Exotic Phases in Geometrically Frustrated Quantum Magnets · 2014-01-08 · My work has also bene...

Exotic Phases in Geometrically Frustrated QuantumMagnets

by

Tyler Dodds

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2013 by Tyler Dodds

Abstract

Exotic Phases in Geometrically Frustrated Quantum Magnets

Tyler Dodds

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

Quantum magnetic materials provide pathways to exotic spin-disordered phases. We

study two broad classes of quantum spin systems and their ground states. The first class

is that of spin-dimer systems, which form valence-bond-solid states. In such systems,

competition between interactions among the dimers can lead to interesting magnetization

behaviour. We explain the magnetization of Ba3Cr2O8 as a Bose-Einstein condensate of

spin-carrying excitations. Furthermore, we investigate possible dimerized and nearby

magnetically ordered states in the Shastry-Sutherland compound (CuCl)LaNb2O7.

The second class of spin systems feature geometric frustration, which may stabilize

spin-liquid states without any order or particular dimerization. We argue the proximity

of the face-centred-cubic double perovskite La2LiMoO6 to such a phase, to explain its

lack of long-range order. We argue for the coexistence of such a state, along with spiral

magnetic order, to explain the anomalous thermodynamic measurements in the spin-

density-wave phase of powder samples of Volborthite, a distorted kagome-lattice spin

system. Finally, we study spin liquid phases that have spin correlations consistent with

those found from inelastic neutron scattering of the disordered kagome-lattice material

Herbertsmithite. We predict electron spin resonance absorption lineshapes associated

with these phases.

ii

Acknowledgements

In my six years in Toronto, I have been fortunate to have received the support of so

many through my endeavours. First, I am thankful to my advisor Professor Yong Baek

Kim for providing guidance and perspective on my journey through academia. I have

benefited greatly from our discussions and his clear, physical motivation of ideas. Second,

I am thankful to the members of my supervisory committee, Professors Hae-Young Kee,

Kenneth Burch, Young-June Kim, and Roger Melko. Their insightful questions have

been important in helping me to consider my work from different points of view.

I have also been very fortunate to have been studying with a group of talented and

friendly graduate students and postdoctoral fellows. This has produced an excellent com-

munal learning environment that has been of great benefit. I have had the opportunity to

collaborate with several postdoctoral fellows, and would like to thank Bohm-Jung Yang,

Ting-Pong Choy, Shunsuke Furukawa, and Subhro Bhattacharjee for their teaching and

advice during our work together. Much of my learning was shaped by discussions with

Jean-Sebastien Bernier, So Takei, Michael Lawler, Daniel Podolsky, Christoph Puetter,

Ganesh Ramachandran, Si Wu, Brandon Ramakko, William Witczak-Krempa, Jean-

Michel Delisle Carter, Matthew Killi, Jeff Rau, Eric Lee, Vijay Venkataraman, Robert

Schaffer, Mingxuan Fu, Kyusung Hwang, Sungbin Lee and Ashley Cook.

The support of my family and friends has been instrumental. I am especially grateful

to my parents Ted and Loretta, my brother Colin, and my constant ally Angie for their

encouragement.

My work has also benefited from many helpful discussions with Seung-Hun Lee, Tsu-

tomu Momoi, Leon Balents, John Greedan, Sung-Sik Lee, and O. Starykh. I thank

Seung-Hun Lee for providing the experimental data on Ba3Cr2O8. Finally, I would like

to thank The American Physical Society c© 2013 for allowing me to include material that

was previously published in Physical Review B.

iii

Contents

1 Introduction 1

1.1 Quantum Spins in Solid State Systems . . . . . . . . . . . . . . . . . . . 2

1.2 Spin-Dimer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Double-Perovskite Spin Systems . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Kagome Lattice Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Slave-Particle Mapping of Quantum Magnetism . . . . . . . . . . . . . . 11

2 Slave-Particle Representation of Spins 13

2.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Slave-Particle: Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Spin Liquid States 22

3.1 Gauge Structure of Slave-Particle Representations . . . . . . . . . . . . . 22

3.2 Projective Symmetry Group . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Translationally-Invariant Z2 Spin Liquid . . . . . . . . . . . . . . . . . . 27

4 Magnetic-Field-Induced Bose-Einstein Condensation of Triplons in Ba3Cr2O8 30

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Triplon Dispersion via Bond-Operator Approach . . . . . . . . . . . . . . 32

4.3 Hartree-Fock Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . 36

4.4 Critical Density Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Ferromagnetically-coupled dimers on the distorted Shastry-Sutherland

lattice 49

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

iv

5.2 Quantum Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Schwinger Boson Mean-Field Theory . . . . . . . . . . . . . . . . . . . . 52

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Interplay Between Spin-Orbit Coupling and Lattice Distortion in Dou-

ble Perovskites 59

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3 Classical Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.4 Sp(N) Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.5 Planar Anisotropy Model Results . . . . . . . . . . . . . . . . . . . . . . 72

6.6 General Anisotropy Model Results . . . . . . . . . . . . . . . . . . . . . . 75

6.7 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Spinon Fermi Surface and Spin Density Wave Order in Coupled Chain

Model for Volborthite 82

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.2 The coupled-chain Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . 85

7.3 Classical Ground state and Schwinger-boson mean field theory . . . . . . 86

7.4 Slave Fermion Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.5 Spinon Spin-Density-Wave Order . . . . . . . . . . . . . . . . . . . . . . 93

7.6 Comparison to Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8 Quantum Spin Liquids in the Absence of Spin Rotation Symmetry 102

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.2 Spin-12

Model on the Kagome Lattice . . . . . . . . . . . . . . . . . . . . 107

8.3 Slave-Fermion Construction of Quantum Spin Liquid States . . . . . . . 109

8.4 Projective Symmetry Group and Mean-Field Ansatze . . . . . . . . . . . 112

8.5 Spin Correlations in U(1) and Z2 Spin Liquids . . . . . . . . . . . . . . . 116

8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9 Conclusion 130

9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.2 The Path Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

v

A Classical and Saddle-Point Solutions of the Schwinger Boson Model of

La2LiMoO6 133

A.1 Classical O(N) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.2 Saddle-Point Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

B Mean-Field Hamiltonians, Fourier Transform, and RPA Theory of Vol-

borthite 136

B.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.2 Explicit Forms of Mean-Field Hamiltonians . . . . . . . . . . . . . . . . . 137

B.3 Generalized RPA Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C U(1) Ansatze, Projective Symmetry Group, Mean-Field and Dynamical

Structure Factors Results for Herbertsmithite 141

C.1 U(1) Spin Liquid Ansatze . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.2 Projective Symmetry Group . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.3 Mean-Field Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

C.4 Dynamical Structure Factor of Singlet+Triplet Ansatze . . . . . . . . . . 148

Bibliography 150

vi

Chapter 1

Introduction

Materials with permanent magnetic moments, called ferromagnets, have been known

since as long as as 1000 BC. [1] Other materials with no permanent moments may still

show magnetic phenomena, such as paramagnets, which are attracted to either pole

of a magnet, or diamagnets, which are repelled. However, despite the great success of

electromagnetic theory, as exemplified by Maxwell’s equations, it is unable to account for

the magnetic properties in materials. As early as 1911, it was known (from the Bohr-van

Leeuwen theorem) that classical systems in equilibrium must have no net magnetization.

[2] Then, in 1922, the Stern-Gerlach experiment showed that magnetic moments of silver

atoms are quantized. With this, the stage was set for magnetism to be understood as a

purely quantum mechanical phenomenon.

Macroscopic magnetism requires two properties of quantum mechanics. One is the

statistics of particles, and the second is spin, intrinsic angular momentum (of “magnitude”

S) to each fundamental particle. A significant portion of modern condensed matter

physics is concerned with magnetism due to spins of electrons and ions in a crystal.

While ferromagnetism, paramagnetism and diamagnetism can all be explained within

the proper quantum framework, systems of interacting spins are also studied for the

possible exotic phases with collective behaviour that they may host. Most of the interest

lies in low-temperature behaviour, where thermal effects are small and entanglement

between particles becomes important.

A great deal of interest in exotic phases of spin systems came from the discovery of

high-temperature superconductivity, [3] one of the most important problems in condensed

matter physics. The precise role and importance of the magnetism in these supercon-

ductors is still undecided. However, many of the theoretical frameworks have taken their

own life in the study of insulators where the magnetism of localized spins is paramount

to the behaviour of the material. [4] This connection can be illustrated by the organic

1

Chapter 1. Introduction 2

insulator κ-(ET)2Cu2(CN)3, which becomes superconducting under pressure. [5] Unlike

most magnetic materials, it shows no order down to temperatures as low as 32 mK, and

is thought to be an example of an exotic spin-disordered state called a spin liquid.

We will continue by overviewing the emergence of quantum spin systems from the

complicated arrangement of nuclei and electrons in solid state systems. Afterwards,

we will discuss the emergence of exotic phases in spin systems, detailing the particular

systems of interest in this thesis.

1.1 Quantum Spins in Solid State Systems

1.1.1 The Route to Localized Spins

Much of condensed matter physics is aimed at understanding the electronic properties

of solids. Since the properties of nuclei in solids are more easily characterized, (due to

much higher masses compared to electrons) the study of systems of interacting electrons

has produced a broad range of interesting physics, ranging from semiconductors with

vast technological applications, to exotic ground states such as fractional Quantum Hall

states or high-temperature superconductivity.

Electrons carry a spin-12

moment, and provide an excellent opportunity to study

interacting quantum magnetism. The spin degrees of freedom take sole relevance in

a class of materials known as Mott insulators, in which the effective Coulomb repulsion

between electrons is large enough to suppress most electron motion through the material,

and leave electrons essentially confined to their corresponding nuclei.

The Hubbard model is the simplest model that produces this behaviour. It assumes

a basis of electronic wavefunctions that are localized around the nuclei, (a tight-binding

approximation) and a Hamiltonian allowing electrons to hop from site to site on the

lattice. In this model, the electron-electron Coulomb interaction is also considered as

localized, leading to an effective on-site repulsion of electrons. When this repulsion U

dominates over the hopping matrix elements t, the electron band-width becomes smaller

and smaller, with electrons becoming increasingly localized.

Virtual hopping processes are responsible for effective antiferromagnetic interactions

between the spins of neighbouring electrons. [6] For the case of d orbitals, this energy scale

J = t2/U is the dominant one in the problem, far larger than, for instance, the direct

dipole-dipole interaction between the magnetic moments of the electron. [7] However, for

localized f orbitals of magnetic rare-earth ions, the direct dipole-dipole interactions can

be significant.


In many Mott insulators, the magnetic ions are coordinated (often octahedrally) by

full-shell ions, such as O2− ions, accounting for the reduced conductivity of the electrons

between these magnetic ions. In this case, the effective interactions between localized

spins come from even high-order virtual exchange processes through these O ions, called

superexchange. [7] The Goodenough-Kanamori rules describe the details of the interaction

depending on the orbital geometry and intermediate ions.

Next, we will see how to consider localized magnetic ions as carriers of spin-12

moments.

1.1.2 Magnetic S = 12 Ions

The magnetic ions we will be considering in this work will all be transition-metal ions,

where the relevant electrons are in the d-orbitals, on top of a stable closed-shell config-

uration. Such a closed-shell configuration has total spin of zero, and displays only weak

diamagnetism. [8] Contributions from nuclear magnetic moments are similarly weak.

A simple case is a d1 configuration of a single electron in a d orbital, where the

total magnetic moment of the ion must consider the spin of one electron, and its orbital

angular momentum. Here, the neighbouring O2− ions again become important due to

their charge. The geometry of the resulting crystal electric fields splits the energy of the

five d orbitals, depending on the symmetry surrounding the magnetic ion. Electrons will

have higher energy when occupying orbitals that lie closer to the O ions.

Often, magnetic ions and O ions will form octahedra, in which case the d orbitals are

split into two higher-energy eg orbitals, and three lower-energy t2g orbitals. In the cases

of interest, Ba3Cr2O8 and La2LiMoO6, (Cr5+ and Mo5+) there is a further reduction in

symmetry due to a weak Jahn-Teller distortion as a result of degenerate ground states.

As a result, the t2g orbitals split in energy. In the case where the lowest orbital is non-

degenerate, the contribution of the orbital angular momentum is zero, 〈Lx,y,z = 0〉. This

occurs when the symmetry around the magnetic ion is low enough, and is referred to as

the quenching of orbital angular momentum. [7]

A similar scenario occurs when Cu2+ are the magnetic ions, which have a 3d9 config-

uration, meaning that for octahedral configurations, all orbitals are filled, except for one

half-filled eg orbital. Such a doubly-degenerate configuration will undergo Jahn-Teller

distortion to lift this degeneracy, and orbital angular momentum quenching is complete.

In these cases, we can think of the electronic configuration as having only two states,

simply due to the S = 12

degree of freedom. Spin-orbit coupling will affect this picture

slightly, and is discussed in the 4d case of Mo in Chapter 6. For 3d ions Cr and Cu, the

other magnetic ions considered in this work, this picture is not qualitatively altered. It


will, however, alter the virtual exchange processes, allowing for anisotropic interactions

between spins. The magnitude of such antisymmetric terms, D ∝ ∆ggJ , where ∆g is the

deviation from the free-electron value, so that D ∝ λ, the spin-orbit coupling strength. [9]

Magnitudes of D/J up to 0.1 are not uncommon in our relevant spin systems.

Next, we’ll consider the Heisenberg model, which arises from the most important

superexchange processes between magnetic ions.

1.1.3 Heisenberg Model

The Heisenberg model consists of isotropic (spin-rotationally-invariant) interactions be-

tween spins on a lattice,

HHeisenberg =∑ij

JijSi · Sj. (1.1)

Here, Jij can come from a mixture of direct-exchange and super-exchange processes (for

the materials we consider, direct exchange will be weak). As the magnetic ions get further

apart, super-exchange paths become larger and must include more than intermediate ion,

necessitating higher-order processes. Consequently, Jij will drop off as the distance be-

tween sites i and j increases. Density-functional calculations can give estimates for these

J . In most cases, nearest-neighbour bonds will have significantly higher J (by roughly

an order of magnitude) than next-neighbour bonds, and so on for further neighbours.

Despite its simplicity, a nearest-neighbour Heisenberg model can stabilize many differ-

ent phases, depending on the geometry and symmetry of the lattice. Most conventional

are magnetically ordered phases, which can occur when each interaction Si · Sj can be

simultaneously satisfied – with spins aligned for Jij < 0, or anti-aligned for Jij > 0. In

this case, the ordering follows the classical intuition (S → ∞) where the spins’ mag-

netic moments are treated as vectors. Quantum mechanically, however, the situation

is slightly different. While a ferromagnetic product state is indeed the ground state of

the ferromagnetic Heisenberg model, the same is not true for an antiferromagnetic state.

However, careful analysis for the square lattice and triangular lattice antiferromagnets

show that the ground state has the classically expected magnetic order. [10–12]

1.1.4 Geometric Frustration

With antiferromagnetic interactions, it may not be possible to choose an orientation of

the magnetic moments that satisfy every interaction simultaneously. For bipartite lattices

such as the square lattice, spins can be oppositely aligned on each sublattice; however,


(a) (b)

+ . . .

(c)

+ . . .

Figure 1.1: Schematic diagram of singlet formation for various states without magneticorder on the triangular lattice. (a) In the valence-bond-solid state, translational sym-metry is broken, though each singlet bond has no magnetic order. (b) In a resonatingvalence bond state, a sum over possible configurations restores the lattice symmetries.(c) Singlets need not form over only nearest-neighbour pairs.

for lattices with triangular geometries, all three spins cannot be simultaneously anti-

parallel. This is referred to as geometric frustration, and an extensive classical ground-

state degeneracy is a signature. This degeneracy will be broken by thermal fluctuations

or quantum fluctuations, a process known as order by disorder in which a certain ground

state is selected. [13–17]

However, geometric frustration may also realize interesting non-ordered phases. One

such phase is a valence-bond-solid state, which consists of pairs of spins that combine

to form spin singlets, which have no angular momentum whatsoever. Such states satisfy

the antiferromagnetic interaction within each pair, at the expense of interactions between

pairs. An example of these singlets can be seen in Fig. 1.1.4.

