Algorithm for Vertex Models, PhysicalReview Letters 70, 875 (1993).

[96] G. Barkema and M. Newman, Monte Carlo simulation of ice models, Physical ReviewE 57, 1155 (1998).

[97] H. G. Evertz, The loop algorithm, Advances in Physics 52, 1 (2003).

[98] Y.-J. Kao and R. G. Melko, Short-loop algorithm for quantum Monte Carlo simu-lations, Physical Review E 77, 036708 (2008).

[99] H. Shinaoka and Y. Motome, Loop algorithm for classical Heisenberg models withspin-ice type degeneracy, Physical Review B 82, 134420 (2010).

[100] Y. Wang, H. De Sterck, and R. G. Melko, A generalized Monte Carlo loop algorithmfor frustrated Ising models, Physical Review E 85, 036704 (2012).

[101] P. Mellado, O. Petrova, Y. Shen, and O. Tchernyshyov, Dynamics of MagneticCharges in Artificial Spin Ice, Physical Review Letters 105, 187206 (2010).

[102] L. A. Mol et al., Magnetic monopole and string excitations in two-dimensional spinice, Journal of Applied Physics 106, 063913 (2009).

[103] L. A. S. Mol, W. A. Moura-Melo, and A. R. Pereira, Conditions for free magneticmonopoles in nanoscale square arrays of dipolar spin ice, Physical Review B 82,054434 (2010).

[104] M. C. Kuntz, O. Perkovic, K. A. Dahmen, B. W. Roberts, and J. P. Sethna, Hys-teresis, Avalanches and Noise, Arxiv preprint arXiv:9809122 (1998).

[105] J. G. Deak and R. H. Koch, The effect of edge roughness on magnetization reversalin micron-sized permalloy thin films, Journal of Magnetism and Magnetic Materials213, 25 (2000).

148

[106] J. I. Martın, J. Nogues, K. Liu, J. L. Vicent, and I. K. Schuller, Ordered magneticnanostructures: fabrication and properties, Journal of Magnetism and MagneticMaterials 256, 449 (2003).

[107] M. T. Bryan, D. Atkinson, and R. P. Cowburn, Experimental study of the influenceof edge roughness on magnetization switching in Permalloy nanostructures, AppliedPhysics Letters 85, 3510 (2004).

[108] M. T. Bryan, D. Atkinson, and R. P. Cowburn, Edge roughness and coercivity inmagnetic nanostructures, Journal of Physics: Conference Series 17, 40 (2005).

[109] D. K. Koltsov, R. P. Cowburn, and M. E. Welland, Micromagnetics of ferromagneticequilateral triangular prisms, Journal of Applied Physics 88, 5315 (2000).

[110] S. H. Liou, R. F. Sabiryanov, S. S. Jaswal, J. C. Wu, and Y. D. Yao, Magnetic domainpatterns of rectangular and elliptic arrays of small permalloy elements, Journal ofMagnetism and Magnetic Materials 226-230, 1270 (2001).

[111] C. A. F. Vaz, C. Athanasiou, J. A. C. Bland, and G. Rowlands, Energetics ofmagnetic ring and disk elements: Uniform versus vortex state, Physical Review B73, 054411 (2006).

[112] Y. Imry and S.-K. Ma, Random-field Instability of the Ordered State of ContinuousSymmetry, Physical Review Letters 35, 1399 (1975).

[113] M. J. Alava, P. M. Duxbury, C. F. Moukarzel, and H. Rieger, Exact CombinatorialAlgorithms: Ground States of Disordered Systems, in Phase Transitions and CriticalPhenomena, edited by C. Domb and J. L. Lebowitz, chap. 2, pp. 143–317, Academic,2001.

[114] S. Pollard, Personal correspondence, 2010.

[115] R. P. Cowburn, Probing antiferromagnetic coupling between nanomagnets, PhysicalReview B 65, 092409 (2002).

[116] A. Imre, G. Csaba, G. H. Bernstein, W. Porod, and V. Metlushko, Investigation ofshape-dependent switching of coupled nanomagnets, Superlattices and Microstruc-tures 34, 513 (2003).

[117] T. Thomson, G. Hu, and B. D. Terris, Intrinsic Distribution of Magnetic Anisotropyin Thin Films Probed by Patterned Nanostructures, Physical Review Letters 96,257204 (2006).

[118] V. Basso and A. Magni, Field history analysis of spin configurations in the random-field Ising model, Physica B 343, 275 (2004).

[119] M. J. Alava et al., Ground-state optimization and hysteretic demagnetization: Therandom-field Ising model, Physical Review B 71, 064423 (2005).

[120] R. Albert and A.-L. Barabasi, Statistical mechanics of complex networks, Reviewsof Modern Physics 74, 47 (2002).

[121] M. E. J. Newman, The Structure and Function of Complex Networks, SIAM Review45, 167 (2003).

[122] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex net-works: Structure and dynamics, Physics Reports 424, 175 (2006).

149

[123] S. N. Dorogovtsev, Lectures on Complex Networks (Oxford University Press, Oxford,2010).

[124] A. Scala, L. A. Nunes Amaral, and M. Barthelemy, Small-world networks and theconformation space of a short lattice polymer chain, Europhysics Letters 55, 594(2001).

[125] D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks,Nature 393, 440 (1998).

[126] E. Ravasz, S. Gnanakaran, and Z. Toroczkai, Network structure of protein foldingpathways, Arxiv preprint arXiv:0705.0912 (2007).

[127] F. Rao and A. Caflisch, The protein folding network, Journal of Molecular Biology342, 299 (2004).

[128] Z. Toroczkai and K. E. Bassler, Jamming is limited in scale-free systems, Nature428, 716 (2004).

[129] Z. Toroczkai, B. Kozma, K. E. Bassler, N. W. Hengartner, and G. Korniss, Gradientnetworks, Journal of Physics A: Mathematical and Theoretical 41, 155103 (2008).

[130] L. da F. Costa and G. Travieso, Strength distribution in derivative networks, Inter-national Journal of Modern Physics C 16, 1097 (2005).

[131] F. Li, T. Long, Y. Lu, Q. Ouyang, and C. Tang, The yeast cell-cycle network isrobustly designed, Proceedings of the National Academy of Sciences of the UnitedStates of America 101, 4781 (2004).

[132] Y. Han, Phase-Space Networks of Geometrical Frustrated Systems, Physical ReviewE 80, 051102 (2009).

[133] Y. Han, Phase-space networks of the six-vertex model under different boundaryconditions, Physical Review E 81, 041118 (2010).

[134] Y. Peng, F. Wang, M. Wong, and Y. Han, Self-similarity of phase-space networks offrustrated spin models and lattice gas models, Physical Review E 84, 051105 (2011).

[135] E. P. Borges, D. O. Cajueiro, and R. F. S. Andrade, Mapping dynamical systemsonto complex networks, The European Physical Journal B 58, 469 (2007).

[136] F. Kyriakopoulos and S. Thurner, Directed Network Representation of DiscreteDynamical Maps, in Computational Science ICCS 2007, edited by Y. Shi, G. vanAlbada, J. Dongarra, and P. Sloot, Lecture Notes in Computer Science Vol. 4488,pp. 625–632, Springer 2007.

[137] L. Angelani, G. Parisi, G. Ruocco, and G. Viliani, Connected Network of Minimaas a Model Glass: Long Time Dynamics, Physical Review Letters 81, 4648 (1998).

[138] L. Angelani, G. Parisi, G. Ruocco, and G. Viliani, Potential energy landscape andlong-time dynamics in a simple model glass, Physical Review E 61, 1681 (2000).

