Magnetic Monopoles in Spin Ice - University of Oxford · PDF fileMagnetic Monopoles in Spin...

Magnetic Monopoles in Spin Ice

Claudio CastelnovoUniversity of Oxford

Roderich MoessnerMax Planck Institut

Shivaji SondhiPrinceton University

ISIS Seminars, Rutherford Appleton Laboratory, Didcot (UK),September 30, 2008

Nature 451, 42 (2008)

Outline

General context

Introduction, from frustrated magnetism to spin ice

Dipolar interactions and the ice rule

Excitations: monopoles = fractionalised dipoles

Experimental evidence:I: Searching for monopolesII: The magnetic Coulomb liquid

The emergent gauge structure of spin ice

Conclusions

Collective phenomena and complexity

Complementary fundamental questions:

I What are the fundamental building blocksof matter, and how do they interact?

⇒ high energy + particle physics

I Given building blocks and interactions: what is the resultingcollective behaviour?

⇒ many-body physics and complexity

Magnetic monopole search

Monopole passes through ring

⇒ magnetic flux through ring changes

⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ QD is set up

Magnetic monopole search



⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ QD is set up

Cabrera 1982

Outline

General context






Conclusions

Conventional vs frustrated Ising models

I Consider classical Ising spins, pointing eitherup or down: σi = ±1

I Simple exchange (strength J):

H = J∑〈ij〉

σiσj

I J < 0: ferromagnetic – spins align

I J > 0: antiferromagnetic – spins antialign

I . . . but only where possible: ‘frustration’

=⇒ What happens instead?

Frustration leads to (classical) degeneracy

Not all terms in H = J∑

〈ij〉 σiσj can simultaneously be minimised

I But we can rewrite H:

H =J

2

(4∑

i=1

σi

)2

+ const.

which can be minimised

I for a single tetrahedron:∑

i σi = 0

⇒ Ngs = (42) = 6 ground states

Degeneracy is the hallmark of frustration

Zero-point entropy on the pyrochlore lattice

I Pyrochlore lattice = corner-sharingtetrahedra

Hpyro =J

2

∑tet

(∑i∈tet

σi

)2

I Pauling estimate of ground stateentropy S0 = ln Ngs:

Ngs = 2N

(6

16

)N/2

⇒ S0 =N

2ln

3

2

I microstates vs. constraints;N spins, N/2 tetrahedra

Mapping from ice to spin ice

I In ice, water molecules retain their identity

I Hydrogen near oxygen ↔ spin pointing in

150.69.54.33/takagi/matuhirasan/SpinIce.jpg

Pauling entropy in spin ice Anderson 1956; Harris+Bramwell 1997

Ho2Ti2O7 (and Dy2Ti2O7) are pyrochlore Ising magnets

Pauling entropy measured by Ramirez as predicted

The real (dipolar) Hamiltonian of spin ice Siddharthan+Shastry

I The nearest-neighbour model Hnn for spin ice is not correctdetails

I Leading term is dipolar energy (µ0µ2/4πa3 > J):

H = Hnn +µ0

4π

∑ij

~µi · ~µj − 3(~µi · r̂ij)(~µi · r̂ij)r3ij

I Both give same entropy (!!!) Gingras et al.

Wrong model → right answer . . .

WHY???

Outline

General context






Conclusions

The ‘dumbbell’ model (1)

Dipole ≈ pair of opposite charges (µ = qa):

I Sum over dipoles ≈ sum over charges:

H =

2Ndip.∑i ,j=1

v(rij) =

2Ndip.∑i ,j=1

µ0

4π

qiqj

rij

The ‘dumbbell’ model (2)

Choose a = ad , separation between centres of tetrahedra

I v ∝ q2/r is the usual Coulomb interaction (regularised):

v(rij) =

µ04π

qiqj

rijrij 6= 0

±vo(µa )2 = ±

[J3 + 4D

3 (1 +√

23)

]rij = 0,

Origin of the ice rules

Resum tetrahedral charges Qα =∑

i∈α qi :

H ≈∑ij

v(rij) −→∑αβ

V (rαβ) =

{µ04π

QαQβ

rαβα 6= β

12voQ2

α α = β

I Ice configurations (Qα ≡ 0) degenerate ⇒ Pauling entropy!

Outline

General context






Conclusions

Excitations: dipoles or charges?

I Ground-state

I no net charge

I Excited states:

I flipped spin ↔ dipole excitation

I same as two charges?

Fractionalisation in d = 1

Excitations in spin ice: dipolar or charged?

Single spin-flip (dipole µ)

≡

two charged tetrahedra(charges qm = 2µ/ad)

Are charges independent?⇒ Fractionalisation in d = 3?