Quantum spin liquids are states that break no symmetries, though spins continue

to fluctuate, even down to zero temperature. The resonating valence bond state was

the first such proposed state, and is essentially a superposition over all arrangements of

valence-bond-solid states. [18] These singlets need not exist over only nearest-neighbour

pairs. A schematic diagram of these concepts is shown in Fig. 1.1.4.

With no broken symmetries, a quantum spin liquid (QSL) state has no order param-

eter that can characterize the state. However, there are many possible type of spin liquid

states with a given symmetry, (see Chapter 3) so there must be some other underlying

physical characterization. This is referred to as topological order, which describes states

beyond the Landau symmetry-breaking description, and for which quantum entanglement

plays a key role. [19–21]

One of the most interesting features of spin liquid states is the prediction of fractional-

ized excitations called spinons, which are charge-neutral, but carry S = 12. [22–24] These

can be either bosonic or fermionic, and in Chapter 2 we will see some of the possible

ways to write down a theory of such excitations. In contrast, excitations in magnetically

ordered or valence-bond-solid systems are S = 1 bosons, called magnons.


1.1.5 Quantum Antiferromagnets

The exotic phases discussed in the previous subsection naturally find their homes in

various types of spin systems.

Spin-dimer systems feature antiferromagnetic spin-spin interactions that are intrinsically

stronger between pairs of sites. The lattice naturally separates into pairs of these dimers;

in the limit of strong intra-dimer interactions, the system forms a spin singlet on each

of these dimers, a valence-bond-solid state. As inter-dimer interactions increase, dimers

become correlated and the singlet order begins to “melt”. These systems are discussed

in further detail in Sec. 1.2.

Triangles and tetrahedrons form the basic two- and three-dimensional building blocks of

frustrated lattices. These are, in essence, the smallest and simplest units of frustration,

and a lattice composed of these units may be heavily frustrated. In particular, corner-

sharing arrangements have been the focus of investigations as possible quantum spin

liquid candidates.

Corner-sharing triangles in two dimensions form the kagome lattice, discussed in

detail in Sec. 1.4, which has seen a long and concerted effort, both theoretically and

experimentally, to realize a quantum spin liquid ground state. In three dimensions, they

form the hyperkagome lattice, as in Na4Ir3O8, which is also expected to form a spin

liquid. [25–28]

Corner-sharing tetrahedra form the pyrochlore lattice, which has many different re-

alizations [29] and easy-axis terms often lead to spin-ice behaviour, where two spins

point in or out of each tetrahedron. The degeneracy of such configurations leads to

finite ground-state entropy, and spin-flip excitations may take the form of a magnetic

monopole. [30] In contrast, other pyrochlores such as Yb2Ti2O7 also show evidence of

spin-liquid behaviour. [31]

While most of the work on frustrated magnetism has focused on maximizing frus-

tration and corner-sharing geometries, other frustrated systems have been studied, each

with a slightly different focus. For instance, spin-1 systems are studied often for the

effect of the biquadratic term∑

ij Jij (Si · Sj)2 in addition to the Heisenberg term (1.1),

despite reducing the effects of quantum fluctuations. [32–36] Orbital degeneracy and spin-

orbital effects are studied in the slightly-less-frustrated face-centred-cubic lattice of the

double-perovskite spin systems, as discussed in Sec. 1.3.

We will not elaborate on the rich physics of even unfrustrated spin chains in one

dimension, as exemplified by the highly anisotropic copper pyrazine dinitrate, a S = 12

Heisenberg chain. [37] In one dimension, the domain wall created by a magnon is a S = 12

spinon, [38] showing how fractionalization can arise relatively straightforwardly in one


dimension. Instead, we will focus now on the history of spin-dimer, double-perovskite,

and kagome lattice spin systems, to set the stage for the work done in this thesis.

1.2 Spin-Dimer Systems

Magnetic-field-induced quantum phase transitions in spin dimer systems have provided

excellent playgrounds for the investigation of novel universality classes of zero temper-

ature quantum phase transitions. [39] These systems possess non-magnetic spin singlet

ground states with a spin gap to triplet excitations (triplons). When a magnetic field

H is applied, a quantum phase transition occurs at a critical field Hc, where the spin

gap closes and the lowest triplet excitation condenses. At H > Hc, the average triplon

density is finite and can be controlled by the applied magnetic field.

The resulting ground states are determined by a delicate balance between kinetic en-

ergy and the repulsive interaction between triplons. [40] On one hand, if the triplon hop-

ping processes are suppressed by frustration or the repulsive interaction dominates, the

condensed triplons may form a superlattice with broken translational symmetry, leading

to magnetization plateaus. This is known to occur, for instance, in SrCu2(BO3). [41, 42]

On the other hand, if the magnetic interaction does not have much frustration or the

kinetic energy dominates, the ground state can be described as a Bose-Einstein conden-

sate (BEC) of triplons and form a homogeneous magnetically ordered state. In this case,

the magnetically ordered state at H > Hc supports a staggered magnetization trans-

verse to the field direction, leading to a canted antiferromagnetic state (until the system

eventually becomes fully polarized as H increases). This type of behaviour has been ob-

served in three-dimensionally coupled spin dimer systems including TlCuCl3 [43,44] and

BaCuSi2O6, [45] which exhibits unconventional critical behaviour. [46–53] Furthermore,

recent discoveries of A3M2O8, [54] where A = Ba or Sr, and M = Cr or Mn, have pro-

vided a lot of excitement for spin dimer system research, as these systems may represent

a variety of different spin dimer interactions and quantum ground states.

In two dimensions, the S = 12

layered copper oxyhalides (CuX)An−1BnO3n+1 offer

an interesting family of S = 12

frustrated magnets with a rich variety of behaviours

reminiscent of the aforementioned behaviour. Extensive experimental investigations have

uncovered a collective singlet ground state with a spin gap in (CuCl)LaNb2O7, [55–58]

a collinear stripe magnetic order in (CuBr)LaNb2O7, [59] and a magnetization plateau

at 13

of the saturated moment in (CuBr)Sr2Nb3O10. [60] A density-functional study by

Tassel et al. [61] has proposed a remarkable microscopic structure in (CuCl)LaNb2O7,

where the strongest fourth-neighbour create an effectively dimerized Shastry-Sutherland


lattice, similar to SrCu2(BO3), though ferromagnetic. The physics of spin-dimer systems

is not relegated solely to physically dimerized systems.

A notable feature of the Shastry-Sutherland lattice is the strong suppression of the

triplon hopping due to frustration. [62, 63] In the case of antiferromagnetic interac-

tions between dimers, the localized nature of the triplons gives rise to various frac-

tional plateaus in the magnetization process, [64–66] which are experimentally observed

in SrCu2(BO3)2. [42, 67,68]

1.3 Double-Perovskite Spin Systems

One class of frustrated antiferromagnets is found in the double perovskite oxides, which

host a wide range of interesting behaviour. [69–74] These compounds of chemical formula

A2BB′O6 feature ordered, interpenetrating face-centred cubic (FCC) lattices of the B

and B′ ions when the charge difference between these ions is large. [75] Both B and B′

transition metal ions are octahedrally coordinated by oxygen. A geometrically frustrated

FCC lattice is obtained when only the B′ ions are magnetic.

A conventional picture of isotropic antiferromagnetic superexchange is insufficient for

these materials. Altering this picture are two important effects considered in our work.

The first effect is spin-orbit coupling, which is relevant for the 4d and 5d transition

metal ions that comprise the magnetic sites. Spin-orbit coupling has been seen to lead

to increased correlation effects, particularly in materials containing 5d Ir ions. This

is responsible for topological insulating behaviour, [76] particularly in the pyrochlore

iridates, [77–82] the Mott insulator ground state of Sr2IrO4, [83–89] and the potential

spin-liquid ground state of Na4Ir3O8 [25–28, 90–94] and honeycomb compounds A2IrO3.

[95]

The second effect is geometrical distortion from the cubic case; monoclinic distortion

is commonly seen in double perovskites. [75] Lowered symmetry from the monoclinic

distortion will spoil the exchange isotropy directly, and introduce new exchange pathways.

Many of these systems show magnetic ordering transitions and little frustration. This

is true for nominal S = 12, d1 systems such as Ba2CaReO6, [96] Ba2LiOsO6, [97] and

Ba2NaOsO6, [69] or the nominal S = 1 Ba2CaOsO6. [96] We note that these systems do

not have pure spin moments, due to the presence of spin-orbit coupling. Spin-freezing

behaviour is observed in the cubic Ba2YReO6 [98] around 50 K, and in the distorted

Sr2CaReO6 at 14 K, [99] as well as Sr2MgReO6 at 50 K. [100]

Other interesting behaviour is seen in materials that show no evidence of spin ordering

or freezing. The cubic Ba2YMoO6 shows evidence of a valence-bond glass [101] or collec-


Figure 1.2: The kagome lattice, a network of two-dimensional corner-sharing triangles.

tive spin-singlet ground state. [102] The monoclinically distorted 4d2 La2LiReO6, while

possibly hosting a single-ion non-Kramers singlet state, shows a large difference between

zero-field and field-cooled samples, indicating a possibility of an exotic ground state. [98]

The 4d1 La2LiMoO6 shows evidence of short-range correlations developing below 20 K,

and no long-range order down to 2 K. [102]

The importance of spin-orbit coupling and orbital degeneracy has been studied by

Chen et. al. for cubic d1 and d2 configurations. [103,104] In particular, it leads to effective

four-spin and six-spin interactions terms, giving rise to interesting new physics. Among

the phases found are several magnetically ordered phases, which may correspond to

some of those seen in experiment. Furthermore, a spin-quadrupolar state is stabilized at

intermediate temperatures, which may explain the two Curie regimes in the susceptibility

of Ba2YMoO6. [102]

1.4 Kagome Lattice Spin Systems

Spin systems with corner-sharing-triangle geometries host a high degree of geometric

frustration, and as such are recognized as good candidates to realize quantum spin liquid

states. [105, 106] Consequently, these systems have been extensively investigated, par-

ticularly the kagome lattice, as shown in Fig. 1.2. [22, 23, 105–108, 108–131] However,

the fate of the nearest-neighbour (NN) S = 12

Heisenberg model on the kagome lat-

tice (HKAF) is unclear, despite the long history of proposals for the ground state.1

Such states include valence-bond solids, [123,124,133–135] gapless [136–138] and gapped

QSLs. [117,125,129,139–141]

Of these, a gapped QSL was found recently by Density Matrix Renormalization Group

1For a pedagogical review of the earlier history, see Ref. [132].


(DMRG) calculations. [112, 114, 131] However, variational Monte Carlo (VMC) studies

have found a gapless spin-disordered ground state with energy very similar to that of

the DMRG. [111] These and other studies also seem to suggest that the strictly NN-

HKAF may be close to a quantum phase transition. It may sit close to a quantum

phase transition between a Z2 spin liquid and a valence-bond-solid state upon tuning the

second-neighbour interaction, [113, 114, 142, 143] or a magnetically ordered state upon

tuning the Dzyaloshinsky-Moriya interaction. [108,118,119,144]

The search for realizations of such a Heisenberg kagome antiferromagnet model has

unearthed several candidate materials, each suffering, however, from some sort of disorder

or distortion. The main candidates are as follows:

• Vesignieite. Vesignieite features a slightly distorted kagome lattice composed of

isosceles triangles, where the ratio of the two interaction strengths J ′/J ' 1.

[128] Furthermore, there is a significant Dzyaloshinsky-Moriya (DM) interaction

of strength D ∼ 0.13J . [145] Vesignieite features long-range magnetic order at 9

K, [146] demonstrating the ability of these frustration-relieving perturbations to

quickly favour long-range order.

• Volborthite. Powder samples of Volborthite indicated an unusual distorted structure

of isosceles triangles, with a J ′ coupling along one direction that is ferromagnetic.

[147] NMR studies also showed an unusual ordered state at low temperature. [127]

However, single-crystal results indicate that an orbital switching effect leads to a

structural transition at T = 310 K, leading to a nearest-neighbour model with three

different antiferromagnetic interaction strengths, and long-range magnetic order at

low temperatures. [148]

• Kapellasite. In contrast to the above cases, Kapellasite has a structurally perfect

isotropic kagome lattice. [149] However, it suffers from a large amount of disorder,

with ∼ 27% Zn2+ in the kagome plane and ∼ 12% Cu2+ on the Zn sites. [150]

Furthermore, there is a large third-neighbour coupling on the same strength as

the nearest-neighbour coupling. [151] Thus, it makes for a poor realization of the

HKAF. Experimentally, there is no observed long-range order down to 20 mK, so

it nonetheless remains a candidate quantum spin liquid. [151]

• Herbertsmithite. Herbertsmithite is again a perfect realization of the isotropic

kagome lattice. [130] In this case, no long-range order is seen down to 50 mK,

a small fraction of the Curie-Weiss temperature θCW ∼ −300 K. [121] In particu-

lar, µSR sees no signs of spin freezing. [122] However, with application of 2.5 GPa


of pressure, Neel ordering is seen below TN = 6 K. [152] While Herbertsmithite

suffers disorder, it comes mostly in the form of 14% occupation of inter-layer non-

magnetic Zn ions by magnetic Cu2+ ions, with < 5% non-magnetic impurities in

the Cu2+ kagome planes. [153] Furthermore, inelastic neutron scattering has shown

diffuse magnetic scattering for a broad range of momenta and energy, down to tem-

peratures as low as 1.6 K. [116, 121, 154] This suggests the existence of deconfined

S = 12

“spinon” excitations, as opposed to dispersive S = 1 spin-wave excitations

expected in magnetically ordered states.

1.5 Slave-Particle Mapping of Quantum Magnetism

1.5.1 Mapping Spins to Particles

As has been overviewed in this Chapter, quantum magnetism provides a rich set of ground

states, and excitations of varying statistics and angular momentum. The relationship

between the spin operators is complicated, and we would like to find a simpler way

to represent them. The language of creation and annihilation operators of bosons or

fermions allows a direct mapping of the excitations in terms of quasi-particle operators.

One such example appears in the Holstein-Primakoff approach, which considers the

effect of quantum fluctuations on a particular classical order of the spins. [155] In some

sense, this is mapping between spin raising and lowering operators with bosonic annihi-

lation and creation operators. In the semi-classical limit, the resulting quadratic bosonic

Hamiltonian yields the dispersion of these S = 1 bosonic “spin wave” excitation, describ-

ing fluctuations of the classical magnetic order.

The so-called Schwinger boson description, in contrast, is better suited for repre-

senting states that have explicit spin-rotation invariance. It maps the spin raising and

lowering operators to quadratic annihilation and creation operator terms. [156] As a re-

sult of splitting up the spin operators into two halves, we can describe S = 12

excitations.

Many of the representations used in Chapter 2 – both bosonic and fermionic – will follow

this style.

1.5.2 Overview of the Thesis

Various slave-particle techniques will be used throughout this thesis, and we will begin in

Chapter 2 with an overview of the mappings and physical meanings of the resulting states.

Particularly, Chapter 3 will discuss the symmetries and emergent gauge redundancies

brought about by these descriptions. With the formalisms understood, we will proceed


to tackle understanding novel phenomena in quantum magnetism. In Chapter 4. we try

to understand the magnetization process of a spin-dimer system Ba3Cr2O8 in terms of

Bose-Einstein condensation of S = 1 excitations. In contrast, in Chapter 5, we consider

the application of the S = 12

Schwinger boson description in (CuCl)LaNb2O7. A similar

technique is used in Chapter 6 to investigate the possibility of a spin-liquid state in

La2LiMoO6, to explain the lack of long-range order. We use a fermionic description in

Chapter 7 to understand the unconventional magnetic order of the powder samples of

Volborthite. Finally, in Chapter 8 we use this technique to characterize spin correlations

of possible spin liquid ground states of Herbertsmithite, possibly in the absence of spin-

rotation symmetry.

The work in Chapter 4 has been published in Physical Review B [157]. The work of

Chapter 5 forms part of a paper that has been published in Physical Review B. [158] The

work in Chapter 6 has been published in Physical Review B. [159] Finally, the work in

Chapter 8 has been submitted for publication in Physical Review B.