[139] Z. Burda, A. Krzywicki, and O. C. Martin, Network of inherent structures in spinglasses: Scaling and scale-free distributions, Physical Review E 76, 051107 (2007).

[140] H. Seyed-allaei, H. Seyed-allaei, and M. R. Ejtehadi, Energy-landscape networks ofspin glasses, Physical Review E 77, 031105 (2008).

150

[141] S. Carmi, S. Havlin, C. Song, K. Wang, and H. A. Makse, Energy-landscape networkapproach to the glass transition, Journal of Physics A: Mathematical and Theoretical42, 105101 (2009).

[142] J. P. K. Doye, Network Topology of a Potential Energy Landscape: A Static Scale-Free Network, Physical Review Letters 88, 238701 (2002).

[143] J. P. K. Doye and C. P. Massen, Characterizing the network topology of the energylandscapes of atomic clusters, The Journal of Chemical Physics 122, 084105 (2005).

[144] F. Noe, D. Krachtus, J. C. Smith, and S. Fischer, Transition Networks for theComprehensive Characterization of Complex Conformational Change in Proteins,Journal of Chemical Theory and Computation 2, 840 (2006).

[145] F. Noe and J. C. Smith, Transition networks: A unifying theme for molecularsimulation and computer science, in Mathematical Modeling of Biological Systems,Volume I, edited by A. Deutsch et al. chap. 11, pp. 121–137, Birkhauser, 2007.

[146] D. Gfeller, P. De Los Rios, A. Caflisch, and F. Rao, Complex network analysisof free-energy landscapes, Proceedings of the National Academy of Sciences of theUnited States of America 104, 1817 (2007).

[147] D. Gfeller, D. Morton de Lachapelle, P. De Los Rios, G. Caldarelli, and F. Rao,Uncovering the topology of configuration space networks, Physical Review E 76,026113 (2007).

[148] G. Bertotti, P. Bortolotti, A. Magni, and V. Basso, Topological and energetic aspectsof the random-field Ising model, Journal of Applied Physics 101, 09D508 (2007).

[149] P. Bortolotti, V. Basso, A. Magni, and G. Bertotti, Oriented graph structure oflocal energy minima in the random-field Ising model, Physica B 403, 398 (2008).

[150] P. Bortolotti, V. Basso, A. Magni, and G. Bertotti, Organization and energy prop-erties of metastable states for the random-field Ising model, Journal of Magnetismand Magnetic Materials 322, 1402 (2010).

[151] A. Baronchelli, A. Barrat, and R. Pastor-Satorras, Glass transition and randomwalks on complex energy landscapes, Physical Review E 80, 020102(R) (2009).

[152] R. C. Silva, F. S. Nascimento, L. A. S. Mol, W. A. Moura-Melo, and A. R. Pereira,New Journal of Physics 14, 015008 (2012).

[153] V. Kapaklis et al., Melting artificial spin ice, New Journal of Physics 14, 035009(2012).

[154] X. Xu, J. Hu, and F. Liu, Empirical analysis of the ship-transport network of China,Chaos 17, 023129 (2007).

[155] Y. Hu and D. Zhu, Empirical analysis of the worldwide maritime transportationnetwork, Physica A 388, 2061 (2009).

[156] J.-P. Onnela et al., Analysis of a large-scale weighted network of one-to-one humancommunication, New Journal of Physics 9, 179 (2007).

[157] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, Classes of small-worldnetworks, Proceedings of the National Academy of Sciences of the United States ofAmerica 97, 11149 (2000).

151

[158] P. Erdos and A. Renyi, On random graphs I, Publ. Math. Debrecen 6, 290 (1959).

[159] J. Villain, R. Bidaux, J.-P. Carton, and R. Conte, Order as an effect of disorder,Journal de Physique 41, 1263 (1980).

[160] Y. Shokef, A. Souslov, and T. Lubensky, Order-by-disorder in the antiferromagneticIsing model on an elastic triangular lattice, Proceedings of the National Academyof Sciences 108, 11804 (2010).

[161] R. Tarjan, Depth-First Search and Linear Graph Algorithms, SIAM Journal onComputing 1, 146 (1972).

[162] R. Milo et al., Network motifs: simple building blocks of complex networks, Science298, 824 (2002).

[163] S. Maslov and K. Sneppen, Specificity and stability in topology of protein networks,Science 296, 910 (2002).

[164] N. Guelzim, S. Bottani, P. Bourgine, and F. Kepes, Topological and causal structureof the yeast transcriptional regulatory network, Nature Genetics 31, 60 (2002).

[165] K. A. Eriksen, I. Simonsen, S. Maslov, and K. Sneppen, Modularity and ExtremeEdges of the Internet, Physical Review Letters 90, 148701 (2003).

[166] S. Maslov, K. Sneppen, and A. Zaliznyak, Detection of topological patterns incomplex networks: correlation profile of the internet, Physica A 333, 529 (2004).

[167] M. Kitsak et al., Identification of influential spreaders in complex networks, NaturePhysics 6, 888 (2010).

[168] S. B. Seidman, Network structure and minimum degree, Social Networks 5, 269(1983).

[169] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrarydegree distributions and their applications, Physical Review E 64, 026118 (2001).

[170] S. Dorogovtsev, J. Mendes, and A. Samukhin, Giant strongly connected componentof directed networks, Physical Review E 64, 025101(R) (2001).

[171] N. Schwartz, R. Cohen, D. ben-Avraham, A.-L. Barabasi, and S. Havlin, Percolationin directed scale-free networks, Physical Review E 66, 015104(R) (2002).

[172] M. Boguna and M. Angeles Serrano, Generalized percolation in random directednetworks, Physical Review E 72, 016106 (2005).

[173] M. Angeles Serrano and P. De Los Rios, Interfaces and the edge percolation map ofrandom directed networks, Physical Review E 76, 056121 (2007).

[174] J. G. Restrepo, E. Ott, and B. R. Hunt, Weighted Percolation on Directed Networks,Physical Review Letters 100, 058701 (2008).

[175] M. E. J. Newman, Properties of highly clustered networks, Physical Review E 68,026121 (2003).

[176] I. Horcas et al., WSXM: a software for scanning probe microscopy and a tool fornanotechnology, Review of Scientific Instruments 78, 013705 (2007).

152

[177] A. Schumann, P. Szary, E. Y. Vedmedenko, and H. Zabel, Magnetic dipole con-figurations in honeycomb lattices: order and disorder, New Journal of Physics 14,035015 (2012).

[178] A. Benassi and S. Zapperi, Barkhausen instabilities from labyrinthine magneticdomains, Physical Review B 84, 214441 (2011).

153

154

Appendix A

Numerical simulation methods

A.1 Code used in simulations

Included here is code for the simulation of a simple rotating protocol.

//-----------------------

// Include these libraries:

//-----------------------

#include <iostream>

#include <fstream>

#include <sstream>

#include <string.h>

#include <stdio.h>

#include <math.h>

#include <vector>

#include <MersenneTwister.h> // random number generator

// see http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html

using namespace std;

//-----------------------

// ’utilities’

//-----------------------

template<class T> struct vec T x, y; ;

// ’position vector’ type, typically a pair of doubles.