Deconfined magnetic monopoles

The dumbbell Hamiltonian gives

E (r) = −µ0

4π

q2m

r

I magnetic Coulomb interaction

I deconfined monopoles

I monopoles in H, not B

I charge qm = 2µ/ad =(2µ/µB)(αλC/2πad)qD

≈ qD/8000

Intuitive picture for monopoles

Simplest picture does not work: disconnect monopoles

Next best thing: no string tension between monopoles:

Two monopoles form a dipole:

I connected by tensionless ‘Dirac string’

I Dirac string is observable

⇒ qm ≈ qD/8000 not in conflict with quantisation of e

Outline

General context






Conclusions

Experiment I: Stanford monopole search



⇒ e.m.f. induced in the ring ⇒ countercurrent ∝ qm is set up

I ‘Works’ for both fundamental cosmic and spin ice monopoles

I signal-noise ratio a problem

How do we know if a particle is elementary?

Outline

General context






Conclusions

Experiment II: interacting Coulomb liquid

Monopoles form a two-component Coulomb liquid

I any characteristic collective behaviour?

I interaction strength Γ ∝ (q2m/〈r〉)/T ∼ exp[−cv0/T ]/T

vanishes at both high and low T

I solution: [111] magnetic field acts as chemical potential

⇒ can tune 〈r〉 and T separately details

~B⇑

Liquid-gas transition in spin ice in a [111] field

I Hnn predicts crossover to maximally polarised state

I dipolar H: first-order transition with critical endpointFisher et al.

I observedexperimentallySakakibara+Maeno

I confirmednumerically

Outline

General context






Conclusions

Conventional order and disorder

Gas-crystal (e.g. rock salt):

Paramagnet-ferromagnet (e.g. fridge magnet)

In between: critical points

Anything else???

Is spin ice ordered or not? details

No order as in ferromagnet

I deconfined monopoles (in 3d)

Not disordered like a paramagnet

I ice rules

⇒ ‘conservation law’

Consider magnetic moments ~µi

as (lattice) ‘flux’ vector field

I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A

I Local constraint⇒ ‘emergent gauge structure’

I Bow-tie motif in neutron scattering

I Algebraic (but not critical!) correlations

Is spin ice ordered or not? details

No order as in ferromagnet

I deconfined monopoles (in 3d)

Not disordered like a paramagnet

I ice rules ⇒ ‘conservation law’

Consider magnetic moments ~µi

as (lattice) ‘flux’ vector field

I Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~A

I Local constraint⇒ ‘emergent gauge structure’

I Bow-tie motif in neutron scattering

I Algebraic (but not critical!) correlations

Bow-ties in neutron scattering

proton correlations in waterice Ih Li et al.

spin correlations in kagome iceFennell+Bramwell

Outline

General context






Conclusions

Emergent particles and new order in spin ice

Spin ice is an interesting model system (and material!)

I frustrated magnet with ‘ground-state entropy’

I emergent gauge structure; (dimensional reduction in a field)

Magnetic monopoles as excitations

I fractionalisation / deconfinement in 3d material

I magnetic Coulomb law (felt by external test particle)

I would show up in monopole search

Picture credits

Iceberg:windows.ucar.edu/tour/link=/earth/polar

/images/NOAA iceberg jpg image.html

Levitation:math.ucr.edu/home/baez/physics/General

/Levitation/levitation.html

Field lines:mcatpearls.com/master/img911.png

NaCl:greenfacts.org/images/glossary/crystal-

lattice.jpg

NATUREJOBSNew Year’s resolutions

A magnetic north–south divide in spin ice

POLES APARTPOLES APART

GEOPOLITICS Turf wars on

the ocean bed

ARCTIC CLIMATEWarming with

altitude

CANCER SUPPRESSION

The Down’s syndrome link

GEOPOLITICS Turf wars on

the ocean bed

ARCTIC CLIMATEWarming with

altitude

CANCER SUPPRESSION

The Down’s syndrome link

3 January 2008 | www.nature.com/nature | £10 THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE

451, 1–10

6 3 January 200

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.nature.com/nature

no.7174

9 770028 083095

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[artwork by AlessandroCanossa]

Kagome ice: dimensional reduction in a field

Ising axes are not collinear back

I [111] field pins one sublattice ofspins

I Other sublattices form kagomelattice

I Kagome lattice: two-dimensional

I How many dimensions are there?

~B

⇑

Emergent gauge structure back

I Ground states differ by reversingspins around closed loops, forwhich the average 〈~µ〉 = 0

I Upon coarse-graining: lowaverage 〈~µ〉 preferred⇒ E ∼ (∇× ~A)2 ⇒ artificialmagnetostatics

Ansatz: upon coarse-graining, obtain energy functional of entropicorigin:

Z =

∫D~A exp[Scl], Scl = −K

2

∫(∇× ~A)2

The resulting correlators are transverse and algebraic:

∝ −q2⊥

q2↔

(3 cos2 θ − 1

)r3

Energy scale hierarchy in spin ice materials

(Dy, Ho magnetic moment ∼ 10µB) back

Energy scales:

I crystal field in the local[111] direction ∼ 200 K

I exchange interaction∼ 1− 2 K

I dipolar interaction∼ 2.5 K (at nn distance)

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