Chapter 2

Slave-Particle Representation of

Spins

2.1 Spin

The quantum spin operator Sa (a ∈ x, y, z) is intrinsic angular momentum to parti-

cles; as such, the spin operators have the angular momentum fundamental commutation

relations, namely [160]

[Sx, Sy] = i~Sz [Sy, Sz] = i~Sx [Sz, Sx] = i~Sy. (2.1)

Since S · S and Sa commute, they can be simultaneously diagonalized (a = z is the

conventional choice), giving the eigenstates labelled |Sm〉, where

S2 |Sm〉 = ~2S(S + 1) |Sm〉 Sz |Sm〉 = ~m |Sm〉

S = 0,1

2, 1, . . . m = −S,−S + 1, . . . , S. (2.2)

These eigenstates for a given spin quantum number S are connected by the so-called

“raising” and “lowering” operators given by S± = Sx ± iSy,

S± |Sm〉 = ~√S(S + 1)−m(m± 1) |S(m± 1)〉 . (2.3)

Particles with non-zero spin quantum also carry a magnetic dipole moment µ =

−gµBS/~, where µB = e~2me

is the Bohr magneton and g ∼= −2.0023 for an electron,

which has S = 1/2. [8]

For the case of S = 1/2, which will appear throughout the rest of this work, there

13

Chapter 2. Slave-Particle Representation of Spins 14

are only two different states within the Hilbert space,∣∣∣∣12 , 1

2

⟩≡ |↑〉 ,

∣∣∣∣12 ,−1

2

⟩≡ |↓〉 , (2.4)

corresponding to spins polarized up or down along the z-axis. We can take linear com-

binations of these to find spins polarized along the x- and y-axes,∣∣∣∣Sx =1

2

⟩=|↑〉+ |↓〉√

2

∣∣∣∣Sx = −1

2

⟩=|↑〉 − |↓〉√

2∣∣∣∣Sy =1

2

⟩=|↑〉+ i |↓〉√

2

∣∣∣∣Sy = −1

2

⟩=|↑〉 − i |↓〉√

2. (2.5)

Within this two-site basis, the spin operators are written in simple matrix form Sa =σa

2, where σ are the 2× 2 Pauli matrices.

2.1.1 Addition of Angular Momentum

The eigenstates in (2.2) may be useful in describing the state of a multiple-spin system.

It is particularly useful in the case where a many-spin magnetically-ordered wavefunction

can be described as a product state of |↑〉 or |↓〉 on different sites. For instance, consider

a ferromagnetic state |↑↑ · · · ↑〉 = |↑〉1⊗|↑〉1⊗· · · |↑〉N , where spins 1 through N are each

described up the |↑〉 eigenstate.

However, when we take linear combinations of such product wavefunctions, the state

of each spin may no longer be separate from the others in the system. As a canonical (and

quite useful) example, we will consider the four-state Hilbert space of a system of two

S = 12

spins. We can take the simplest Hilbert space to be |↑↑〉 , |↑↓〉 , |↓↑〉 , |↓↓〉. It

turns out that all of these states can be written as eigenstates of the total spin operator

Stot = S1 + S2. By applying S2tot and Sztot, we can see that |↑↑〉 = |1, 1〉 and |↓↓〉 =

|1,−1〉. Through application of the raising operator S+tot on |↓↓〉 we find that the linear

combination |↑↓〉+|↓↑〉2

= |1, 0〉. These three states are referred to as the triplet of states

with S = 1. There is one additional state orthogonal to these, which completes the

Hilbert space, |↑↓〉−|↓↑〉2

= |0, 0〉; it is referred to as the singlet state.

In general, when considering the combined states of two angular momenta, the result-

ing states are eigenstates of the total angular momentum. [160] The new spin operator

S may range from |S1− S2|, |S1− S2|+ 1, · · · , S1 + S2, and for each such S, m ranges as

usual from −S,−S + 1, . . . , S.

The distinction between triplet and singlets states becomes most apparent when con-

sidering the Heisenberg interaction term S1 ·S2, for which these are also eigenstates. The


triplet states each have the eigenvalue ~24

, while the singlet state has eigenvalue −3~24

.

Singlet states being favoured by antiferromagnetic interactions is an important notion

that underlies much of the thinking about geometrically frustrated spin systems.

For the remainder of the work, we will take units where ~ = 1.

2.2 Slave-Particle: Formalisms

2.2.1 Spins in Dimerized Systems

As discussed in Sec. 2.1, a singlet-triplet basis on a pair of sites is a natural basis when

the sites are interacting with a Heisenberg term. Interactions in dimerized system consist

mainly of strong antiferromagnetic interactions between pairs of sites (dimers), greatly

favouring the formation of singlet states on these dimers. When these pairs are isolated,

the ground state consists of a singlet on each pair. Other interactions can then be viewed

as interactions between the singlets, altering the ground states as a result. This is often

referred to as the bond-operator formalism. [161–165]

We can represent which of these four states is found on a given dimer by placing

precisely a single hard-core boson s†, t†0, t†−, or t†+. These are creation operators for the

states

|s〉 = s† |0〉 =|↑↓〉 − |↓↑〉√

2= |0, 0〉 ,

|t0〉 = t†0 |0〉 =|↑↓〉+ |↓↑〉√

2= |1, 0〉 ,

|t+〉 = t†+ |0〉 = − |↑↑〉 = − |1, 1〉 ,

|t−〉 = t†− |0〉 = |↓↓〉 = |1,−1〉 . (2.6)

The triplet states |tm〉 (m = 1, 0,−1) are chosen as the Sz eigenstates satisfying

Sz|tm〉=m~|tm〉. The hard-core constraint s†s + t†0t0 + t†+t+ + t†−t− = 1 is enforced on

each dimer, ensuring that the physical state is exactly one of the four above.

The two spin operators constituting a dimer can be rewritten in terms of the bosonic

bond operators as

S1α =1

2

(s†tα + t†αs− iεαβγt

†βtγ

),

S2α =1

2

(−s†tα − t†αs− iεαβγt

†βtγ

), (2.7)

with α ∈ x, y, z and ε the totally antisymmetric tensor. Summation over repeated


indices is henceforth assumed. This form gives the correct matrix elements in the singlet-

triplet Hilbert space. We define |tα〉 triplons as eigenstates with Sα |tα〉 = 0 so that

tz = t0, tx = 1√2

(t− + t+) and ty = i√2

(t− − t+). Here, “triplon” names the S = 1

bosonic excitation given by t†α. In this form, the intra-dimer interaction can be written

as

J0S1i · S2i = J0

(−3

4s†isi +

1

4t†iαtiα

), (2.8)

where i labels the dimer.

When inter-dimer interactions are weak and dimers are nearly isolated, most dimers

will be in the singlet state. Such a macroscopic occupation is represented by the Bose-

Einstein condensation singlet s† bosons. In doing so, we are neglecting fluctuations in the

singlets by replacing the singlet operators s and s† with a complex number s, creating

a uniform condensate. As a result, the singlet bosons drop out of the problem, leaving

only the singlet number occupancy given by |s|2 to be determined. What remains is

a Hamiltonian describing the triplet bosons, where inter-dimer interactions lead to a

dispersion of the triplet excitations, as well as triplet-triplet interaction terms.

If the dispersion for the triplons touches zero at some point in the Brillouin zone, they

may also undergo Bose-Einstein condensation. In this case, condensation of these spin-1

particles at that wavevector leads to long-range order, since we ignore the fluctuations of

these spin-carrying excitations. This will we worked out in detail in Sec. 4.6.

2.2.2 Spin-12

As with the dimer case, we will continue with the strategy of representing each state in

the spin Hilbert space with a single hard-core particle. In the case of S = 12, our Hilbert

space consists of only two states, |↑〉 = p†↑ |0〉 and |↓〉 = p†↓ |0〉, where |0〉 is the vacuum

state with no particles. Here, we take the so-called parton p† to have either fermionic or

bosonic statistics; in these cases we write p† as f † or b†, respectively. The commutation

relations are

fi, f †j = δij fi, fj = 0 [bi, b†j] = δij [bi, bj] = 0. (2.9)


Our Hilbert space consists of exactly one parton on each spin site; p†αpα = 1. Conse-

quently, these are hard-core partons. The spin operator can be written as

Sa =p†ασ

aαβpβ

2, (2.10)

where α ∈ ↑, ↓. With single occupancy of hard-core partons, both Sa and S2 have the

correct matrix elements, as well as the appropriate commutation relations (2.1).

The single-particle constraint also implies pairing constraints on each site.

p†αpα = 1 pαεαβpβ = 0 p†αεαβp†β = 0. (2.11)

εαβ =

(0 1

−1 0

)is the 2× 2 totally antisymmetric tensor.

2.2.3 Spin > 12

To represent S > 12, we could use 2S + 1 species of partons, one to represent each

state in the Hilbert space. This would be an analogous construction to the slave-particle

representation on the dimer, which featured one parton per state of the dimer. For low

values of S, particularly S = 1, such a construction has been used in serveral different

approaches, particularly on the triangular lattice, from exotic S = 1 spin liquids [32, 33]

to the general bilinear-biquadratic model. [34–36]

For our studies, which are focused on S = 12

systems, we will consider an alternate,

more general approach that retains two different types of partons, but allows for many

particles per site. [129, 166, 167] In this case, the number of partons determines the spin

quantum number, 2S = b†αbα. Due to Pauli exclusion, this is a bosonic construction. The

main advantage of this formulation is that is allows the S = 12

limit discussed in Sec.

2.2.2 to be connected with larger values of S within the same framework; as S →∞, the

limit of semi-classical ordering is reached. This framework captures the evolution from

magnetically ordered to spin-disordered states as the number of particles is lowered. As

in the dimer case, condensation of b†↑ or b†↓ bosons, which each carry S = 12, leads to

long-range order.

2.2.4 Single-Particle Constraint

As it stands, the slave-particle transformations discussed in Secs. 2.2 and 2.2.3 yield

a difficult problem. Spin-spin interactions, such as a Heisenberg Hamiltonian H =


∑ij JijSi · Sj, are quartic in the partons. This form is what we aim to take advantage

of, given the vast toolbox of many-body physics; however, the single-particle constraint

is not readily amenable to solution.

To see this, we will enforce the constraint term on every site with a Lagrange multiplier∑i

µi

(p†iαpiα − 2S

). (2.12)

This considers the ground state average of these terms,⟨p†iαpiα − 2S

⟩= 0. We will

treat this constraint term on average in two ways. In the first, we will ignore the time-

dependence of µi in (2.12). In the second, we will treat the constraint on average over

the lattice, so that the µ become sublattice-dependent chemical potentials. The result-

ing states do not satisfy the single-particle constraint. Gutwiller projection, as used in

Variational Monte Carlo studies, can project these states to the physical subspace of one

particle per site.

The starting point, before considering fluctuations (in a field-theoretic approach) or

projection (in a numerical approach) is to consider a mean-field Hamiltonian that is

quadratic in the partons. Having a proper understanding of such states is an important

first step to understand qualitative features one can expect from slave-particle treatments.

2.3 Quadratic Hamiltonians

2.3.1 Quadratic Terms

Here, we will consider the characteristics of all possible quadratic terms composed of the

partons p↑ and p↓. First, we will consider terms with partons on a single site. We have

the hopping terms

p†↑p↑ + p†↓p↓ = n p†↑p↓ + p†↓p↑ = 2Sx (2.13)

ip†↑p↓ − ip†↓p↑ = 2Sy p†↑p↑ − p

†↓p↓ = 2Sz. (2.14)

These describe the on-site density and the spin operators. Due to the constraint, most

on-site pairing terms are zero, except for the S > 12

bosonic case, where these terms do

not arise naturally.

Including spin operators in a quadratic Hamiltonian provides a direct method to create

a state with magnetic ordering, and may be useful in the case where a fermionic slave-

particle description captures important properties, but long-range order is also desired.


Term Singlet Triplet

Hopping χij = p†iαpjα Eaij = p†iατ

aαβpjβ

Pairing ηij = piα(iσy)pjβ Y aij = piα(iσyσa)αβpjβ

Table 2.1: Quadratic hopping and pairing operators between sites i and j. Parton indicesα, β ∈ ↑, ↓, a ∈ x, y, z and σa are the Pauli matrices.

In Chapter 7, we will postulate such a scenario for Volborthite, which has Fermi-surface-

like behaviour and long-range order.

Next, we will consider the hopping and pairing quadratic terms between two sites,

as listed in Table 2.1. There are one singlet (S = 0) and three triplet (S = 1) terms

of both hopping and pairing types. These correspond to spin-singlet and spin-nematic

order parameters, respectively. To see the former, we note that χ and η are unchanged

under a global spin rotation. To see the latter, we can re-write the bond quadrupolar

and vector chiral order parameters in the fermionic case as [168]

Kαβij ≡

⟨Sαi S

βj

⟩− δαβ

3〈Si · Sj〉 ;

Qij;αβ ≡Kαβij +Kβα

ij

2= Eα

ijEβij

∗ − δαβ3|Eij|2 + h.c. = Y α

ij Yβij

∗ − δαβ3|Yij|2 + h.c.

P γij ≡ εγαβKαβ

ij = i(χijE

γij∗ − χ∗ijE

γij

)= i(ηijY

γij∗ − η∗ijY

γij

). (2.15)

Due to the details of the mean-field decoupling for the systems of interest (see Sec. 2.3.2),

we will not encounter the bosonic triplet terms in this work. The vectors Eij and Yij

mirror the forms of “director” vectors for nematic order. [155]

2.3.2 Mean-Field Channels

As seen for the case of the spin-quadratic tensor (2.15), we can re-write interaction terms

that are quadratic in spin as a product of only hopping or only pairing terms. We will

do this for the three types of spin-spin interactions encountered in this work, Heisenberg,

Ising and Dzyaloshinsky-Moriya (DM) interactions. From the form taken after this re-

writing, a natural mean-field decoupling can be identified.


We first begin by re-writing the Heisenberg, DM, and Ising interactions as

Si · Sj =1

8

(−3χ†ijχij + E†ij · Eij

)=

1

8

(−3η†ij ηij + Y†ij · Yij

), (2.16)

z · Si × Sj =i

4(χ†ijE

zij − E

z†ij χij) =

i

4(η†ijY

zij − Y

z†ij ηij), (2.17)

Szi Szj =

1

8(−χ†ijχij − E

z†ij E

zij + Ex†

ij Exij + Ey†

ij Eyij)

=1

8(−η†ij ηij − Y

z†ij Y

zij + Y x†

ij Yxij + Y y†

ij Yyij), (2.18)

for the case of fermionic partons, where x and y components can be found by cyclic

permutation of indices. We suppress the site indices i, j for the rest of the section.

Singlet hopping and pairing interactions can be combined by rewriting pairing terms as

hopping terms:

η†η =χ†χ

2− E† · E

2+ n

Y a†Y a = − χ†χ

2− E† · E

2+ Ea†Ea + n. (2.19)

For the Heisenberg term, we can thus re-write

Si · Sj = −1

4

(χ†χ+ η†η − 1

)=

1

4

(E†E + Y†Y − 3

). (2.20)

We can see that, for different signs of the interaction, either the singlet or triplet terms

are favoured.

Next, we turn to the case of bosonic partons, where

Si · Sj =1

8

(+3χ†χ− E† · E

)=

1

8

(−3η†η + Y† · Y

), (2.21)

Again, we try re-writing pairing terms as hopping terms:

η†η = − χ†χ

2+

E† · E2− n

Y a†Y a =χ†χ

2+

E† · E2− Ea†Ea − n. (2.22)

As in the fermionic case, we can write Si · Sj in terms of only the singlet or triplet

channels; however, we are unable to write it in terms of only attractive channels – the

terms always come with mixed positive and negative signs. To get around this issue, we

will make an approximation for the operator ninj, which equals 1 in the physical Hilbert


space, but will fluctuate from that in a mean-field theory. So, we will re-write

−Si · Sj = −χ†ijχij

2+ni2

+ninj

4∼= −

χ†χ

2+ S(S + 1),

Si · Sj = −η†ij ηij

2+ninj

4∼= −

η†η

2+ S2. (2.23)

2.3.3 Mean-Field Decoupling

With the re-organization done in Sec. 2.3.2, a mean-field decoupling will follow naturally

from an interaction term written as Hterm = −CO†O for C > 0. Then both a mean-

field decoupling and the saddle-point solution of a Hubbard-Stratonovich transformation

yield Hterm → C(−O†O −O∗O + |O|2

). The self-consistent mean-field equation (or

Hubbard-Stratonovich saddle-point solution) gives⟨O⟩

= O.