155

// This is used for island locations (i, j)

//-----------------------

// Global variables

//-----------------------

int * moments;

double * field_dip_para;

double * field_dip_perp;

vec <double> * locations;

double * axes;

double * Hc_values;

// arrays describing the spins/fields

int HSIZE;

int VSIZE;

int NUMB;

// spin array dimensions. NUMB = HSIZE*VSIZE

double h;

double hx;

double hy;

double theta;

// field parameters

double Hc_mean;

// mean switching threshold for all spins

double PI=3.141592654;

MTRand mt;

// random number generator

156

//-----------------------

// Set the island axes

//-----------------------

void islandAxesDiagonal( double angle_std_dev )

for (int i=0; i<HSIZE; i++)

for (int j=0; j<VSIZE; j++)

int k = i + j * HSIZE ;

double theta = mt.randNorm(0., angle_std_dev);

if ((j % 2) == (i % 2))

axes[k] = PI/4.0 + theta;

else

axes[k] = -(PI/4.0 + theta);

//-----------------------

// Initialise the moments so the array is saturated and has +x net

// polarisation

//-----------------------

void initOpenEdgePlusXPolarised( )

for( int k=0; k<NUMB; k++ )

moments[k] = 1;

//-----------------------

// Set the spin interactions

//-----------------------

void localFields( double pos_std_dev, double energy_std_dev )

157

// set the island locations

for(int j=0;j<VSIZE;j++)

for(int i=0;i<HSIZE;i++)

int k = i + j*HSIZE;

double delta_rx = 0.;

double delta_ry = 0.;

if(std_dev>1e-6)

delta_rx = mt.randNorm (0., pos_std_dev);

delta_ry = mt.randNorm (0., pos_std_dev);

locations[k].x = double(i) + delta_rx;

locations[k].y = double(j) + delta_ry;

//set the island energies

for(int k=0; k<NUMB; k++)

for(int kk=0; kk<NUMB;kk++)

field_dip_para[k + kk*NUMB] = 0.;

if(k != kk)

double rx = locations[kk].x - locations[k].x;

double ry = locations[kk].y - locations[k].y;

// x and y components of r_k, kk

double r_para = rx * cos(axes[k]) + ry * sin(axes[k]);

// component of r_k, kk parallel to spin k

double r_para_2 = rx * cos(axes[kk]) + ry * sin(axes[kk]);

// component of r_k, kk parallel to spin kk

double distance = sqrt( rx*rx + ry*ry );

field_dip_para[k + kk*NUMB] =

h_dip*(- 1./pow(distance, 3) * cos(axes[kk]-axes[k])

+ 3./pow(distance, 5) * r_para * r_para_2);

158

if(kk<=k && energy_std_dev>1e-6)

double energy_disorder = mt.randNorm(1., energy_std_dev);

int k1 = k+kk*NUMB;

field_dip_para[k1] = energy_disorder * field_dip_para[k1];

field_dip_para[k1] = energy_disorder * field_dip_para[k1];

//-----------------------

// Return the total dipolar energy

//-----------------------

double energy_dip( )

double etot=0.;

for ( int k=0; k<NUMB; k++ )

for ( int kk=0; kk <NUMB; kk++ )

etot += - field_dip_para[k + kk*NUMB] * moments[kk] * moments[k];

// energy is -h.m

return etot/2.0; // divide by 2 for double-counting

//-----------------------

// Return a list of spins that are able to flip

//-----------------------

vector<int> flippable()

vector<int> can_flip;

for( int k = 0; k < NUMB; k++ )

if( moments[k] !=0 )

double field_para = (hx*moments[k]*cos(axes[k])

159

+ hy*moments[k]*sin(axes[k])) ;

// contributions to parallel field from the applied field

for(int kk = 0; kk<NUMB; kk++)

field_para += field_dip_para[k + kk*NUMB]*moments[kk]*moments[k];

// sum of dipolar field contributions

if ( field_para <= -Hc_values[k] )

// simple min-h-para flipping

can_flip.push_back(k);

// if we encounter an un-aligned spin, add a reference to it to the

// flippable spins list

return can_flip;

//-----------------------

// Align spins with local field, picking spins randomly

//-----------------------

void spinAlignerRandom( )

vector<int> to_flip = flippable();

while( to_flip.size()>0 )

if( to_flip.size()>1)

// if more than one spin could be flipped, we flip one which is

// chosen randomly from the set of flippable spins

double u = mt.rand();

int i = int( u * double(to_flip.size()) );

moments[ to_flip[i] ] = - moments[ to_flip[i] ];

else if( to_flip.size()==1 )

// if only one spin is flippable, we flip that one

moments[ to_flip[0] ] = - moments[ to_flip[0] ];

to_flip = flippable();

160

//-----------------------

// Output vertex populations and dipolar energy to file

//-----------------------

void output( std::string filename )

fstream OUTPUT;

OUTPUT.open( (filename).c_str(), ios::out | ios::app);

int n_1 = 0;

int n_3 = 0;

int n_4 = 0;

// populations of types 1, 3, 4. n2=1-n1-n3-n4

int less;

int more;

// iterate over all rows

for(int j=0; j<VSIZE-1; j++ )

less = 0;

more = 0;

// on odd-numbered rows (counting from 0), need to offset starting column

// to count the ’in between’ vertices

if( j % 2 == 1 )

less = 1;

more = 1;

//iterate over the columns.

for(int i=more; i<HSIZE-1 - less; i+=2)

int thisrow = i + j * HSIZE;

int nextrow = i + (j+1) * HSIZE;

// site ’thisrow’ is always the lower-left spin in a vertex

161

if(

moments[thisrow] != 0 && moments[thisrow+1] != 0 &&

moments[nextrow] !=0 && moments[nextrow+1] !=0 &&

abs(moments[thisrow] + moments[thisrow+1]) == 2 &&

abs(moments[nextrow] + moments[nextrow+1]) == 2 &&

moments[thisrow] + moments[nextrow] == 0 &&

moments[thisrow+1] + moments[nextrow+1] == 0 )

n_1++;

if(

moments[thisrow] != 0 && moments[thisrow+1] != 0 &&

moments[nextrow] !=0 && moments[nextrow+1] !=0 &&

abs(moments[thisrow] + moments[nextrow]) == 2 &&

abs(moments[thisrow+1] + moments[nextrow+1]) == 2 &&

moments[thisrow] + moments[thisrow+1] == 0 &&

moments[nextrow] + moments[nextrow+1] == 0 )

n_4++;

if(

moments[thisrow] != 0 && moments[thisrow+1] != 0 &&

moments[nextrow] !=0 && moments[nextrow+1] !=0 &&

abs(moments[thisrow] + moments[nextrow]

+ moments[thisrow+1] + moments[nextrow+1]) == 2)

n_3++;

if(OUTPUT.is_open() )

OUTPUT.seekp( 0, ios::end );

162

OUTPUT << h << " " << n_1<< " " << n_3 << " ";

OUTPUT << n_4 << " " << energy_dip() << "\n";

else

cout << "writing output failed!" << endl;

OUTPUT.close();

//-----------------------

// Put it all together in main()

//-----------------------

int main()

// array-related constants:

HSIZE = 20;

VSIZE = HSIZE;

NUMB = HSIZE*VSIZE;

Hc_mean=11.25;

//disorder

double posn_std_dev = 0.;

double angle_std_dev = 0.;

double Hc_std_dev = 0.6;

double energy_std_dev = 0.;

// field protocol related constants:

double t_step = 0.01;

double cycles=10;

h = 10.;

// setup things that don’t change from trial to trial:

locations = new vec<double> [NUMB];

163

moments = new int [NUMB];

field_dip_para = new double [NUMB*NUMB];

axes = new double [NUMB];

Hc_values = new double [NUMB];

islandAxesDiagonal( angle_std_dev );

localFields( posn_std_dev, energy_std_dev );

for (int trial = 0; trial < 10; trial++)

cout << "trial: " << trial << endl;

// set Hc values

for (int k=0; k<NUMB; k++)

double Hc_dev = mt.randNorm(0., Hc_std_dev);

Hc_values[k] = Hc_mean + Hc_dev;

// initialise array moments:

initOpenEdgePlusXPolarised(0);

// simulate dynamics of a rotating field

for(theta=0.; theta<cycles*2.*PI; theta+=t_step)

hx = h*cos(theta);

hy = h*sin(theta);

spinAlignerRandom();

output("output.dat");

// in reality, file name is a list of the parameters used, but this is

// not shown here, to conserve space

164

delete [] locations;

delete [] moments;

delete [] field_dip_para;

delete [] axes;

delete [] Hc_values;

return 0;

A.2 Switching criteria

Throughout this Thesis, we model island magnetisation reversal as occurring when

h‖ < −hc, (A.1)

where h‖ is the component of the field parallel to the island magnetisation and hc > 0.