The Heisenberg terms (both ferromagnetic and antiferromagnetic) have already been

re-written in this manner, as in (2.20) and (2.23). Such decouplings provide the basic

mean-field framework used throughout this work. The other interactions will be re-visited

in Sec. 8.3.1.

2.3.4 Large-N Generalization

For the bosonic or fermionic slave-particle descriptions mentioned above, generalizations

can be made with a number of flavours of partons larger than two (↑ and ↓). While

different values of N do not match the physical N = 2 problem, they reproduce the

N = 2 saddle-point solution as their exact ground states when N → ∞. Corrections

to these mean-field solutions can be considered in powers of 1/N . While several such

generationalizations can be made, we will discuss the ones that will reproduce our mean-

field theories as N →∞.

Our bosonic representation of the Heisenberg model, for arbitrary filling, can be

generalized to a model with Sp(N) symmetry, where Sp(1) ∼ SU(2). [129,166,167] Such

a model can still connect to the semi-classical limit as the boson filling becomes large.

Similarly, an arbitrary-N generalization reproducing the fermionic mean-field state of a

Heisenberg model has been recently introduced. [169]

While we will not explicitly consider the effects of 1/N corrections, they present

the most obvious way to extend our results, especially in cases where mean-field theory

predicts pathological behaviour, such as strictly zero correlations on certain bonds.

In the next Chapter, we will discuss the gauge-redundancy of the slave-particle rep-

resentations, and the mean-field quadratic Hamiltonians describing spin liquid states.

Chapter 3

Spin Liquid States

3.1 Gauge Structure of Slave-Particle Representa-

tions

3.1.1 Local U(1) Gauge Redundancy

Besides the on-site density constraint, the slave-particle representation has another dif-

ficulty, which is a gauge redundancy of different parton descriptions of the same spin

operator. Since the spin operator is the only physical observable of interest, we may

perform a gauge transformation on the partons – if this transformation leaves the spin

operator and Hilbert space invariant, then we have a different description of the same

physical state.

By looking at Si = p†iασαβpiβ/2, we see that the U(1) transformation piα → piαeiθi

leaves Si invariant, and does not change the occupancy. We note that this is a local

transformation performable with a different phase θi at every spin site in the system.

However, for fermions, the fact that f is a spin-12, chargeless particle means that a

particle-hole transformation retains all the essential properties of the parton. For bosons,

such a transformation does not exist. A particle-hole transformation fα → f †α flips the

sign of Sx and Sz, corresponding to a spin rotation by π around the y-axis. While this does

not leave the physical problem unchanged, such a spin rotation can leave spin-disordered

or nematic-ordered states unchanged. In the case of quadratic Hamiltonians, we will

see precisely how this extra degree of gauge redundancy arises. A better understanding

of the gauge structure of these quadratic Hamiltonians will also reveal the structure of

fluctuations beyond mean-field theory. [170, 171] For the remainder of this section, we

will focus on the fermionic case; the bosonic case is a simpler subset of this.

22

Chapter 3. Spin Liquid States 23

3.1.2 Fermionic SU(2) Gauge Redundancy

Here, we consider writing a general quadratic fermionic Hamiltonian, deriving from the

terms in Table 2.1. With eight real and imaginary terms, we can write the quadratic

terms as [172–174]

HQ =∑ij

∑a

tr(σaFiUaijF†j ), (3.1)

Fi =

(f i↑i f †i↓fi↓ −f †i↑

), Ua

ij = iCa0 + Can · σn, Can =

can a = 0

ican a > 0, can ∈ R. (3.2)

We can explicitly write out these singlet terms as

H0Q = ic0

0

(χij − χ†ij

)+ c0

1

(ηij + η†ij

)+ c0

2

(−iηij + iη†ij

)+ c0

3

(−χij − χ†ij

)= χij

(−c0

3 + ic00

)+ ηij

(c0

1 − ic02

)+ h.c. (3.3)

and we can see a clear contribution to the real and imaginary parts of the singlet hopping

and pairing parameters, χ and η. For the triplet terms, we have the following:

H1Q = c1

0

(−Ex

ij − Exij†)+ ic1

1

(−Y x

ij + Y xij†)+ ic1

2

(iY xij + iY x

ij†)+ ic1

3

(Exij − Ex

ij†)

= Exij

(−c1

0 + ic13

)+ Y x

ij

(−ic1

1 − c12

)+ h.c.

H2Q = c2

0

(−Ey

ij − Eyij†)

+ ic21

(−Y y

ij + Y yij†)

+ ic22

(iY yij + iY y

ij†)

+ ic23

(Eyij − E

yij†)

= Eyij

(−c2

0 + ic23

)+ Y y

ij

(−ic2

1 − c22

)+ h.c.

H3Q = c3

0

(−Ez

ij − Ezij†)+ ic3

1

(−Y z

ij + Y zij†)+ ic3

2

(iY zij + iY z

ij†)+ ic3

3

(Ezij − Ez

ij†)

= Ezij

(−c3

0 + ic33

)+ Y z

ij

(−ic3

1 − c32

)+ h.c. (3.4)

The constraint terms can be written in this basis as∑

i,a µaiG

ai , where the constraint

and spin operators are written as

Gai =

tr(Fiσ

aF †i

)4

, Sai = −tr(σaFiF

†i

)4

. (3.5)

In this basis, we can write down two types of transformations of the Fi basis matrix.

Spin: Fi → ViFi, Vi ∈ SU(2), Gauge: Fi → FiWi, Wi ∈ SU(2). (3.6)


The first generates a spin rotation, as seen by the trace in (3.5). Under such a global spin

transformation, the x, y and z components of Uaij are rotated accordingly. The second

transformation leaves the spin operator unchanged. Thus, the quadratic Hamiltonian has

an emergent local SU(2) gauge redundancy, as such a transformation does not change

the physical spin operators.

A gauge transformation Wi, while leaving the physical problem unchanged, may (sig-

nificantly) change the quadratic Hamiltonian. However, the Hamiltonian is invariant

under a particular subset of these transformations, called the invariant gauge group

(IGG).

3.2 Projective Symmetry Group

3.2.1 Definition

With this large amount of gauge redundancy, it can be difficult to tell if two quadratic

Hamiltonians are related through some (non-trivial) gauge transformation. These ansatze

leave no local order parameters with which to write down a description of these states. In

particular, this is an issue when considering symmetry properties of these Hamiltonians,

since a symmetry operation may take a state to a different gauge, but make no physical

difference.

To determine which states are allowed under a symmetry group SG, we define the

projective symmetry group (PSG) for SG, which enforces particular gauge choices to

construct states obeying the physical symmetry of SG. The PSG characterizes the possible

distinct sets of gauge transformations GS for each operation S ∈ SG, where

(GSS)Uij(GSS)† = Uij. (3.7)

The PSG itself is defined as GSS for S ∈ SG. We note that (3.7) implies that GS

is defined only up to an element of the IGG. A particular set GS characterizes a

corresponding HQ and hence the mean field ground state. Therefore, elements of the

IGG are the gauge transformations GI associated with the identity operation I of the

symmetry group SG. In other words, the symmetry group SG = PSG/IGG. One can

choose a gauge where the elements of the IGG are site-independent, namely GI(i) = ±Ifor Z2 transformations, GI(i) = eiθτ

3(θ ∈ [0, 2π)) for U(1) transformations, and GI(i) =

eiθn·~τ (n ∈ S2, θ ∈ [0, 2π)) for SU(2) transformations, characterizing Z2, U(1) or SU(2)

spin liquids, respectively. [175]

Determination of the allowed sets GS, and thereafter the ansatze HQ, comes from


the group multiplication table of SG. For every group multiplication rule written in the

formABC−1 = I, a similar constraint is put on the PSG, namely (GAA)(GBB)(GCC)−1 ∈IGG, a pure gauge transformation.

Each different PSG may correspond to a physically distinct quadratic Hamiltonian

obeying the symmetry group SG. These Hamiltonians possess topological order, which

distinguishes among these states, all of which have the same physical symmetry. [175,176]

3.2.2 Fluctuations and the Invariant Gauge Group

Here, we consider the low energy fluctuations about the mean-field solution. The gauge

redundancy of the spinon description (Eq. (3.6)) immediately suggests that the low

energy effective theory contains spinons minimally coupled to a lattice gauge field. The

structure of the gauge group can be determined from the structure of the mean field QSL

ansatz for HQ [176,177] in the following way. Each mean-field ansatz is characterized by

Uaij ∈ SU(2) on the bonds.

Let us consider the singlet-only case, Uaij = U0

ij. Taking the product of such Uij over

various closed loops (C) starting from a specific base site (i), we obtain the “flux” of the

Uij link fields which has the following form: [175]

WC(i) =∏

(jk)∈C

Ujk =∑

b=0,x,y,z

AbC(i)σb (3.8)

where AbC(i) are numbers specific to the loop and σa are the identity (b = 0) and Pauli

matrices (b = 1, 2, 3) in the gauge space. For U(1) QSLs, Ab(i) for different loops are

proportional to each other for the same b and at least one of Ab(i) is non-zero for b =

1, 2, 3. Then, it is possible to write [175]

U(1) QSL : WC(i) ∝ σ3 ∀C (3.9)

This means that there exists a gauge in which all Uij ∝ σ3 and hence HQ of the form

in Eq. (3.2) is invariant under all gauge transformation Fi → Fieiθzi σ

3where θi ∈ [0, 2π].

Thus the gauge group is U(1) and accordingly we have a U(1) QSL. [175]

For Z2 spin liquids, two or more loops based at the same site have AbC(i) that are

not proportional to each other, and a similar analysis as above shows that the gauge

transformation of the following form is only possible: Wi = ±1. [175] This is a Z2 gauge

group describing Z2 spin liquids.

For U(1) states, the gauge field fluctuations are gapless photonic excitation, due to

the coupling to the emergent U(1) gauge field. As a result, in two dimensions, gapped


U(1) spin liquids are unstable to a confinement transition due to instanton proliferation,

[178] and enters a conventional magnetically-ordered phase. In contrast, gauge field

fluctuations in Z2 spin liquids are gapped visons, and these states may remain in the

deconfined phase when spinons are gapped or gapless. [106]

Triplet Ansatze The above arguments can be extended to the case of the most general

triplet ansatz with a four-component spinon representation. [177] However, we can show

within this Fi basis that this PSG holds equally for the singlet and triplet mean-field

parameters.

We consider the symmetry operation Fi → VSFS(i), which consists of a global spin

rotation VS as well as a lattice symmetry operation i→ S(i). Then we have

H =∑ij

tr(σaFiU

aij → F †j

)→∑ij

tr(V †Sσ

aVsFS(i)UaijF†S(j)

). (3.10)

Now, V †σV rotates the vector of Pauli matrices, so we can re-write it as Rabσb. Here,

we can consider R is a 4× 4 matrix rotating the x, y and z-components, with σ0 = I,

R =

(I 0

0 R3D

), H →

∑ij

tr(RabσbFS(i)U

aijF†S(j)

). (3.11)

Next, we will re-write the Hamiltonian in order to determine the relationship between Uaij

matrices on the lattice. We will rotate the a-indices for both the Rabσb and FS(i)Uaij →

F †S(j) components, and transform the spatial indices to get

H →∑ij

tr(σaFi(R

−1)abU bS−1(i)S−1(j)F

†j

). (3.12)

We want the Hamiltonian to be equivalent up to a gauge transformation, which is

applied after the symmetry operation.

H →∑ij

tr(σaFiGS(i)(R−1)abU b

S−1(i)S−1(j)G†S(j)F †j

). (3.13)

Here, we aim to show that a single GS suffices for all singlet and triplet terms. This can

be achieved if we have the relation

Uaij = GS(i)(R−1)abU b

S−1(i)S−1(j)G†S(j) (3.14)


for all bonds related by the symmetry operation S.

To show that this is consistent, let’s write S = S2S1, where S, S1, S2 ∈ SG. We’ll first

perform the above procedure twice, acting first S1 to get

Uaij = GS1(i)GS2(S

−11 (i))(R−1

1 )ab(R−12 )bcU c

S−12 (S−1

1 (i)),S−12 (S−1

1 (j))G†S2

(S−11 (j))G†S1

(i)

= GS1(i)GS2(S−11 (i))(R−1

1 R−12 )abU b

S−1(i),S−1(j)G†S2

(S−11 (j))G†S1

(i). (3.15)

We note that the rotation matrix R = R2R1, so comparing the above expressions we

have

GS1(i)GS2(S−11 (i))(R−1)abU b

S−1(i),S−1(j)G†S2

(S−11 (j))G†S1

(i)

= GS(i)(R−1)abU bS−1(i),S−1(j)GS(j). (3.16)

Hence, GS1(i)GS2(S−11 (i)) = GS(i)Ωi for Ωi ∈ IGG, since by definition Ωi leaves the

quadratic Hamiltonian invariant. However, this is the same constraint as in the singlet-

only case; the main difference is that the determination of symmetry-related Uij (3.14)

must involve the rotation effects as well.

3.3 Translationally-Invariant Z2 Spin Liquid

To illustrate the concepts behind the PSG classification, we will study the simple case

of Z2 spin liquids where the symmetry group contains only two translations T1 and T2.

For concreteness, we’ll take T1 = x and T2 = y on the square lattice, and not enforce any

other symmetries (such as rotation or reflection) of the square lattice. We will also take

the spin liquid to be singlet: Uaij = U0

ij.

The multiplication table of the symmetry group SG can be encapsulated in the relation

T1T2T−11 T−1

2 = I (in this simple case, there is only one relation). The corresponding

relation for the GS (since G−1 = G†) is

GT1(i)T1GT2(i)T2T−11 G†T1(i)T

−12 G†T2(i) = η12

=GT1(i)GT2(T−11 (i))T1T2T

−11 T−1

2 G†T1(T−12 (i))G†T2(i)

=GT1(i)GT2(T−11 (i))G†T1(T

−12 (i))G†T2(i)

=⇒ G†T2(T−11 (i))G†T1(i)GT2(i)GT1(T

−12 (i)) = η12 = ±1. (3.17)

In order to solve this, we will choose a particular gauge in which to work, noting that


under a gauge transformation Wi, we have

GS(i)→ WiGS(i)W †S−1(i). (3.18)

Using this, we can first choose (for each value of x) a particular gauge transformation

along . . . , (x, y), (x, y + 1), . . . so that GT2(i) = I, the identity. We note that the

implementation this gauge transformation under periodic boundary conditions can be

subtle, so we will consider open boundary conditions for this example. [179]

Any further gauge transformations Wi that are equivalent if x at i is equivalent will

mean that, by (3.18), GT2(i) is unchanged. Then our condition reduces toG†T1(i)GT1(T−12 (i)) =

η12. We’ll perform another gauge fixing: for a given y, we’ll choose a transformation along

. . . , (x, y0), (x + 1, y0), . . . so that GT1(y = y0) = I. Now, since GT1 changes by η12

upon moving from (x, y) to (x, y + 1), we have that

GT1(x, y) = ηy12I, GT2(x, y) = I. (3.19)

Now, note that we have two different expression for possible PSGs GSS. One PSG

is given for η12 = I, the other for η12 = −I. With this done, we will look at the possible

mean-field parameters, which must satisfy the symmetry relation (3.14). Here, we have

the simplified condition

Uij = GS(i)US−1(i)S−1(j)G†S(j). (3.20)

Let’s look first at the nearest-neighbour bonds. For bonds along the x-direction,

both points always have the same y value, even after translation, so U(x,y),(x+1,y) =

U(x+1,y),(x+2,y) = U(x,y+1),(x+1,y+1). Uij for horizontal lines is invariant. In contrast, the

vertical bonds have both points differing by a T2 translation, so we have U(x,y),(x,y+1) =

U(x,y+1),(x,y+2) but U(x,y),(x,y+1) = η12U(x+1,y),(x+1,y+1). The resultant sign structure is

shown in Fig. 3.1.

From these relations among the Uij, we can build a quadratic Hamiltonian with the

symmetry of the square lattice and a Z2 IGG. We note that for particular choices of

parameters (as in some cases when Uij is uniform) the IGG has actually a larger U(1)

symmetry. For instance, constant magnitude real hopping with the signs of Fig. 3.1

describe a U(1) spin liquid. These two ansatze, η12 = ±1, are often referred to as “0-

flux” and “π-flux” ansatze, respectively – moving around a plaquette, a relative phase of

0 or π can be picked up.