In other words, switching when the component of the total field acting antiparallel to the

island magnetisation is greater than some threshold hc. An alternative switching criterion

is Stoner-Wohlfarth switching [90, 91]:

|h‖|2/3 + |h⊥|2/3 > h2/3SW (A.2)

and

h‖ < 0 (A.3)

The two switching criteria can be interpolated between, by replacing condition (A.3)

with

h‖ < −hmin (A.4)

where hmin ≥ 0 can be varied between 0 and hc. Figure A.1 shows n1 vs h curves

for rotating fields acting on 20 × 20 spin open edge arrays, without any disorder, and

switching described by (A.2) and (A.4), with hSW = 10 and hmin ranging between 0 and

10. Although the details of the n1 vs h curves are affected by the value of hmin, the general

shape and the existence of two field regimes (see Section 2.4) is not.

165

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Increasing hmin

4 6 8 10h

0.2

0.4

0.6

0.8

n1,final

Figure A.1: How hmin affects array response. Mean final type 1 population for a20 × 20 spin open edge arrays, without any disorder, subject to a rotating applied field.Switching is described by (A.2) and (A.4), with hSW = 10 and hmin = 1 (black), 2 (blue), 3(red), 4 (green), 5 (orange), 6 (brown), 7 (purple), 8 (grey), 9 (pink) and 10 (i.e: switchingof type (A.1)) (light blue).

Of course, the response of a physical island to fields will not be described exactly

by the simple switching criteria described here. An extension of this work would be to

use, e.g, OOMMF [80] to characterise the switching of a realistic island, and use those

characteristics in simulations. However, the results presented here suggest that details

of switching are relatively unimportant. This is likely to be even more true when strong

quenched disorder is present and the thresholds for spin flips have as much variation arising

from stochasticity as from interactions [67].

A.3 Range of interactions

In this Appendix, we comment briefly on how the range of interactions affects dynamics in

ideal systems. The range of interactions is important because if interactions are sufficiently

short range spins in the same local configuration are exactly equivalent. Under long range

interactions, this equivalence becomes only approximate, because the fields acting on spins

from the rest of the lattice depend on their location. This is counter to our simplified

assumption throughout this Thesis that in ideal systems, bulk spins and edge spins form

two classes of spins that are exactly equivalent within the class but inequivalent between

classes.

This position dependence is in fact quite weak and edge effects are far more important.

Long range interactions lead to similar dynamics to short range (nearest and next-nearest

166

Regime 1 Regime 2

Regime 1

Regime 2

Long range Short range

Vertex-only

Figure A.2: Final type 1 populations vs field strength for an open edge arrayunder a rotating field protocol, with long range, short range and vertex-onlyinteractions. There are two regimes with the same n1 for vertex-only interactions, andtheir n1 vs θ dynamics are also shown.

neighbour) interactions. However, the next nearest neighbour ‘transverse’ (T) interactions

that couple islands that are not in the same vertex do play an important role. We show

here that their omission can lead to qualitatively different dynamics.

Figure A.2 shows the final n1 vs h for an ideal open edge array subject to a rotating

field protocol, with three types of point dipole interactions: long range, in which every

spin interacts with every other spin; short-range, in which jN = 3/2, jL = 1/√

2 and jT =

1/(2√

2) but other interactions are neglected; and vertex-only interactions, for which jT is

also set to zero. The system with short range interactions exhibits the same field regimes

as the system with long range interactions, but the system with vertex-only interactions

does not.

The reason for this difference is that the T type interactions that are neglected in

the vertex-only scheme serve to couple diagonal lines of vertices. In the absence of these

couplings, if the process 3© 2© → 2© 3© occurs, it drives type 3 vertices all the way across

the array. If the field is not strong enough for type 3 expulsion to occur, these vertices

move in ‘spiral’ paths around the array, creating type 1 vertices each time they change

propagation direction. These dynamics are very regular, like the low field dynamics, and

167

similarly lead to large final type 1 populations.

At first, it might seem that the failure of vertex-only interactions to approximate the

dynamics of a system with long range interactions poses a problem for the interacting

vertex model for dynamics used in this Thesis. However, the neglect of T interactions is

effectively an assumption that on average their effects cancel. Sometimes this assumption

is indeed unjustified: for example, in a uniform polarised configuration all T-neighbour

pairs of spins are aligned so that the mean T interaction energy is non-zero. However, if

vertices are well mixed, the assumption holds.

A.4 Comparison of periodic and free boundary conditions

In this Section, we compare simulation results for ideal arrays with periodic and free

boundary conditions. Every spin in the polarised configuration of an array with periodic

boundary conditions (or an infinite array) is equivalent in terms of interactions with the

other spins. On the other hand, arrays with free boundary conditions have edge spins

that have fewer neighbours than bulk spins do, so that there are two inequivalent ‘classes’

of spins. Systems with the two types of boundary conditions exhibit can very different

dynamics.

Note that for simplicity in implementing periodic boundary conditions and to avoid

array size effects, the systems studied in this section have only short range interactions,

limited to nearest and next-nearest neighbour spins. In subsequent simulation studies of

arrays with open boundary conditions we use long range interactions, but find that these

give qualitatively the same dynamics as short range interactions do (see Appendix A.3),

justifying our decision to truncate the dipole sums here.

The arrays are initialised in a polarised configuration (as illustrated in Figure 1.7) and

are then subject to rotating applied fields of constant amplitude. For the arrays with

periodic boundary conditions, two field regimes are observed, as shown in Figure A.3,

where the net x component of the magnetisation is plotted against field angle. For field

strengths below a threshold set by the near neighbour interactions and Equation (1.5),

the spins are unable to respond to the external field because the inter-spin coupling and

switching barriers are stronger than the field. Above the threshold, spins are able to

respond. However, all spins with the same alignment relative to the external field respond

168

Figure A.3: Magnetisation states available to an ice with periodic boundaryconditions. Normalised x component of net array magnetisation as a function of fieldangle for a rotating field of strength h = 11.97 (left) and h = 11.98 (right). The thresholdfield for dynamics is approximately h = 11.97. For h = 11.97, no dynamics occur and mx

remains constant. For h = 11.98 the magnetisation tracks the rotating applied field, seenin the oscillation of mx. This transition between two behaviours is what we expect fromthe strength of the field acting on spins in the polarised configuration.

in the same way, since they are equivalent, so the magnetisation tracks the applied field.

There are four configurations available to the system, corresponding to the four possible

polarisation directions, i.e, (mx,my) = (1, 0), (0, 1), (−1, 0), (0,−1).

When the boundary conditions are changed to free boundary conditions, spins at the

array edge interact with fewer neighbours, which reduces the required external field for

flipping. The exact field required depends on the particular edge geometry. As will be

discussed in more detail in Section 3.2, there are three possible edge geometries. For

the time being, we focus on so-called ‘open’ edges and show that they induce non-trivial

dynamics. As will be seen in Chapter 3, the other edge types also induce non-trivial

dynamics that are different to those induced by open edges.