With this, we conclude our general discussion of the structure of spin-liquid states.


Figure 3.1: Sign structure of Uij for a singlet Z2 ansatz with the translational symmetryof the square lattice. Solid lines have a sign of +1, while dashed lines have a sign of η12,+1 or −1 for the two different possible ansatze, respectively. Uij on horizontal bonds areuniform, while Uij on vertical bonds alternate sign when η12 = −1.

In the next two chapters, we present two different formalisms to study two different spin-

dimer systems, the aniostropically-coupled Ba3Cr2O8 and the ferromagnetically-coupled

Shastry-Suthland lattice (CuCl)LaNb2O7.

Chapter 4

Magnetic-Field-Induced

Bose-Einstein Condensation of

Triplons in Ba3Cr2O8

4.1 Introduction

In this chapter, we present a theory of the magnetic field-induced quantum phase tran-

sition discovered in the spin-dimer system Ba3Cr2O8, where Cr5+ carries an S = 12

moment (3d1). [180] Low temperature bulk susceptibility shows that this compound does

not have any magnetic long-range order down to 1.5 K in the absence of an external

magnetic field. [181, 182] When the external magnetic field H reaches Hc1 ∼ 12 T, a

field-induced transition to a magnetically ordered state occurs and a fully polarized state

arises at H > Hc2 ∼ 23 T. [180] In this compound, two neighbouring S = 12

Cr5+ ions

lying along the c direction form a singlet dimer. In the ab-plane, these dimer singlets are

coupled into triangular lattices, which are stacked along the c direction (see Fig. 4.1).

According to recent elastic and inelastic neutron scattering measurements, [54] Ba3Cr2O8

is an excellent model system for weakly coupled S = 12

quantum spin dimers, featuring

strong intra-dimer spin coupling of J0 = 2.38 meV and weak inter-dimer couplings less

than 0.52 meV. [54] Because of the orbital degeneracy of the Cr5+ ion, there is a struc-

tural transition around 70 K via a Jahn-Teller distortion, relieving the frustration. As a

consequence, spatially anisotropic inter-dimer couplings arise. The relative orientations

of the anisotropic inter-dimer couplings are described in Fig. 4.2. It was confirmed that

the magnetically ordered state has a commensurate and collinear transverse spin compo-

nent for Hc1 < H < Hc2. [180] This is in contrast to the case of Ba3Mn2O8 with orbitally

30

Chapter 4. Bose-Einstein Condensation of Triplons 31

Figure 4.1: Schematic diagram showing two neighbouring triangular lattice planes ofdimers in Ba3Cr2O8. Two primitive lattice vectors a and b are shown in the lower plane.The third primitive lattice vector c′ connects central dimers in the neighbouring planes.We set the vertical distance between neighbouring planes by c/3. Here we use JP toindicate the in-plane nearest-neighbour inter-dimer spin coupling. J1 (J2) denotes thenearest-neighbour (next nearest-neighbour) inter-plane dimer spin coupling.

non-degenerate S=1 Mn5+ ions (3d2), where the geometric frustration and single-ion

anisotropy lead to incommensurate spiral order upon triplon condensation. [183]

We first consider the Heisenberg spin Hamiltonian using the spin exchange couplings

determined by inelastic neutron scattering measurements. [54] Applying the bond opera-

tor formalism, we obtain the dispersion of the lowest energy triplet excitations. [165,184]

We confirm that the Hamiltonian written in terms of bond operators at the quadratic

level, neglecting singlet fluctuation, leads to the same triplon dispersion as determined

in experiment in Ref. [54].

We then use the Hartree-Fock-Popov (HFP) approximation combined with the realis-

tic triplon dispersion to interpret two different experimental data sets, those of M. Kofu

et al. in Ref. [180] and A. Aczel et al. in Ref. [185]. In particular, we would like to under-

stand to what extent Ba3Cr2O8 is a good candidate for the BEC of triplons in compari-

son to other three-dimensionally coupled spin-dimer systems such as TlCuCl3. [186–188]

Within the HFP analysis, we determine the effective inter-triplet repulsion U and the

zero-field spin gap ∆ in Ba3Cr2O8. It turns out that the strength of the effective repulsive

interaction U between triplons in Ba3Cr2O8 is an order of magnitude smaller than that

of TlCuCl3, [187] and smaller than the bandwidth of the triplons as well. This suggests

that the system is indeed in the regime where the kinetic energy dominates. In addition,

the shape of the dispersion near the triplon band minimum results in the large effective

mass of triplons in Ba3Cr2O8. As a result, the relation [Hc(T ) −Hc(0)] ∝ T 3/2 in three

dimensions for quadratic triplon dispersion works only at T < 0.06 K, while it works

at T < 0.6 K in TlCuCl3. The HFP approach is used to describe other experimental


measurements such as the longitudinal and transverse staggered magnetizations and the

heat capacity. Despite the simplicity of the theoretical approach, the HFP approximation

is found to explain these physical properties even quantitatively.

The rest of the chapter is organized as follows. In Sec. 4.2 we use the bond operator

approach to obtain the triplet dispersion from the microscopic Hamiltonian. Theoretical

description of the triplon Bose-Einstein condensation within the HFP approximation is

discussed in detail in Sec. 4.3. In Sec. 4.4, we apply the HFP approach to explain

the experimental data and draw the phase diagram in the plane of Hc versus T. The

HFP approach is applied to describe the heat capacity in Sec. 4.5 and the magnetization

measurements in Sec. 4.6. Finally, in Sec. 4.7, we summarize our results, and discuss

possible limitations and extensions of the current work.

4.2 Triplon Dispersion via Bond-Operator Approach

Taking into account the dimerized nature of the ground state, we consider a Heisenberg

spin Hamiltonian given by

H =∑i

J0Si1 · Si2 +∑i

9∑m=1

JmSi,1 · Si+δrm,1 (4.1)

where i denotes locations of dimers. J0 indicates the intra-dimer spin coupling while inter-

dimer couplings are described by Jm (m = 1, . . . , 9). The two spins within a dimer are

labelled by the subscripts 1 and 2. Since a dimer is made of two neighbouring spins, the

interaction between two dimers contains four different spin couplings. In describing the

interaction between two neighbouring dimers lying at i and j, we assume that Si1 ·Sj1 =

Si2 · Sj2 = −Si1 · Sj2 = −Si2 · Sj1. In terms of the bond operators to be introduced in

this section, this approximation is equivalent to neglecting interaction between triplets;

this is valid when the singlet ground state is robust, with small triplet density. Under

this condition, all of the potential spin-spin Heisenberg interactions between two dimers

reduce to a single effective interaction Jm. Henceforth, this is referred to as the inter-

dimer coupling.

We include nearest neighbour interactions between dimers on the same plane. Be-

tween adjacent planes, we include interactions between first and second-nearest neigh-

bouring dimers. The tetrahedrally-coordinated 3d1 electron in Cr5+ has eg orbital de-

generacy and undergoes Jahn-Teller distortion. This gives rise to spatially anisotropic

inter-dimer interactions. [54] The in–plane projection of the anisotropic inter-dimer in-


m Jm (meV) ∆rm

1 J ′P = 0.1 a2 J ′′P = 0.07 b3 J ′′′P = −0.52 −a− b4 J ′1 = 0.08 2a/3 + b/3 + c/35 J ′′1 = −0.15 −a/3 + b/3 + c/36 J ′′′1 = 0.1 −a/3− 2b/3 + c/37 J ′2 = 0.04 −4a/3− 2b/3 + c/38 J ′′2 = 0.1 2a/3− 2b/3 + c/39 J ′′′2 = 0.09 2a/3 + 4b/3 + c/3

Table 4.1: Strength Jm and relative distance ∆rm for the nine different anisotropic inter-dimer couplings as shown in Fig. 4.2.

teractions are described in Fig. 4.2.

The intra-dimer coupling has a strength of J0 = 2.38 meV. Table 4.1 displays the

values and directions of the couplings to nearby dimers Jm, depicted in detail in Fig.

4.2. These strengths have been determined in Ref. [54] by fitting a random-phase ap-

proximation (RPA) dispersion to the triplon dispersion measured by inelastic neutron

scattering.

Figure 4.2: In-plane projection of the strengths and directions of the different anisotropicinter-dimer couplings considered in this work. The couplings JP are in-plane nearestneighbour couplings, which are represented by dashed lines. J1 (J2) is the first (second)nearest-neighbour inter-plane coupling.

Since the intra-dimer exchange interaction J0 dominates all the other inter-dimer

couplings in Ba3Cr2O8, it is natural to take advantage of the bond-operator slave-particle

representation of the singlet and triplet dimer states discussed in Sec. 2.2.1.


We will begin by assuming there is a large occupation of condensed singlets. Our

analysis considers the low triplon density limit, so we retain terms up to quadratic order

in the t operators. The interaction terms neglected by this will be phenomenologically re-

introduced in Sec. 4.3. To impose the single-particle constraint on average we introduce

a site-dependent chemical potential µi. This adds to the Hamiltonian the constraint term∑i

µi

(1− s†isi − t

†iαtiα

). (4.2)

To make a tractable analysis, the above constraint is enforced in a mean-field man-

ner, taking µi = µ, on average over the entire lattice. In the momentum represen-

tation, this gives |s|2 +∫

d3k(2π)3

t†kαtkα = 1. At T = 0, the values of µ and s are de-

termined from the saddle point condition. First,⟨∂H∂µ

⟩= 0 enforces the constraint

0 =∑

i

(1−

⟨s†isi

⟩−⟨t†iαtiα

⟩). Second, the condition

⟨∂H∂s

⟩= 0 minimizes the ground-

state energy with respect to the singlet density.

Our bond-operator Hamiltonian obtains a quadratic form in the momentum space:

HBO = Ndε0 +H0 +H±. (4.3)

Here Nd denotes the number of dimers on the lattice. The t0 triplons interact with them-

selves but not the other triplon species. They contribute with the quadratic Hamiltonian

H0 =1

2

∑k

(t†k0 t−k0

)(Ak Bk

B∗k A∗k

)(tk0

t†−k0

),

Ak =J0

4− µ+Bk, Bk = −|s|2

∑m

Jm cos (k ·∆rm). (4.4)

H0 can be diagonalized by the Bogoliubov transformation γk0 = uktk0 + vkt†−k0, with

quasiparticle energy [189]

ωk =√A2

k −B2k =

√(J0

4− µ)2 + 2(

J0

4− µ)Bk. (4.5)

Neither t0 nor γ0 are subject to Zeeman splitting by the external field. However, the

t+ and t− triplons are split. Furthermore, they interact with each other. The resultant


quadratic Hamiltonian H± is

H± =1

2

∑k

Ψ†k

Ak − h 0 0 Bk

0 Ak + h Bk 0

0 Bk Ak − h 0

Bk 0 0 Ak + h

Ψk, (4.6)

where h = gµBH and Ψ†k=(t†k+ t†k− t−k+ t−k−). H± is diagonalized into quasiparticles

γk± with energy ωk ∓ h, where

γ†k+ = uk+t†k+ + vk+t−k−,

γ†k− = uk−t†k− + vk−t−k+, (4.7)

uk+ =Bk√

2ωk(Ak − ωk), vk+ =

Ak − ωk√2ωk(Ak − ωk)

,

uk− =Ak + ωk√

2ωk(Ak + ωk), vk− =

Bk√2ωk(Ak + ωk)

. (4.8)

The γ+ triplon, with spin along the quantization axis, will be the focus of the HFP

treatment, since it interacts with no other species and lowers its energy from the Zeeman

splitting. Finally, the constant part of the Hamiltonian is given by

ε0 = −3

4J0|s|2 + µ(1− s2)− 3

2Nd

∑k

Ak. (4.9)

In the limit of vanishing inter-dimer interactions (in this case, all J ′, J ′′, J ′′′ → 0), the

saddle-point solution gives s = 1 and µ = −34J0. This limit serves as a good starting

point in Ba3Cr2O8, where J0 is much larger than the inter-dimer couplings. Furthermore,

with these values of s and µ, the triplon dispersion matches the RPA form fitted to

experimental values in Ref. [54]. Solving the saddle-point conditions with the couplings

in Table 4.1 taken as bare values, we find s = 0.992 and µ = −0.775J0, showing a high

degree of dimerization. We will take s to be real henceforth, without loss of generality.

The bare couplings will be slightly renormalized (compared to the s = 1, µ = −3J0/4

case) as a result. In particular, J0 → J0 − ∆µ and Jm → |s|2Jm for m = 1, . . . , 9.

However, we can take these renormalized couplings to have the values in Table 4.1, since

only the final dispersion will be used in our Hartree-Fock calculation. We furthermore

assume that the dispersion is not temperature dependent within the low-temperature


regime considered.

4.3 Hartree-Fock Effective Hamiltonian

We turn our focus to the γ+ quasiparticle; being the field-aligned quasiparticle, it will con-

dense with sufficiently large Zeeman splitting. We consider field strengths large enough

that we may ignore the higher-energy γ0 and γ− quasiparticles. The typical splitting is of

energy gµbHc(0) ∼= 15.4 K. This scale is significantly larger than the highest temperature

(around 2.7 K) where the BEC transition occurs. [185] Consequently, ignoring terms in

the Hamiltonian with γ0 and γ− is a safe approximation to make in this external field

regime.

The Hamiltonian we take for the b ≡ γ+ triplons is H = HK +HU , which is the sum

of the kinetic and inter-triplon interaction terms. Here,

HK =∑k

(εk − µ) b†kbk, (4.10)

HU =1

2Nd

∑k,k′,q

Uqb†kb†k′bk+qbk′−q, (4.11)

where µ = gµBH − ∆ is the chemical potential and ∆ is the zero-field gap (1.37 meV

from the bond-operator theory). εk + ∆ is the zero-field dispersion with εk determined

from the bond-operator theory. The quartic terms from the bond-operator theory give

rise to interactions between the b triplons. Combined with the interaction from the hard-

core constraint, this gives an approximate form for Uq. However, in the low-temperature

limit where the excited triplons lie near the band minimum at Q, we may approximate

Uq ≈ UQ as a constant, U . The value of the interaction parameter U will be determined

from a fit to the experimental data.

The condensate will form at the dispersion minimum Q = 12(u + v). [180] We define

the reciprocal lattice vectors u, v, and w in the conventional way. For example, u =

2π b×ca·(b×c)

. We follow the Hartree-Fock-Popov approach of Ref. [187] by condensing the

triplons at Q: b†Q, bQ →√Ndnc, where nc is the condensate density (the condensed

boson fraction per dimer). Introducing the summation∑′, which excludes any terms

containing creation or annihilation operators at momentum Q, we decompose HU as


follows:

HU =U

2Nd

N2c +

UNc

Nd

∑q

′

bqb−q + b†−qb

†q

2+ 2b†qbq

+U√Nc

Nd

∑k,q

′b†kbk+qbQ−q + h.c.