As seen in Figure A.4, for ‘very strong’ and ‘very weak’ fields the same trivial dynamics

are seen as for the periodic array, but for intermediate field strengths other dynamics are

possible. We will discuss the dynamical processes occurring in Section 2.4, but for now

we note that the wider range of mx values taken for the open edge arrays indicates that

a larger number of spin configurations are being accessed by the system.

Finally, we note that the results of this Section depend on the system being ideal, and

that quenched disorder modifies the situation significantly, by making all spins inequiv-

alent to one another and allowing dynamics to start in the array bulk even when free

boundary conditions are used. Similar results are seen by other authors, such as Benassi

and Zapperi [178], who report an unrealistic sudden magnetisation reversal in simulations

of a thin magnetic film with periodic boundary conditions when the quenched disorder

169

Figure A.4: Free boundary conditions allow more magnetisation states to beaccessed. Normalised x component of net magnetisation as a function of field angle forrotating fields of strengths between h = 10 and h = 13. For fields of intermediate strength,non-trivial dynamics are possible, reflected in the range of mx values taken for h = 11 andh = 12. These data are taken from a single simulation run, but repeating the simulationgives similar results.

strength is zero, but not when disorder is present.

A.5 Field pulse ramp rates

As mentioned in Section 3.6, random field protocols can be implemented in experiments

by rotating a sample through a random angle (in zero field), stopping the rotation, and

applying a field pulse. Unless the field pulse is ramped up on a much faster time scale

than the response of the islands’ magnetisations, the finite ramp rate of the field can have

an effect on the magnetisation configuration. This is because as the field ramps up, it

first ‘selects’ spin flips that are allowed at lower field strengths and thereby ‘guides’ the

evolution of the system, whereas if a field of strength h is instantaneously applied, any of

the spin flips that are energetically allowed for field amplitude h can occur and alternative

dynamical pathways are opened.

We illustrate this in Figure A.5 by comparing configurations attained by an array

initially in a +x polarised configuration, subject to fields that have either been instan-

taneously applied with amplitude h, or ramped up from 0 to h in steps dh = 0.05. We

170

Ideal system With switching field disorder

Fiel

d a

t π

Fiel

d a

t π

-0.3

Figure A.5: Whether the field is ramped or applied instantaneously can have astrong effect on the system’s response. The red data points represent fields that haveramped up in steps of dh = 0.05, the blue data points represent fields that have rampedinstantaneously. The fields are applied an angle π and π − 0.3 to the x axis; the initialconfiguration is a +x polarised configuration. Results are shown for both ideal systemsand systems with switching field disorder with σ = 1.875. In all cases, averages are madeover 10 simulation runs and error bars represent one standard deviation.

examine both ideal and strongly disordered systems, and look at two field angles: θ = π

(that is, the −x direction) and an off-axis field θ = π− 0.3. In the ideal system, a slowly-

ramped field always gives different results to an instantaneous field, but when disorder is

present the two protocols are closer – indeed, for the off-axis fields the two protocols give

type 1 populations with overlapping error bars for all field strengths.

We note that in the simulations presented here, the initial configuration is a polarised

configuration, whereas all but the first field application in a random θ protocol are applied

to different configurations. What effect this has on the effects of field ramping is not yet

known.

A question raised by the importance of finite field ramp rates is how small the field step

needs to be to accurately simulate a continuously increasing field. In fact, even relatively

large field steps lead to configurations close to those obtained by fields that ramp up with

small step dh. This can be seen in Figure A.6, where a field applied in the −x direction

to an array initially in the +x polarised configuration gives similar type 1 populations to

171

a) b)

Figure A.6: How the n1 vs h curve depends on the field ramp rate. (a) shows anideal system, (b) shows a system with switching field disorder with σ = 1.875. The fieldis applied at an angle of π to the initial net magnetisation. Blue represents dh = 0.01, redrepresents dh = 0.05 and green represents dh = 1. In all cases, averages are made over 10simulation runs and the shaded regions represent ± 1 standard deviation.

a dh = 0.01 protocol even when dh = 1. As a result, we can be confident that the single-θ

protocols with dh = 0.05, discussed in Sections 3.3, 3.4 and 4.3, lead to similar outcomes

to a continuously rotating field.

172

Appendix B

Vertex dynamics

B.1 All vertex processes

In this Appendix we give a systematic and general discussion of one- and two-vertex pro-

cesses. We calculate the applied fields required for each process under the approximation

that spins only interact within vertices1 (‘vertex-only’ interactions). This allows prediction

of dynamics at different field strengths, as seen in Chapter 2.

In the bulk of an array, every spin is a member of two vertices so single spin flips are

described by vertex pair processes. These are listed in Table B.1. Table B.1 also gives

the energy cost of the spin flip required for each vertex pair process. Comparison of these

energy costs with the external field strength gives an indication of the processes that can

occur, allowing prediction of spin ice dynamics.

Array edges present a more complicated situation. Under the approximation of vertex-

only interactions, edge processes of 4-island edge arrays can be described in terms of

isolated vertex processes. Open edge arrays also have single vertex processes occurring

at their edges, but because of the geometry, the spin that flips has a nearest neighbour

belonging to a separate vertex. This needs to be accounted for. Table B.2 summarises all

the possibilities for edge processes for these geometries and again gives energy costs.

Closed edge arrays have edge vertices containing only three spins. These three-island

vertices can be classified into three types based on energy, see Section 3.2. They can

participate in vertex processes with their neighbouring four-island vertices or with each

other. Table B.3 lists all these possibilities and the associated energies.

1See Appendix A.3 for a discussion of the role of the range of vertex interactions.

173

Process hd Process hd4© 4©→ 3© 3© −6−

√2 3© 4©→ 4© 3© 0

3© 4©→ 1© 3© −6 2© 2©→ 3© 3©√

23© 3©→ 1© 1© −6 +

√2 3© 3©→ 1© 4©

√2

3© 4©→ 2© 3© −3−√

2 1© 3©→ 3© 2© 3−√

23© 3©→ 1© 2© −3 1© 2©→ 3© 3© 32© 4©→ 3© 3© −3 3© 3©→ 2© 4© 32© 3©→ 3© 1© −3 +

√2 2© 3©→ 3© 4© 3 +

√2

1© 4©→ 3© 3© −√

2 1© 1©→ 3© 3© 6−√

23© 3©→ 2© 2© −

√2 1© 3©→ 3© 4© 6

1© 3©→ 3© 1© 0 3© 3©→ 4© 4© 6 +√

22© 3©→ 3© 2© 0

Table B.1: Vertex processes in the bulk. All possible two vertex processes and thestrength of the vertex-only dipolar field projected onto the shared spin. The list is orderedby the dipolar field strength. The required external field for the process is given by|ha,‖| ≥ hc + hd.

Open edges

Process hd Process hd4©→ 3© −3− 1/

√2± 3/2 2©→ 3© 1/

√2± 3/2

3©→ 1© −3 + 1/√

2± 3/2 1©→ 3© 3− 1/√

2± 3/23©→ 2© −1/

√2± 3/2 3©→ 4© 3 + 1/

√2± 3/2

4-island edges

Process hd Process hd4©→ 3© −3− 1/

√2 2©→ 3© 1/

√2

3©→ 1© −3 + 1/√

2 1©→ 3© 3− 1/√

23©→ 2© −1/

√2 3©→ 4© 3 + 1/

√2

Table B.2: Vertex processes open and 4-island edges. The required external fieldfor the process is given by |ha,‖| ≥ hc + hd, where hd is calculated in the vertex-onlyapproximation.

174

Closed edges

Process hd Process hd3e© 4©→ 1e© 3© −6− 1/

√2 1e© 3©→ 3e© 1© 1/

√2

3e© 3©→ 1e© 1© −6 + 1/√

2 2e© 2©→ 2e© 3© 1/√

22e© 4©→ 2e© 3© −3− 1/

√2 3e© 3©→ 1e© 4© 1/

√2

3e© 3©→ 1e© 2© −3− 1/√

2 1e© 3©→ 3e© 2© 3− 1/√

22e© 3©→ 2e© 1© −3 + 1/

√2 2e© 1©→ 2e© 3© 3− 1/

√2

3e© 2©→ 1e© 3© −3 + 1/√

2 1e© 2©→ 3e© 3© 3 + 1/√

21e© 4©→ 3e© 3© −1/

√2 2e© 3©→ 2e© 4© 3 + 1/

√2

3e© 1©→ 1e© 3© −1/√

2 1e© 3©→ 3e© 4© 6 + 1/√

22e© 3©→ 2e© 2© −1/

√2 1e© 1©→ 3e© 3© 6− 1/

√2

3e© 3e©→ 2e© 2e© −3−√

2 2e© 3e©→ 3e© 2e© 02e© 2e©→ 1e© 1e© −3 +

√2 2e© 2e©→ 1e© 3e©

√2

2e© 3e©→ 1e© 2e© −3 1e© 1e©→ 2e© 2e© 3−√

21e© 3e©→ 2e© 2e© −

√2 1e© 2e©→ 2e© 3e© 3

1e© 2e©→ 2e© 1e© 0 2e© 2e©→ 3e© 3e© 3 +√

2

Table B.3: Vertex processes at closed edges. Unlike open and 4-island edges, closededges contain vertices that have only 3 islands, and edge processes may involve both 3and 4 island vertices (upper section of Table) or 3 island vertices only (lower section ofTable). The required external field for the process is given by |ha,‖| ≥ hc + hd, where hdis calculated in the vertex-only approximation.

We note that although many different processes are topologically possible as evidenced

by Tables B.1, B.2 and B.3, only a few of these processes are observed in simulations with

polarised initial configurations. For example, processes involving type 4 vertices rarely

occur because of the large energy costs involved in type 4 creation.

B.2 Vertex population equations from random initial con-

figuration

Here we present the equations describing the vertex population dynamics of an open

edge array initially in a completely random configuration and evolving under a rotating

applied field. When all spins are randomly aligned, all vertex types (1–4) are present with

population fractions determined entirely by their multiplicities, so the initial conditions

are n1(0) = 1/8, n2F (0) = 1/4, n3F (0) = 1/2 and n4(0) = 1/8, with n2T,3T (0) = 0.

When all energetically-allowed processes are included, the population dynamics equa-

175

Process ν

3© 3©→ 1© 1© a(1−N/2)18f(h,−6 +

√2)

3© 3©→ 1© 2© a(1−N/2)14f(h,−3)

2© 3©→ 3© 1© a(1−N/2)12f(h,−3 +

√2)

3© 3©→ 2© 2© a(1−N/2)14f(h,−

√2)

4© 4©→ 3© 3© a(1−N/2)2f(h,−√

2− 6)4© 3©→ 3© 2© a(1−N/2)2f(h,−3)4© 3©→ 3© 1© a(1−N/2)f(h,−6)4© 2©→ 3© 3© a(1−N/2)4f(h,−3)4© 1©→ 3© 3© a(1−N/2)4f(h,−

√2)

2©→ 3© aN 14f(h, 1/

√2− 3/2)

3©→ 1© aN 116

(f(h, 1/

√2− 9/2) + f(h, 1/

√2− 3/2)

)3©→ 2© aN 1

8

(f(h,−1/

√2− 3/2) + f(h,−1/

√2 + 3/2)

)4©→ 3© aN 1

2

(f(h,−1/

√2− 9/2) + f(h,−1/

√2− 3/2)

)Trapping a(1−N )

Table B.4: Rates of vertex processes for Equations (B.1). Rates are calculated usingEquations (2.3), as described in Section 2.5.

tions are

n1 = ν 3© 2©→ 1© 3©n2Fn3F + 2ν 3© 3©→ 1© 1©n23F + ν 3© 3©→ 1© 2©n2

3F + ν 3©→ 1©n3F

+ ν 4© 3©→ 3© 1©n4n3F − ν 4© 1©→ 3© 3©n4n1

(B.1a)

n2F = −ν 3© 2©→ 1© 3©n2Fn3F + ν 3© 3©→ 1© 2©n23F + 2ν 3© 3©→ 2© 2©n2

3F

− ν 2©→ 3©n2F + ν 3©→ 2©n3F − νTn2Fn41

+ ν 4© 3©→ 3© 2©n4n3F − ν 4© 2©→ 3© 3©n4n2F

(B.1b)

n3F = −2(ν 3© 3©→ 1© 1© + ν 3© 3©→ 1© 2© + ν 3© 3©→ 2© 2©

)n2

3F

+ ν 2©→ 3©n2F −(ν 3©→ 2© + ν 3©→ 1©

)n3F − νTn3Fn

41

+ 2ν 4© 4©→ 3© 3©n24 + 2ν 4© 1©→ 3© 3©n4n1 + 2ν 4© 2©→ 3© 3©n4n2F + ν 4©→ 3©n4

(B.1c)

n4 = −ν 4© 3©→ 3© 1©n4n3F − ν 4© 3©→ 3© 2©n4n3F − 2ν 4© 4©→ 3© 3©n24

− ν 4©→ 3©n4 − ν 4© 2©→ 3© 3©n4n2F − ν 4© 1©→ 3© 3©n4n1

(B.1d)

n2T = νTn2Fn41 (B.1e)

n3T = νTn3Fn41, (B.1f)

with rates ν as given in Table B.4. These can be solved numerically using standard

algorithms.

176

Appendix C

Results relating to disorder

C.1 Disordered array with field applied at 90 to initial mag-

netisation

In this Appendix we discuss how a system with quenched disorder responds to a field

at 90 to its initial magnetisation. As was seen in simulations of magnetisation reversal

(Section 4.3), disorder removes the differences between edges and causes dynamics to occur

over a wider range of field strengths, with the magnetisation changing smoothly. This is

seen in the plot of mx vs h in Figure C.1. Again, as seen in Figure C.2, the smoothing and

broadening of the mx vs h curves and the loss of distinction between edges occurs because

type 3 vertices are nucleated in the array bulk as well as at the edges. These propagate

along diagonal lines in the process 3© 2© → 2© 3©, thereby rotating the net magnetisation

of the array to align it with the field.

C.2 Orientational disorder reduces nearest-neighbour cou-

pling

In Figure 4.1, we saw that the mean energy – and not simply the standard deviation in

energy – of type 1 and 4 vertices was affected by orientational disorder. Here we show this

is because orientational disorder reduces the strength of nearest-neighbour coupling.

The energy of a nearest-neighbour pair of islands, with island orientations perturbed

177

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

ÈhÈ

Xmx\

Figure C.1: Normalised net x component of the magnetisation for open edgearrays (orange dot-dash line), closed edge arrays (red dashed line) and 4-islandedge arrays (solid blue line). The field is applied at 90 to the initial net magnetisationand is increased in strength in steps of 0.05 in our reduced units. Averages are made over10 simulation runs, with each run having a different disorder realisation.

Figure C.2: Disorder causes bulk nucleation of type 3 vertices. Vertex configura-tions at h = 12, h = 14 and h = 17 for an open edge array of 20 × 20 spins subject to afield applied at 90 to the original (+x) magnetisation.