+

U

2Nd

∑k,k′,q

′b†kb†k′bk+qbk′−q, (4.12)

using the fact that 2Q is a reciprocal lattice vector, so that b−k+Q = b−(k+Q). Performing

a mean-field quadratic decoupling of the quartic terms yields the following mean-field

quadratic Hamiltonian:

HMF = E0 +∑k

′εkb†kbk +

Unc2

∑q

′bqb−q + b†−qb†q, (4.13)

where

E0 = −µnc + UNd

(n2c

2− (n− nc)2

), εk = εk − µ, µ = gµBH −∆− 2Un. (4.14)

This decoupling is valid so long as the triplon densities⟨b†kbk

⟩are small. In the non-

condensed (normal) phase, the Hamiltonian is already diagonalized. The triplon density

is given by the Bose distribution function, and must be determined self-consistently:

n =

∫d3k

(2π)3fB(εk). (4.15)

In the condensed phase, we must perform another Bogoliubov transformation, which

leads to the following diagonalized Hamiltonian:

HMF =∑k

′Ek

(ϕ†kϕk +

1

2

)− 1

2

∑k

′εk + E0, (4.16)

where

Ek =√ε2k − (Unc)2, ϕk = ukbk + vkb

†−k,

uk =

√εk

2Ek

+1

2, vk =

√εk

2Ek

− 1

2. (4.17)


The number of uncondensed triplons is

n− nc =

∫d3k

(2π)3

[εkEk

(fB(Ek) +

1

2

)]− 1

2. (4.18)

For the final ϕ quasiparticles to be condensed at Q, they must be gapless. This

constrains the effective chemical potential as µ = −Unc, so that the field in the condensed

phase is given by

gµBH = ∆ + U (2n− nc) . (4.19)

Between these two phases is the transition curve Hc(T ) where triplons begin to con-

dense. The b triplons are gapped in the non-condensed phase. However, at the transition

point to the condensed phase, they become gapless. This constrains the effective chemical

potential as µ = 0 to give the critical field

Hc(T ) =∆

gµB+

2U

gµBncr(T ). (4.20)

Since εk = εk in this case, we determine the critical boson density at the transition, ncr,

by the integral

ncr =

∫d3k

(2π)3fB(εk). (4.21)

4.4 Critical Density Phase Diagram

In the HFP approach the critical field Hc depends on the critical boson density ncr linearly

as shown in Eq.(4.20). A linear fit of Hc to ncr determines the interaction parameter

U from the slope. In addition, the zero-temperature critical field Hc(0) gives another

estimate for the gap ∆. With given experimental data (Hc, T ), we obtain (Hc, ncr) pairs

making use of the Eq.(4.21). In the low-temperature, low-density regime where HFP

approach is valid, we expect Hc to be linear in ncr.

We present three different fits of Hc to ncr. These are based on the two different

experimental data sets obtained from Ref. [180] and Ref. [185]. Lines of best fit neglect

the high-temperature (density) regimes where Hc loses linearity in ncr. A linear relation

between Hc and ncr is achieved for the case of H parallel to the c-axis, as shown in

Fig. 4.3 From the linear relation, we estimate the interaction constant U ∼= 6.4 K and

zero-field spin gap ∆ ∼= 1.35 meV from the data of M. Kofu et al. in Ref. [180], while


12

13

14

15

16

17

18

19

0 0.05 0.1 0.15 0.2H

c (

T)

ncr

Data (M. Kofu et al.)Low-density best fit

Data (A. Aczel et al.)Low-density best fit

Figure 4.3: The critical field Hc as a function of critical density ncr with the applied fieldH parallel to the c-axis. ncr(T ) is calculated, using the HFP approach, from the givenexperimental temperatures. A linear fit of Hc to ncr is performed in a low-density range.The experimental data is from M. Kofu et al. in Ref. [180] for diamonds, and from A.Aczel et al. in Ref. [185] for circles.

the data of A. Aczel et al. in Ref. [185] gives an estimate of U ≈ 8.7 K and ∆ ∼= 1.37

meV, which is very close to the value obtained from the bond-operator approach. Since

this experimental data shows smaller deviations from the linear fit, over a larger range

of temperatures, we use the estimate of U ≈ 8.7 K and ∆ ∼= 1.37 meV in the following

analysis. In Fig. 4.4 we display the low temperature phase diagram Hc(T ), which is

again obtained using the data of Ref. [185].

12

12.5

13

13.5

14

14.5

0 0.5 1 1.5 2

Hc (

T)

T (K)

Best Fit Critical FieldExperimental Data

Figure 4.4: Phase diagram giving the critical field Hc as a function of temperature.Experimental data is from A. Aczel et al. in Ref. [185], with applied field H parallel tothe c-axis. The solid line shows the theoretical result obtained from the HFP approachusing the linear fit displayed in Fig. 4.3.

On the other hand, the data forH perpendicular to the c-axis features low-temperature


12

13

14

15

16

17

0 0.05 0.1 0.15 0.2 0.25H

c (

T)

ncr

DataBest Fit

0 0.02 0.04

12

12.2

Figure 4.5: The critical field Hc as a function of critical density ncr with the applied fieldH perpendicular to the c-axis. Here we use the data in Ref. [180] . Inset: discrepancybetween experiment and the best-fit line in the small ncr regime.

behavior inconsistent with the general linear trend as shown in Fig. 4.5 and its inset.

We think that the existence of a Dzyaloshinsky-Moriya (DM) interaction is one possi-

ble explanation of this low-temperature discrepancy from the linear behavior. Further

discussion in this direction is given in Sec. 4.7.

Since a full-dispersion treatment successfully reproduces the phase diagram for the

dimerized spin system TlCuCl3, [187] it is instructive to use it in comparison with the

HFP approach for Ba3Cr2O8. Triplons in Ba3Cr2O8 have a smaller self-interaction con-

stant, of U ≈ 8.7 K, compared to TlCuCl3, which has U ≈ 320 K. [187] However, triplon

densities are significantly higher in Ba3Cr2O8 than in TlCuCl3, by over an order of mag-

nitude. This makes the Hartree-Fock critical field shift Uncr greater in Ba3Cr2O8 than

in TlCuCl3. In Fig. 4.6 we plot Uncr in these two systems with varying temperature.

Within the Hartree-Fock-Popov approach, the term −2Un acts as a shift in the effective

chemical potential as shown in Eq.(4.14). A decrease in U will increase the effective

chemical potential, causing an increase in the triplon density as seen with Ba3Cr2O8.

The shape of the dispersion affects the temperature range in which the power-law

behavior of ncr ∝ T32 is satisfied. At very low temperature, the quadratic approximation

to the minimum of the dispersion becomes increasingly accurate. As T → 0, the quadratic

dispersion εk = k2

2myields ncr ∝ T

32 by evaluating Eq.(4.21) exactly [186] to give

limT→0

ncr(T ) =ζ 3

2

2

(Tm

2π

) 32

. (4.22)

In Ba3Cr2O8, a low-temperature T32 fit deviates from the full dispersion critical density


0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4U

ncr (K

)

T (K)

Ba3Cr2O8

TlCuCl3

Figure 4.6: Comparison of overall interaction energy scale Uncr as a function of tempera-ture between Ba3Cr2O8 (solid line) and TlCuCl3 (dashed line) systems. ncr is the densityof triplons at the condensate transition and U the inter-triplon interaction strength.

ncr at around 0.06 K as shown in Fig. 4.7. This is an order of magnitude smaller than

for TlCuCl3, where the T32 behavior persists up to around 0.6 K as described in Fig. 4.8.

The smaller temperature scale of Ba3Cr2O8 is to be expected from the smaller triplon

bandwidth, represented by the large effective mass near the dispersion minimum. From

the power-law fits to Eq.(4.22) in Fig. 4.7 and Fig. 4.8 we find that 1/m ∼= 1.36 K (43.6

K) for Ba3Cr2O8 (TlCuCl3 [187]) showing the narrower bandwidth of Ba3Cr2O8. Here

we have set ~2/kB = 1.

4.5 Specific Heat

We apply the HFP approach to explain the specific heat data measured by M. Kofu et

al. [180] To determine the magnetic contribution to the specific heat, we first find the

expectation value of the energy per dimer. After condensing the triplons at momentum

Q, the diagonalized mean-field Hamiltonian contains only number operators of thermally

distributed bosonic quasiparticles (see Eqs. (4.13) and (4.16)). By differentiating the

energy with respect to temperature, we find the specific heat per dimer. In the normal


0

0.0005

0.001

0.0015

0.002

0.0025

0 0.02 0.04 0.06 0.08 0.1

ncr

T (K)

Low-T FitCalculated ncr

Figure 4.7: Critical density ncr of Ba3Cr2O8 calculated from the HFP theory with thefull dispersion (cross points). The solid line is T

32 power-law fit coming from the simple

quadratic dispersion. The points obtained from the full dispersion begin to deviate fromthe power-law fit around 0.06 K.

0

5e-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0 0.2 0.4 0.6 0.8 1

ncr

T (K)

Low-T FitCalculated ncr

Figure 4.8: Same plot as Fig. 4.7 but for the compound TlCuCl3. Note that the resultsobtained from the full dispersion follow the power-law fit up to 0.6 K.


phase,

〈E〉Nd

= −Un2 +

∫d3k

(2π)3εkfB(εk),

CVNdkB

= −β∫

d3k

(2π)3ε2k∂fB∂εk

+ 2U∂n

∂T

∫d3k

(2π)3εk∂fB∂εk

, (4.23)

∂n

∂T= −β

∫d3k

(2π)3εk

∂fB∂εk

1− 2U∫

d3k(2π)3

∂fB∂εk

.

However, in the condensed phase, we have

〈E〉Nd

= E0 −1

2

∫d3k

(2π)3εk +

∫d3k

(2π)3Ek

(fB(Ek) +

1

2

)CVNdkB

= −β∫

d3k

(2π)3E2

k

∂fB∂Ek

+

2U∂n

∂T

[nc − n−

1

2+

∫d3k

(2π)3

εkEk

(fB(Ek) +

1

2+ Ek

∂fB∂Ek

)], (4.24)

∂n

∂T=

β∫

d3k(2π)3

εk∂fB∂Ek

1− 2U∫

d3k(2π)3

εkE2

k

[−εk ∂fB∂Ek

+ µEk

(fB(Ek) + 1

2

)] .Non-magnetic contributions to the specific heat, such as phonon contribution, will not

change appreciably with the applied field. The difference CV (H)− CV (0) thus captures

the heat capacity contribution from triplons. Currently, there exists no zero-field specific

heat data, preventing proper quantitative comparison.

However, we may still make a comparison, up to an overall scale difference, between

the theoretical specific heat and experimental heat capacity data. Fig. 4.9 shows the

calculated magnetic contribution with the experimentally determined heat capacity in

Ref. [180]. The relative scale is chosen to best show similarity in the peak shape for fields

close to the zero-temperature critical field. Despite the scale difference and non-triplon

contribution, the theoretical result still captures the peak at the critical temperature.

However, the drop in heat capacity is overestimated. Furthermore, it is discontinuous,

which can be considered as an artifact of the HFP approximation. [187]

4.6 Magnetization

When H > Hc, γ+ bosons condense, leading to the macroscopic occupation of the triplet

states with the momentum corresponding to the dispersion minimum. The ground state

wave function is then given by the coherent superposition of the singlet and the Sz=1


Figure 4.9: Comparison of experimental heat capacity to theoretical specific heat perdimer as a function of temperature. Comparisons are made for external fields of 12.5T, 12.7 T, and 13 T. The experimental data is from Ref. [180] with the applied field Hparallel to the c-axis.

triplet states. [39,165] The density of the condensate determines the magnetization along

the z-direction. In addition, the condensate supports the staggered magnetization which

has finite transverse components 〈Six〉 and 〈Siy〉 breaking the continuous U(1) rotational

symmetry around the z-direction.

To determine the magnetic ordering, we begin by rewriting the spin operators in terms

of the t0, t− and t+ operators. Using the bond operator representation we obtain the

following relations:

(S1 + S2)α = −iεαβγt†βtγ, (S1 − S2)α = s†tα + t†αs, (4.25)

for α ∈ x, y, z. Due to Zeeman splitting, the tz = γ0 triplets are negligible, and we

find that 〈(S1 + S2)x〉 = 〈(S1 + S2)y〉 = 〈(S1 − S2)z〉 = 0. As γ− are similarly negligible,

we expand the rest of the triplet t± operators in terms of the γ± operators, and ignore

the terms with γ−, which do not contribute to expectation values. The average spin

component per dimer along the field direction, which is nothing but the fraction of

aligned quasiparticles n, is given by

〈(S1 + S2)z〉 =⟨t†+t+ − t

†−t−

⟩=

1

Nd

∑k

⟨γ†k+γk+

⟩(u2

k− − v2k−) = n. (4.26)

Since we have condensed singlet s, the staggered component of the spin becomes (using


u−k = uk)

〈Si1x − Si2x〉 =s√2

⟨t†i+ + t†i− + ti+ + ti−

⟩=

s√2Nd

∑k

eik·ri⟨t†k+ + t†k−

⟩+ h.c.

=sgn(B)

√2s√

Nd

(uQ− − vQ−)<(eiQ·riΓQ). (4.27)

Here ΓQ =⟨γ†Q+

⟩with |ΓQ|2 = nc. Without loss of generality, we fix the overall

phase by taking ΓQ to be real. Only the coherent condensate contributes to the transverse

magnetization. Similarly, the y-component comes from the imaginary component of the

condensate,

〈Si1y − Si2y〉 =sgn(BQ)

√2s√

Nd

(uQ− − vQ−)=(eiQ·riΓQ). (4.28)

The transverse spin component thus is spatially modulated by the condensate wavevector

Q. The transverse magnetization per dimer can be written as [184]

Mxy ≡1

Nd

∑i

eiQ·ri 〈Si1x − Si2x〉 =sgn(BQ)

√2s

√Nd

3 (uQ− − vQ−)∑i

eiQ·ri cos(Q · ri)ΓQ

=sgn(BQ)s√

2Nd

(uQ− − vQ−) ΓQ. (4.29)

The square of the transverse magnetization per Cr5+ ion is then

M2⊥ = (gµBMxy)

2 = g2µ2B

s2Γ2Q

8Nd

(ωQ + AQ −BQ)2

2ωQ(AQ + ωQ)

= g2µ2B

s2nc4

AQ −BQ

2ωQ

= s2ncJ0g

2

8∆µ2B. (4.30)

Having neglected the γ0 and γ− triplons, we estimate s2 ∼= 1−n, using the overall triplet

boson constraint. The total and condensed triplet densities, n and nc, are determined by

solving Eq.(4.18) and Eq.(4.19) self-consistently.

The transverse magnetization has been measured by the elastic neutron scattering

experiments. [180] The applied field is perpendicular to the c-axis. Fig. 4.10 compares

theoretical squared perpendicular magnetization at T = 0.2 K to the experimental results.

[180] Deviation from the experiment occurs most prominently in the critical field. This

is caused by discrepancy between the linear fit Hc ∝ ncr(T ) and the experimental critical

field Hc(T ). However, the shape of the magnetization curve past the critical field is

properly reproduced. This can be seen in Fig. 4.10, where the theoretical result has also


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

11.5 12 12.5 13 13.5 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

M2 ⊥ (

µB2 /

Cr)

(th

eo

retica

l)

M2 ⊥ (

µB2 /

Cr)

(e

xp

erim

en

tal)

H (T)

Theoretical ResultTheory (Translated)Experimental Data

Figure 4.10: Comparison of experimental and theoretical perpendicular magnetizationssquared (solid line) at T = 0.2 K. Perpendicular magnetization is defined in Eq.(4.29).The experimental data is from Ref. [180], with the applied field H perpendicular to thec-axis. Also shown (dashed line) is the theoretical result translated by 0.07 T to matchthe experimental critical-field behavior. The scale of the theoretical result is 1.13 largerthan that of the data.

been translated to match the experimental critical field. The resulting shape matches

over the entire range of fields, with the theoretical magnetization larger by a factor of

1.13. The magnetization also jumps slightly at the critical field. Like the discontinuity

in specific heat, this is an artifact of the HFP treatment. [190]

The parallel magnetization has been measured as a function of applied field (both

parallel and perpendicular to the c-axis) at the condensate transition. [180] Fig. 4.11

gives the HFP result for H perpendicular to the c-axis, with the magnetization per

dimer M‖ = g2µBn. The saturation field, where all spins are aligned with the field, is

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

11 12 13 14 15 16 17 18

M (

µB/C

r)

H (T)

Hc

Hs

Figure 4.11: Parallel magnetization per Cr atom, as a function of applied field perpen-dicular to the c-axis, at T = 0.4 K. The result is found from the triplon density asdetermined by the HFP approach. Critical field Hc and saturation field Hs are indicated.


severely underestimated by the HFP result of 18 T. Experimentally it is found between

23 T and 24 T, from the derivative of magnetization ∂M∂H

. [180] Underestimation occurs

because the triplon density in the HFP approach grows too quickly with increasing field.

4.7 Discussion

We have applied the HFP approach to understand the triplon BEC in Ba3Cr2O8, using

the full dispersion of the triplons measured in the recent neutron scattering experiments

(which is recast in the form of a bond-operator representation of a Heisenberg model).

We investigated the temperature range where the HFP approach is valid with the full

dispersion, and also located the temperature where the quadratic approximation of the

dispersion breaks down. Using this approach, we computed the transverse magnetization

and specific heat that are favorably compared to available experimental data. Our results

show that the BEC picture overall works reasonably well for Ba3Cr2O8.