φ1

φ2

rm1

m2

Figure C.3: A nearest-neighbour pair of spins, subject to disorder in their ori-entations. The dashed lines represent the spin axes in the undisordered case; the actualspin orientations are perturbed from these by θ1 and θ2.

178

by θ1 and θ2, as defined in Figure C.3, is

EN = D

(~m1 · ~m2

r3− 3

(~m1 · ~r)(~m2 · ~r)r5

)(C.1a)

= D

(cos(

π

2+ θ1 − θ2)− 3 cos(

π

4+ θ1) cos(

π

4− θ2)

), (C.1b)

where D = µ0M2/(4πa3) and in our units m = 1 and r = 1. This expression simplifies to

EN = D

(1

2sin(θ1 − θ2)− 3

2cos(θ1 + θ2)

). (C.2)

Because θ1 and θ2 are both evenly distributed about zero, the first term averages to

zero over many disorder realisations: for every disorder realisation in which sin(θ1− θ2) is

positive, there exists another where it is negative and of equal magnitude.

The second term, however, does affect the magnitude of the energy, even when disorder

realisations are averaged over. When there is no disorder, that is, θ1 = θ2 = 0, it is equal

to −32 . However, for any non-zero value of θ1 or θ2, it is reduced in magnitude, because

cos(x) takes its maximal value of 1 at x = 0 and is < 1 for x 6= 0. Thus, the mean

nearest-neighbour coupling EN is reduced in strength by orientational disorder.

This reduction does not affect the energy of type 2 and 3 vertices, because they have no

net contribution to their energy from nearest-neighbour interactions, since they have equal

numbers of satisfied and unsatisfied nearest-neighbour interactions. However, for type 1

and 4 vertices, which have all nearest-neighbour interactions satisfied and unsatisfied,

respectively, the magnitude of their energy is reduced.

179

180

Appendix D

Methods and results relating to

networks

D.1 Simulation results for a 4× 4 array

The dynamics of a 4 × 4 array are less rich than those of the 20 × 20 array, but key

similarities in disorder effects can be seen. Figure D.1 shows the Type 1 vertex population

for rotating and random field protocols acting on perfect and disordered systems. As

for the 20 × 20 system, and as expected from our networks analysis, disorder (both in

protocol and switching fields) allows non-trivial dynamics to occur over a wider range of

field strengths. Furthermore, when switching field disorder is present, protocol disorder

allows larger Type 1 populations to be attained, as is the case for 20 × 20 arrays. Thus,

even though edge effects are far more prominent in 4× 4 arrays than 20× 20 arrays, the

smaller system is still useful for studying disorder effects.

D.2 Choice of field angle step for networks

In Section 5.3, we stated that the networks are constructed out of possible transitions

for a field h = 11.5 taking angles θ = 0, π/128, 2π/128 . . . 2π − π/128. In Figure D.2 we

plot the number of links in the network against the number of field angle steps π/(∆θ)

for a perfect and a typical disordered network. Finer field angle steps allow more links

to be found. As the angular step is decreased, the number of links approaches a limiting

value. We have chosen to use a step of π/128 as a balance between accuracy and speed of

181

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Figure D.1: Type 1 population vs field strength for rotating and random fieldprotocols. Upright triangles represent a uniform rotating field acting on a perfect sys-tem; reversed triangles represent the same protocol acting on a disordered system; circlesrepresent a random sequence of applied fields acting on a perfect system; squares repre-sent the same field protocol acting on a disordered system. Averages are made over 100disorder realisations, with error bars representing one standard deviation. The large errorbars occur because n1 can only take six possible values.

computation.

In fact, for ideal artificial spin ice, the network properties we study in this work do

not depend on ∆θ for ∆θ < π/16, as seen in Table D.1. On the other hand, for a typical

disordered system, the properties continue to change as ∆θ decreases, but tend towards a

limiting value, as seen in Figure D.2.

∆θ N. links Dsat SCC sizes

π/4 414576 5 1, 65400, 2, 56, 3, 8π/8 461872 5 1, 65400, 2, 56, 3, 8π/16 569892 5 1, 65392, 2, 48, 3, 16π/32 661788 5 1, 65392, 2, 48, 3, 16π/64 707460 5 1, 65392, 2, 48, 3, 16π/128 736720 5 1, 65392, 2, 48, 3, 16

Table D.1: Network properties vs field angle step ∆θ for a perfect system. Dsat

is the number of states reachable from a saturated configuration. The SCC sizes are givenin the form size, count.

182

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Figure D.2: The number of network links as a function of field angle step. Theupper plot is of a perfect system, the lower plot is of a typical disordered system (standarddeviation in switching field σ = 2.0). In both cases, the field strength is h = 11.5, relativeto a mean switching field hc = 11.25.

183

184

List of Figures

1.1 The ice rules for frozen water and three dimensional spin ice. . . . . . . . . 2

1.2 SEM image of artificial spin ice. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The inequivalence of nearest-neighbour and next-nearest-neighbour inter-

actions within a square ice vertex. . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Square, kagome and brickwork artificial spin ice geometries. . . . . . . . . . 7

1.5 The three edge geometries studied in this Thesis. . . . . . . . . . . . . . . . 8

1.6 Types of near-neighbour interactions. . . . . . . . . . . . . . . . . . . . . . . 17

1.7 A polarised configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 The square ice vertex types. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Energy vs type 1 population and magnetisation for random configurations. 26

2.3 Energy vs type 1 population and magnetisation for configurations obtained

by a rotating protocol acting on an initially polarised configuration. . . . . 27

2.4 Single-vertex processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Example vertex pair processes. . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Definitions of x and y directions, and field angle θ. . . . . . . . . . . . . . . 30

2.7 How dynamics start at array edges for an ideal array with open edges subject

to a rotating applied field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Final type 1 populations attained by rotating fields in an open-edge ice with

no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.9 The evolution of vertex populations under a rotating field, for an open-edge

ice with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.10 Vertex dynamics for a rotating field in the low field regime, in an open-edge

ice with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

185

2.11 Vertex dynamics for a rotating field in the crossover and high field regime,

in an open-edge ice with no disorder. . . . . . . . . . . . . . . . . . . . . . . 36

2.12 Vertex population dynamics processes without trapping. . . . . . . . . . . . 39

2.13 Vertex population dynamics processes with trapping. . . . . . . . . . . . . . 40

2.14 How the factor g in the vertex process rates is calculated. . . . . . . . . . . 42

2.15 A single edge vertex does not specify all the nearest neighbours of an edge

spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.16 The evolution of vertex populations according to the population dynamics

equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.17 The evolution of vertex populations according to the population dynamics

equations, with and without trapping of type 3 vertices. . . . . . . . . . . . 46

2.18 The field dependence of steady state type 1 populations, according to the

population dynamics equations. . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.19 Final type 1 population as a function of applied field strength for an open-

edge ice with a random initial configuration, and no disorder. . . . . . . . . 49

3.1 Near-neighbour interactions at the edges of arrays . . . . . . . . . . . . . . 52

3.2 3-island edge vertex types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Projection of the total dipolar field onto each spin of open, closed and 4-

island edge spin ice arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Spin dynamics in the first stage of magnetisation reversal for an open edge

array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Magnetisation vs field strength during reversal in perfect systems with open,

closed and 4-island edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Vertex configurations during magnetisation reversal for an open edge spin

ice, with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.7 Spin configurations during magnetisation reversal for an 4-island edge spin

ice, with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.8 Magnetisation vs field strength during 90 rotation in perfect systems with

open, closed and 4-island edges. . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.9 Vertex configurations during 90 rotation in an open edge array, without

disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

186

3.10 Vertex configurations during 90 rotation in a 4-island edge array, without

disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.11 Final type 1 population as a function of applied field strength for a closed

edge array subject to a rotating field, with no disorder. . . . . . . . . . . . . 64

3.12 Magnetisation configurations for a closed edge array subject to a rotating

field, with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.13 The field-driven evolution of net magnetisation for a 4-island edge array

subject to a rotating field, with no disorder. . . . . . . . . . . . . . . . . . . 66

3.14 Vertex configurations of a 4-island edge array subject to a rotating field of

strength h = 11.75, with no disorder. . . . . . . . . . . . . . . . . . . . . . . 67

3.15 Final dipolar energy vs field strength for a 4-island edge array, subject to

a rotating applied field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.16 Random θ protocol field regimes for an open edge array, with no disorder. . 69

3.17 Type 1 domain growth in the array bulk, caused by a random θ protocol. . 71

3.18 Final configurations for arrays subject to large ∆θ protocols depend on the

initial field angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.19 Type 1 populations attained by large ∆θ field protocols for a range of field

strengths and ∆θ values, in an open edge array with no disorder. . . . . . . 74

3.20 ∆θ dependence of type 1 creation for fixed field strength, in an open edge

array with no disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 The mean energy plus/minus one standard deviation of each type of vertex,

subject to positional, orientational or pairwise interaction disorder. . . . . . 80

4.2 The mean plus/minus one standard deviation of the external field required

for the processes 2©→ 3© and 3©→ 2©. . . . . . . . . . . . . . . . . . . . . 81

4.3 Final type 1 populations vs field strength for systems in the weak disorder

regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Final type 1 populations vs field strength for systems in the strong disorder

regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Scaled n1(h = 10.5) vs disorder strength, for positional and switching field

disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

187

4.6 Magnetisation vs field strength during reversal in strongly disordered sys-

tems with open, closed and 4-island edges. . . . . . . . . . . . . . . . . . . . 85

4.7 Spin configurations during magnetisation reversal for an open edge spin ice,

with strong switching field disorder. . . . . . . . . . . . . . . . . . . . . . . 85

4.8 Final type 1 populations attained by rotating fields in ices with strong

switching field disorder, with open, closed and 4-island edges. . . . . . . . . 88

4.9 Vertex populations for a fixed-amplitude rotating field: experiment and

simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.10 The field dependence of steady state type 1 populations, according to the

population dynamics equations, when disorder is present. . . . . . . . . . . 91

4.11 Type 1 populations attained by random field protocols in an open edge

system with strong switching field disorder. . . . . . . . . . . . . . . . . . . 92

4.12 Type 1 populations attained by large ∆θ protocols in an open edge system

with strong switching field disorder. . . . . . . . . . . . . . . . . . . . . . . 92

4.13 Disorder in switching fields can allow type 1 vertex creation for one sense

of rotation and block it for the other. . . . . . . . . . . . . . . . . . . . . . . 94

4.14 Type 1 populations attained by field protocols with and without reversals,

in ices with and without disorder. . . . . . . . . . . . . . . . . . . . . . . . . 95

4.15 The relationship between polarised and ground state configurations. . . . . 98

4.16 The four groups of spins: A, B, C and D, based on their direction in the GS. 99

4.17 Ground state blocking probability vs applied field strength. . . . . . . . . . 102

4.18 The minimum probability that the ground state is blocked vs array size. . . 102

5.1 A simple network demonstrating the essential features of a network describ-

ing a dynamical system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Network for two ferromagnetically-coupled Ising spins, if the external field

can only point up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3 Joint and separate in- and out-degree distributions for an ideal system at

h = 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4 Separate in- and out-degree distributions for an ideal system at h = 7. . . . 118

5.5 Network node degree vs configuration dipolar energy. . . . . . . . . . . . . . 119

188

5.6 Correlation of in- and out-degrees for the perfect and disordered networks,

with field strength h = 11.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.7 The fraction of nodes (spin configurations) that can be accessed from an

initial polarised configuration, vs field amplitude. . . . . . . . . . . . . . . . 122

5.8 Subnetworks of states accessible from the +x polarised configuration, in a

perfect and disordered system. . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.9 The fraction of nodes (spin configurations) that can be accessed from an

initial polarised configuration, vs disorder strength. . . . . . . . . . . . . . . 123

5.10 Small example networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.11 The fraction of nodes (spin configurations) in the largest strongly connected

component and the strongly connected component that contains the +x

polarised configuration, vs field amplitude, and the fraction of nodes in the

largest strongly connected component vs disorder strength. . . . . . . . . . 125

5.12 Illustration of the difference between rotating and random field protocols

in the network picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.13 The fraction of nodes in the largest strongly connected component (SCC), vs

number of network links for spin ice dynamics networks and ‘test’ networks

with fewer correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.1 Comparison of a field protocol in which a rotating field changes its sense of

rotation at random times with other field protocols, in a disordered system. 137

6.2 Preliminary results from simulations of ac demagnetisation in a disordered

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3 MFM image of an as-grown square artificial spin ice, showing extensive

ground state ordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.4 How a ground state configuration responds to strong external fields at 45

and parallel to an island axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.5 Preliminary results for the response of two ground state domains to a field

applied parallel to a spin axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.6 Networks representing configurations accessible from a +x polarised config-

uration, when all spins are identical and when one spin can be controlled

independently. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

189

A.1 Stoner-Wohlfarth and threshold field switching. . . . . . . . . . . . . . . . . 166

A.2 Final type 1 populations vs field strength for an open edge array under a

rotating field protocol, with long range, short range and vertex-only inter-

actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.3 Magnetisation states available to an ice with periodic boundary conditions. 169

A.4 Magnetisation states available to an ice with free boundary conditions. . . . 170

A.5 n1 vs field h, for fields at θ = π and θ = π − 0.3 to the initial net magneti-

sation, with the field ramping either slowly or instantaneously to amplitude

h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.6 Comparison of n1 vs field h, for fields at an angle θ = π to the initial net

magnetisation, for different field ramp rates. . . . . . . . . . . . . . . . . . . 172

C.1 Magnetisation vs field strength during 90 rotation in strongly disordered

systems with open, closed and 4-island edges. . . . . . . . . . . . . . . . . . 178

C.2 Vertex configurations during 90 rotation in an open edge array, with strong

disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.3 A nearest-neighbour pair of spins, subject to disorder in their orientations. . 178

D.1 Type 1 population vs field strength for rotating and random field protocols

in a 4× 4 spin open-edge array, with and without disorder. . . . . . . . . . 182

D.2 The number of network links as a function of field angle step, for a field of

strength h = 11.5, for perfect and disordered systems. . . . . . . . . . . . . 183

190

List of Tables

2.1 Rates of vertex processes for population dynamics. . . . . . . . . . . . . . . 44

4.1 Combinations of groups of spins (see Fig. 4.16) that can contain loose spins

without blocking the ground state, and the probabilities with which they

occur. In all other cases, the ground state is blocked. . . . . . . . . . . . . . 100

B.1 Vertex pair processes in the bulk. . . . . . . . . . . . . . . . . . . . . . . . . 174

B.2 Vertex processes open and 4-island edges. . . . . . . . . . . . . . . . . . . . 174

B.3 Vertex processes at closed edges. . . . . . . . . . . . . . . . . . . . . . . . . 175

B.4 Rates of vertex processes for population dynamics from a random initial

configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

D.1 Network properties vs field angle step for a perfect system, with field h = 11.5.182

191