In the much-studied three-dimensionally-coupled spin-dimer system TlCuCl3, the

triplon band width W ∼ 87 K and the effective interaction U ∼ 340 K within the

HFP analysis. [187] In contrast, our analysis leads to W ∼ 21 K and U ∼ 8.7 K in

Ba3Cr2O8. Thus it may appear that the HFP would work better for Ba3Cr2O8 because

of the smaller U/W ratio. However, smaller U in Ba3Cr2O8 results in a larger critical

triplon density ncr ∼ 0.1 compared to ncr ∼ 0.002 in TlCuCl3, making the dilute triplon

density approximation less valid. In the end, the combined effect in the form of the

HFP correction to the critical field Hc(T ), Uncr, turns out to be bigger for the case of

Ba3Cr2O8. This means that the temperature range where the HFP approach is valid is

more limited in the case of Ba3Cr2O8. Indeed, it is found that the HFP works up to 8K

in TlCuCl3 while it fits the data up to 2K at best in Ba3Cr2O8.

The triplon dispersion in Ba3Cr2O8 is more flat (or effective mass is larger) than

that of TlCuCl3, which leads to a smaller window of temperatures where the quadratic-

dispersion approximation is valid. This is seen in how the relation [Hc(T )−Hc(0)] ∝ T 3/2

reproduces the phase diagram for T < 0.06 K for Ba3Cr2O8, but for T < 0.6 K in

TlCuCl3.

A useful way to improve the HFP results may be to introduce the hard-core constraint

among the triplons. The so-called Bruckner bond operator approach [49, 191] achieves

this by introducing an infinite on-site triplon repulsion by

HC = V∑iαβ

t†iαt†iβtiαtiβ (4.31)


as V →∞. In the low-density limit, this hard-core interaction may be treated exactly by

a summation of ladder diagrams at the one-loop level in the self-energy. This approach,

when generalized to finite temperature, should lower the triplon densities and extend the

region where low-density approximations are valid. This may be a useful future extension

of our work.

Recent ESR measurements indicate the existence of singlet-triplet mixing in the

ground state of Ba3Cr2O8. [180] In the ground state, singlets mix with t0 for H ⊥ c,

and with t± for H ‖ c. This mixing points to the existence of a Dzyaloshinsky-Moriya

(DM) interaction of the form Dij · Si × Sj, with Dij perpendicular to the c-axis. Since

it breaks the U(1) symmetry of the Heisenberg Hamiltonian, the system is no longer

described by a BEC transition. The result is that triplons are gapped and always con-

densed to some extent, turning the transition into a crossover region. [190, 192] Below

the temperature scale of the DM interaction, then, we expect that a simple BEC picture

of triplons will no longer be sufficient. This could explain, for instance, the nonlinearity

of critical field Hc in critical density ncr at low temperatures for H perpendicular to the

c-axis. An understanding of the magnitude and direction of the Dij vector is important

for a proper and full description of Ba3Cr2O8, especially at very low temperatures.

Chapter 5

Ferromagnetically-coupled dimers on

the distorted Shastry-Sutherland

lattice

5.1 Introduction

Among the frustrated materials mentioned in Chapter 1, the S = 12

layered copper

oxyhalides (CuX)An−1BnO3n+1 comprise a family with interesting behaviour reminiscent

of spin-dimer systems, as mentioned in Sec. 1.2. In this family, each magnetic CuX layer

(X=Cl,Br) forms a two-dimensional structure, separated by non-magnetic ions. In each

layer, the Cu2+ ions form a square lattice of S = 12

spins, and a competition between

ferromagnetic and antiferromagnetic interactions is anticipated from small Curie-Weiss

temperatures. The Heisenberg model up to next-nearest-neighbour (with ferromagnetic

J1 and antiferromagnetic J2) originally proposed for this family does not seem to exhibit

a spin gap or a magnetization plateau. [193]

The underlying dimer structure over fourth-neighbour bonds revealed in (CuCl)LaNb2O7

offers a new viewpoint for understanding this behaviour. [61] Due to distortion of Cu-Cl

bonds, the symmetry of the square lattice is lost and exchange paths become non-trivial.

The resultant structure of the most important exchange interactions is seen in Fig. 5.1(a).

These interactions are topologically equivalent to the distorted Shastry-Sutherland lat-

tice [194] as shown in Fig. 5.1(b). Here, the dimers are coupled by two ferromagnetic

exchanges, Jx and Jy.1

1Another coupled-dimer model for (CuCl)LaNb2O7 has been proposed in A. Tsirlin and H. Rosner,[195] based on a different crystal structure. In this model, the arrangement of dimers are similar to thoseof Ref. [61], and the inter-dimer couplings are non-frustrating.

49

Chapter 5. Dimers on the distorted Shastry-Sutherland lattice 50

Figure 5.1: (a) Schematic picture of a single magnetic layer of (CuCl)LaNb2O7. Thicksolid (J), thin solid (Jx), and broken (Jy) lines indicate the three major couplings revealedin the electronic structure calculation of Ref. [61]. Smaller interactions J2b and J ′4 shownby dotted lines are also considered in Ref. [158]. (b) Topologically equivalent Shastry-Sutherland picture of the three major couplings J , Jx, and Jy. A dotted plaquetteindicates a unit cell. The four sublattices are labeled as A, B, C, and D. Figures fromRef. [158].

To capture the essential physics of (CuCl)LaNb2O7, we consider a S = 12

Heisenberg

model in a magnetic field,

H =∑(i,j)

JijSi · Sj − h∑i

Szi . (5.1)

Here (i, j) runs over all the bonds in Fig. 5.1(b), and Jij = J , Jx, and Jy for diagonal

(dimer), horizontal and vertical bonds, respectively. We are concerned with the case of

antiferromagnetic J > 0 and ferromagnetic Jx, Jy < 0, which create frustrated couplings

between the dimers.

The Schwinger boson calculation presented in this chapter contributed to Ref. [158].

In that paper, exact diagonalization and strong-coupling expansion calculations were

performed by S. Furukawa. The rest of this chapter is organized as follows. We begin by

discussing the results of Ref. [158] in Sec. 5.2, particularly the quantum phase diagram.

In Sec. 5.3 we review the application of the bosonic slave-particle mean-field theory to

the Hamiltonian (5.1), and discuss the results in Sec. 5.4.


Figure 5.2: Sketches of magnetization processes in the dimer singlet phase: (a) a jump,(b) a smooth increase, and (c) a plateau at 1/2 of the saturated moment Ms. (d) Phasediagram of the distorted Shastry-Sutherland model (5.1) in the spin-1

2case, determined

by the strong coupling expansion and exact diagonalization analysis. [158] Dashed linesat Jx = 0, Jy = 0, Jx = Jy are guides for the eye. The dimer singlet (DS) phase is dividedinto three regions, I, II, and III, which are characterized by the magnetization processes in(a), (b) and (c). Broken lines around the origin indicate the region boundaries determinedfrom the the first-order effective Hamiltonian. The square and circular symbols are basedon exact diagonalization. The classical ferromagnetic phase boundary is superposed onthe square symbols, showing good agreement. The diamond symbols are the boundariesbetween the DS-II regions and the stripe phases. Narrow spiral phases may appearbetween the DS-I region and the stripe phases for large |Jx|/J or |Jy|/J . For this reason,the DS-stripe phase boundaries (diamond symbols) are not calculated beyond Jx/J orJy/J ≈ −1.3. Figures from Ref. [158].

5.2 Quantum Phase Diagram

In this section, we review the quantum phase diagram found in Ref. [158], as seen in Fig.

5.2. There are two magnetically ordered phases found, and three dimer singlet phases,

each with a different magnetization profile.

When |Jx|, |Jy| . J , the system stays in a dimer singlet phase with a finite spin

gap, which is adiabatically connected to the decoupled-dimer limit Jx = Jy = 0. The

magnetization process of this phase depends crucially on the spatial anisotropy of the

inter-dimer couplings: the magnetization shows a jump or a smooth increase for weak or

strong anisotropy, respectively, after the spin gap closes at a certain magnetic field hc1.

When |Jx| or |Jy| & J , the spin gap of the dimer singlet phase closes, and quantum phase

transitions to various magnetically ordered phases occur – ferromagnetic for similar Jx

and Jy, stripe when Jx and Jy greatly differ. As we will see, these ordered phases also


exist in the classical limit; however, the classical spiral phase cannot be accounted for

using exact diagonalization or the strong-coupling expansion, necessitating the use of a

different approach, such as a slave-particle mean-field theory.

In Ref. [158], comparison with the details of the magnetization process and triplon

bandwidth indicates that (CuCl)LaNb2O7 is in the DS-II phase, and may have one of

J2 and Jy antiferromagnetic, with the other ferromagnetic. Alternatively, if the triplon

dispersion is interpreted to be larger, both Jx and Jy can be ferromagnetic, and these

dispersions are plotted as an additional means of comparison.

In the slave-particle mean-field theory, we will see possible changes the quantum phase

diagram, particularly the possibility of spiral phases existing in the physical spin-12

case.

5.3 Schwinger Boson Mean-Field Theory

In this section, we analyze the Heisenberg model introduced in Eq. (5.1) using a Schwinger

boson mean-field theory, as discussed in Sec. 2.2.3. The mean-field method can cover

a wide range of circumstances where our other methods (of Ref. [158]) may be limited.

Unlike a strong-coupling expansion, this method is not limited to small Jx/J and Jy/J .

The incommensurate spiral state can be easily found by this mean-field theory, in con-

trast with exact diagonalization. Furthermore, the theory can determine the splitting of

degeneracy in the classical phases due to quantum fluctuations. Finally, we can connect

to the semi-classical limit. We consider the case h = 0.

Using the mean-field theory and decouplings as discussed in Sec. 2.2.3, we next

re-write the Hamiltonian by performing a Fourier transform defined by

bk,X,µ =1√Nuc

∑r∈A

eik·(r+δrX)br,X,µ. (5.2)

X = A,B,C,D labels a position in a unit cell as in Fig. 5.1(b), so we rewrite the label

i → (r, X). Nuc is the number of unit cells in the system. We have used the A site

position r to label a unit cell, and δrX to represent the position of each site relative to

the A site,

δrA = 0, δrB = x, δrC = y, δrD = x+ y. (5.3)

At this point, we assume the chemical equivalence of the four sites in the unit cell,

by taking λi = λ the same for all sites in the lattice. In conjunction, we use an ansatz

for the mean fields consistent with the choice of chemical potential. Namely, we take

the x-direction χ to be equal (χx), and do the same for the y-direction χ (χy). The two


η mean fields are between sites A and D (ηAD), and B and C (ηBC). Furthermore, we

take all mean fields to be real. We take the two η mean fields to have equal magnitude,

so that ηAD = η and ηBC = ±η gives two possible choices for the relative signs of η.

With these assumptions, the mean field theory can still describe all the relevant phases

expected to appear in the model.

We are left, in the case of a gapped dispersion, with the mean-field Hamiltonian

HMF = HC +∑k

b†kHkbk, (5.4)

where

HC =− 2Jxχ2x − 2Jyχ

2y + Jη2 − λ(8S + 4),

bTk =(bkA↑, bkB↑, bkC↑, bkD↑, b

†−kA↓, b

†−kB↓, b

†−kC↓, b

†−kD↓

). (5.5)

The matrix Hk is given by

Hk =

(Ck Dk

D†k Ck

), (5.6)

Ck =

λ Jxχx cos kx Jyχy cos ky 0

Jxχx cos kx λ 0 Jyχy cos ky

Jyχy cos ky 0 λ Jxχx cos kx

0 Jyχy cos ky Jxχx cos kx λ

, (5.7)

Dk =

0 0 0 −J

2ηADe

−i(kx+ky)

0 0 −J2ηBCe

−i(kx−ky) 0

0 J2ηBCe

i(kx−ky) 0 0J2ηADe

i(kx+ky) 0 0 0

, (5.8)

with kx = k · x, ky = k · y, and ηAD, ηBC as defined above.

The quadratic Hamiltonian in (5.4) is diagonalized by the Bogoliubov transformation

bk = Zkγk, with γ defined in the same manner as b:

γTk =(γkA↑, γkB↑, γkC↑, γkD↑, γ

†−kA↓, γ

†−kB↓, γ

†−kC↓, γ

†−kD↓

). (5.9)

This is a canonical transformation, where the operators in γ preserve the bosonic com-


mutation relations of the operators in b. These commutation relations are given by

[γk,γ†k′ ] = δkk′η, where η =

(I 0

0 −I

)(5.10)

is an matrix of 4 × 4 blocks. The proper commutation relation are obtained by taking

the columns of Zk to be the eigenvectors ymk (with eigenvalues ωmk) of ηHk. [189] Here,

m = 1, . . . , 8 labels the eight eigenvectors.

When the Hamiltonian Hk is positive definite, the corresponding spinon dispersions

|ωmk| are gapped. This leads to a disordered ground state. In the case of a gapless

dispersion, ηHk0ym′k0 = 0 for some m′, so that both Hk0 and ηHk0 have a zero eigenvalue

at the dispersion minimum k0. Condensation of such zero-energy bosons gives rise to a

magnetically ordered state. To describe the condensation, we replace the operators bk0

with macroscopic constant values,∑m′

xm′k0 =∑m′

cm′k0

√Nucym′k0 , (5.11)

where y†m′k0ym′k0 = 1. With the condensate contribution, the diagonalized Hamiltonian

is

HMF = HC +∑k

8∑m=1

γ†mk|ωmk|γmk +∑k0

∑m′

x†m′k0Hk0xm′k0 . (5.12)

Spin ordering is found at twice the spinon minimum wavevector, 2k0, governing relative

spin orientation between unit cells.

Given the diagonalized Hamiltonian, we must solve for the mean-field values of

A,Bx, By and λ, as well as any condensate vectors xm′k0 and associated minimum

wavevectors k0. It turns out that the solution depends only on the total condensate

density∑

k=±k0

∑m′ |cm′k|2. We have the mean-field equations

∂ 〈HMF 〉∂η

=∂ 〈HMF 〉∂χx

=∂ 〈HMF 〉∂χy

=∂ 〈HMF 〉

∂λ=∂ 〈HMF 〉∂xm′k0

= 0, (5.13)

which we solve self-consistently for a given set of parameters Jx/J , Jy/J , and S, and the

associated values of m′ and k0.


Figure 5.3: Classical phase diagram of the distorted Shastry-Sutherland model (5.1)with h = 0. Dashed lines at Jx = 0, Jy = 0, Jx = Jy are guides for the eye.

5.4 Results

We present the phase diagram as a function of Jx/J and Jy/J for the cases of S → ∞(classical, Fig. 5.3), S = 0.5 (Fig. 5.4(a)) and S = 0.15 (Fig. 5.4(b)). Since a mean-field

theory is expected to underestimate quantum fluctuations, it can be instructive to look

at spin values smaller than the actual case. The Schwinger boson mean-field theory finds

several magnetically ordered phase, and one disordered phase. The magnetically ordered

phases are the ferromagnetic, spiral and stripe phases seen in the classical limit, while

the disordered phase features isolated dimers. Below we summarize how each phase is

described in the Schwinger boson formalism.

Ferromagnetic Phase. In this state, the antiferromagnetic order parameter A = 0,

while both ferromagnetic order parameters take their maximum values, χx = χy = 2S.

The spinon minimum wavevector k0 = 0, as is the spin ordering wavevector, so that the

spins are fully polarized throughout the entire lattice.

Stripe Phase. In this state, the ferromagnetic order parameter associated with the

smaller of Jx and Jy is zero. The ordering wavevector is 0, and a stripe magnetization

pattern is found in the direction of the larger ferromagnetic interaction. Depending on

the relative signs of the η parameters, the spinon minimum wavevector is either (0, π) or

0, for Jx > Jy, both leading to the same ordered state of same energy.


-2.5

-2

-1.5

-1

-0.5

0

-2.5 -2 -1.5 -1 -0.5 0

J y

Jx

FM

Stripe(0,π)

Stripe(π,0)

y-Spiral

x-Spiral

DS

(a)

-5

-4

-3

-2

-1

0

-5 -4 -3 -2 -1 0

J y

Jx

FM

Stripe(0,π)

Stripe(π,0)

y-Spiral

x-Spiral

DS

(b)

Figure 5.4: Phase diagram of the distorted Shastry-Sutherland model (5.1), determinedby a Schwinger boson mean field theory (Sec. 5.3) for (a) S = 0.5 and (b) S = 0.15. Solidlines indicate second-order transitions, while dashed lines indicate first-order transitions.(a) The isolated dimer singlet (DS) state is found for small |Jx|, |Jy| . 0.5J , while theferromagnetic state stabilizes for larger |Jx| and |Jy|. The stripe state expands from theclassical case to fill |Jx| or |Jy| . 0.55J . The spiral state exists in the middle. The x-spiralappearing for |Jx| < |Jy| has an incommensurate long-range order in the x direction asin Fig. 5.5. The y-spiral similarly appears for |Jx| > |Jy|. The dotted line separates thetwo spiral phases. (b) The disordered dimer state (DS) expands significantly comparedto the S = 0.5 case. The stripe state expands moderately, pushing the ferromagnetic(FM) state to moderately larger Jx and Jy. In between these phases, the spiral state isfound in a significantly reduced area. By an even smaller S = 0.1, the spiral state willhave disappeared entirely.


Figure 5.5: A classical spiral ground state with a pitch angle Qx in the x direction (“x-spiral”). The number at each site indicates the angle (in the xy plane) of the spin, where

Qx and Qy are given by cosQx = −JyJ

+ J4Jx

(JyJx− Jx

Jy

)and cosQy = −Jx

J+ J

4Jy

(JxJy− Jy

Jx

).

Spiral Phase. This is an incommensurate magnetically ordered state with a nonzero

ordering wavevector in the direction of smallest ferromagnetic interaction: k0 = (kx, 0)

for Jy > Jx. All of the mean-field parameters are nonzero. There are two inequivalent

spinon minimum wavevectors ±k0 found in the spiral state. They both lead to the same

type of spiral order, described in Fig. 5.5. The relative sign of the two η mean fields

yields either the x-spiral or y-spiral ordering. We find that there is a classical degeneracy

between x- and y-spiral ordering, which is broken by quantum fluctuations, determining

the ordering direction. For Jx > Jy, the y-spiral state has a lower energy, while the

x-spiral state has lower energy for Jy > Jx.

Disordered Phase. The disordered phase is gapped, with no boson condensate, and

has no ferromagnetic correlations: χx = χy = 0. The antiferromagnetic dimer order

parameter η 6= 0, leaving a decoupled dimer state. This is a mean field description cor-

responding to the dimer singlet state.

We consider the effect on the phase diagram of lowering spin from the large-S semi-

classical limit, by comparing the classical, S = 0.5, and S = 0.15 phase diagrams shown

in Figs. 5.3, 5.4(a) and 5.4(b), respectively. As S decreases, the disordered phase appears

and expands around Jx = Jy = 0, pushing out the magnetically ordered phases to larger


Jx and Jy. Of particular notes is the shift of the ferromagnetic phase boundary to larger

Jx and Jy in contradiction with the exact boundary for Jx = Jy = J . [158] Similarly, the

stripe state boundary shifts to larger Jy for Jx > Jy, and larger Jx for Jx < Jy. Between

these aforementioned phase boundaries, we find the spiral state, which consequently

gets pushed out to larger Jx and Jy. This state shrinks as S decreases, having almost

disappeared as seen for S = 0.15 in Fig. 5.4(b). In contrast with the case of the purely

antiferromagnetic Shastry-Sutherland lattice, [196] we see no disordered states, other

than the dimer one, down to S = 0.1. In particular, there are no short-range ordered

analogs of the ferromagnetic, stripe, or spiral ordered states.

5.5 Conclusion

In general, the phases of the quantum phase diagram are generally reproduced in the

slave-particle mean-field theory, though the boundaries are not. This provides a rough

confirmation of the structure of Fig. 5.2. The biggest difference is the existence of a

spiral order phase, which has its x- and y-direction degeneracy broken from the classical

case. As S decreases, however, the spiral phase is found in a smaller and smaller region

of the phase diagram, and may in fact be difficult to stabilize. Nonetheless, it suggests

that some members of the copper oxyhalide family may be close to a spiral order phase.

In the next three chapters, we will explore possible spin-liquid phases for three dif-

ferent frustrated materials: the face-centered-cubic lattice La2LiMoO6, and the kagome

lattices Volborthite and Herbertsmithite.

Chapter 6

Interplay Between Spin-Orbit

Coupling and Lattice Distortion in

Double Perovskites

6.1 Introduction

As discussed in Sec. 1.3, both spin-orbit coupling and lattice distortions play a competing

role in double perovskite systems. In this chapter we study the possibility of a spin-

disordered ground state in the 4d1 La2LiMoO6 – indeed, no long-range order is seen

down to 2 K. [102] We will see how such a state may be brought about by an anisotropic

psuedo-spin model.

For Mo5+ ions, octahedral crystal fields favor the t2g d -orbitals, which have an effective

orbital angular momentum Leff = 1, up to a sign difference. Combined with S = 12

spin

angular momentum, the pseudo-total angular momentum states of Jeff = 12

and Jeff = 32

result. In this case, the quadruplet of Jeff = 32

states form a lower energy manifold than

the other two states of Jeff = 12. [103]

One particularly important result of monoclinic distortion is the local z-axis com-

pression or expansion of the B′-O octahedra, which we refer to as a tetragonal distortion

of these octahedra. While the octahedral crystal field favors the t2g orbitals over the

eg ones, the tetragonal distortion will split the t2g levels. In the case of a local z-axis

compression, the dxy orbital is favoured to be occupied, while an expansion favors the

dxz and dyz. All of these effects will generate the anisotropic interactions that form the

focus of our models.

In this chapter, we use the Sp(N) generalization of Heisenberg models to describe

59

Chapter 6. Distortion, Spin-Orbit Coupling in Double Perovskites 60

these systems. [129, 166, 167] This provides a unifying framework to study the effect of

spin magnitude, from semi-classical ordering at “large spin” to possible spin liquid phases

for “small spin”. This is a many-flavour version of the arbitrary-S bosonic slave-particle

representation discussed in Sec. 2.2.3.

The ability to capture “large-spin” magnetic order may help to describe the higher-

spin analogues of d1 double perovskites. In particular, the “spin-32” analogue of La2LiMoO6

is the isostructural La2LiRuO6, whose 4d3 configuration occupies all three t2g orbitals.

Since the effective magnetic moment is close to the spin-32-only moment, there is only

slight renormalization due to spin-orbit coupling, and intra-orbital Coulomb repulsion

is the dominant effect in determining orbital occupancy. We model this material with

a spin-32

Heisenberg model, given the lack of orbital degeneracy, providing a test for

Sp(N)-predicted ordering at spin larger than 12. In fact, La2LiRuO6 shows type I anti-

ferromagnetic ordering below 30 K, [197] where spins are aligned on each x-y plane but

antiparallel on the x-z and y-z planes, as seen in Figure 6.1. This is consistent with

the results in the semi-classical (“large spin”) limit of our Sp(N) model. In contrast, an

appropriate pseudo-spin-12

anisotropic Heisenberg model for La2LiMoO6 leads to the con-

clusion that this system must be very close to a spin liquid state. This may be consistent

with the absence of magnetic order down to 2 K seen in experiment. [102]

The rest of the chapter is organized as follows. In Sec. 6.2, we discuss the effects

of monoclinic distortion and spin-orbit coupling. This leads us to consider two different

models, the planar anisotropy and general anisotropy models, each taking the form of

Figure 6.1: Magnetic ordering (Type I antiferromagnetic) of the spin-32

Ru in La2LiRuO6

(blue, with arrows). [197] Also shown are the non-magnetic Li (light green) atoms, andtwo of the Ru-O (purple) octahedra, showing the effects of monoclinic distortion.


a pseudo-spin Heisenberg model. In Sec. 6.3, we solve for the classical spin ordering of

both of these models. In Sec. 6.4, we describe the Sp(N) generalization of the Heisenberg

model and its mean-field treatment. Results of this mean-field treatment are shown in

Sec. 6.5 for the planar anisotropy model, and in Sec. 6.6 for the general anisotropy

model. An extension to finite temperature is discussed in Sec. 6.7. In Sec. 6.8, we

summarize our results and discuss extensions of this work.

6.2 Model

In modelling monoclinically distorted double perovskites with 4d or 5d magnetic ions,

there are two important effects of the monoclinic distortion that should be considered

in conjunction with spin-orbit coupling. The first effect of monoclinic distortion is local

z-axis compression or expansion of the B′-O octahedra, which affects orbital occupation.

The second is the change of orbital orientation due to the geometric distortion, which

affects overlap integrals and the resultant interactions. We will derive our models by

considering the effect of distortion and spin-orbit coupling on the interactions between

t2g orbitals.

One motivation for our models comes from a spin-12

Heisenberg model obtained via

spin-dimer calculation for the isostructural monoclinically distorted double perovskites

La2LiMoO6 and Sr2CaReO6. [102] In this method, the tetragonal compression (or expan-

sion) of these materials was modelled by assuming occupation of only the dxy orbitals

(or equal occupation of only the dxz and dyz orbitals). This method is also sensitive to

the effect of the geometric changes resulting from the distortion. However, spin-orbit

coupling was not considered, so that the assumed orbital occupation will be slightly in-

correct. The result is an anisotropic S = 12

Heisenberg model, with estimates for the

relative strengths of the couplings, seen in Table 6.1.

6.2.1 Interactions

To understand the effects of the monoclinic distortion and spin-orbit coupling, we first

look at the interactions between neighbouring t2g orbitals in the case of cubic symmetry,

as have been considered in detail by Chen et al . [103] To facilitate this, we show the six

nearest-neighbour directions δn for the FCC lattice in Figure 6.2. Without distortion,

the a, b and c-axes are simply the Cartesian x, y and z-axes. The strongest interaction

is antiferromagnetic superexchange, involving sites and orbitals lying in the same plane.

For instance, dxy orbitals on neighbouring sites along the x-y plane will interact antifer-


cb

a

J1

J2

J5

J6

J4

J3

Figure 6.2: Nearest-neighbour lattice vectors δn and associated nearest-neighbour cou-plings Jn for the FCC lattice.

romagnetically. Ferromagnetic interactions between sites on a plane will couple orbitals

lying on that plane to orbitals lying perpendicular to it. [103] Along the x-y plane, dxy

orbitals interact ferromagnetically with neighbouring dyz and dxz orbitals. Quadrupole-

quadrupole interactions also exist between all t2g orbitals on neighbouring sites, due to

different orientations of the quadrupole moments of these orbitals.

6.2.2 Monoclinic Distortion

The first effect of monoclinic distortion is the local z-axis distortion of the B′-O oc-

tahedra, a compression for La2LiMoO6 and an expansion for Sr2CaReO6. This splits

the degeneracy of the three t2g orbitals. The dxz and dyz orbitals will remain degener-

ate, but the dxy orbital will have a lower energy for a compression and a higher energy

for an expansion. Consequently, the occupation of the dxy orbital will be favoured or

disfavoured compared to occupation of the other two orbitals. This is taken as a very

important effect in the spin-dimer calculation to explain the relative anisotropies of the

two materials. [102]

The second important effect of the monoclinic distortion is a global c-axis elongation,

and rotation of the B′-O octahedra, affecting the overlap of the occupied orbitals, which

are now tilted out of plane. An example of this, in the case of La2LiRuO6, is shown in

Figure 6.1. The dxy orbitals, for instance, are tilted out of the a-b plane, and will have

some interaction with dxy orbitals on neighbouring planes. In this fashion, many new

exchange pathways will contribute at the nearest-neighbour level.

These effects generate a significant amount of exchange anisotropy in the spin-dimer

calculation. [102] The relative coupling strengths estimated by spin-dimer calculation for

La2LiMoO6 and Sr2CaReO6 can be seen in Table 6.1. Interactions between x-y planes


Material J1 J2 J3 J4 J5 J6

La2LiMoO6 0.14 1.0 0.014 0.014 0.00043 0.00043Sr2CaReO6 0.87 1.0 0.16 0.16 0.25 0.25

Table 6.1: Relative strengths of Heisenberg couplings, given in Figure 6.2, from thespin-dimer calculation of Aharen et al. [102]

in La2LiMoO6 are relatively weak, as expected from dominant in-plane dxy-dxy antiferro-

magnetic interaction and c-axis elongation. We note that further in-plane anisotropy is

significant, due to the strong effect of Mo-O octahedra rotation upon dxy orbital overlap.

In Sr2CaReO6, intra-plane interactions are still larger than inter-plane interactions, even

though the superexchange between dxy orbitals is not present. The only in-plane superex-

change processes occur through tilted dxz or dyz orbitals. Nevertheless, the inter-plane

interactions are significantly stronger than in La2LiMoO6. The length of the unit cell

along the c axis is significantly larger than along the a or b axes, which could explain

the smaller inter-plane coupling compared to the intra-plane one. For both materials,

however, the planar anisotropy of the couplings is clear, and effects of both geometrical

distortion and orbital occupation are important.

6.2.3 Spin-Orbit Coupling

Beyond monoclinic distortion, we now consider spin-orbit coupling, which can be im-

portant in the 4d and 5d magnetic ions commonly seen in the double perovskites. For

instance, spin-orbit coupling in octahedrally coordinated Mo5+ is estimated to be on the

order of 0.1 eV. [198] The effect of spin-orbit coupling on the t2g orbitals of octahedrally

coordinated ions is a well-studied problem. When the octahedral crystal field splitting is

significantly large compared to the spin-orbit coupling, we may project out the eg states.

Upon projection, the L = 2 orbital angular momentum for the d orbitals looks like a

L = 1 pseudo-angular momentum operator l up to a sign change, where L → −l. This

Leff = 1 pseudo-orbital angular momentum combines with the S = 1/2 angular momen-

tum of the single electron to create states of effective total angular momentum Jeff = 32

and 12. The spin-orbit coupling λL · S breaks the degeneracy of these states, where the

four Jeff = 32

states have an energy 3λ/2 lower than the two Jeff = 12

ones. These Jeff = 32


states are written in terms of the t2g ones as∣∣∣∣32 , 3

2

⟩=

1√2

(− |yz, ↑〉+ i |xz, ↓〉)∣∣∣∣32 , 1

2

⟩=

1√6

(− |yz, ↓〉+ i |xz, ↓〉+ 2 |xy, ↑〉)∣∣∣∣32 ,−1

2

⟩=

1√6

(|yz, ↑〉+ i |xz, ↑〉+ 2 |xy, ↓〉)∣∣∣∣32 ,−3

2

⟩=

1√2

(|yz, ↓〉+ i |xz, ↑〉) . (6.1)

With a d1 configuration, the occupancy of the dxy orbital upon projection to these states

is given by [103]

ni,xy =3

4− 1

3(jzi )

2. (6.2)

The occupation operators for the other t2g orbitals are given by cyclic permutation of the

x, y, z indices, and the single-occupancy constraint ni,xy + ni,xz + ni,yz = 1 is satisfied.

The effect of projection onto this Jeff = 32

subspace, due to large spin-orbit coupling,

has been considered by Chen et al. for the cubic materials. [103] The Hamiltonian can

be written in terms of the orbitally-resolved spin operators, such as Si,xy = Sini,xy. Upon

projecting to the Jeff = 32

states, these orbitally-resolved spin operators contain terms

both linear and cubic in j. The resulting Hamiltonian, containing terms of 4th and 6th

order in j, leads to interesting multipolar behavior. [103]

When spin-orbit coupling is much larger than the local z-axis crystal field, the Jeff = 32

states provide the relevant starting point, rather than the t2g orbitals. However, one can

consider the general splitting of t2g orbital degeneracy in the presence of both spin-orbit

coupling and the local z-axis distortion. We can model each site with a local Hamiltonian

Hloc = ∆ [(lz)2 − 2/3]−λl·S, where ∆ > 0 is the strength of the crystal field splitting due

to local z-axis compression. The case for a local z-axis expansion has been considered by

Jackeli and Khaliullin. [199] We proceed in a similar manner, identifying the relevant low-

energy eigenstates of Hloc. Diagonalization of Hloc determines the lowest-energy Kramers


pair to be given by

|↑〉G =sin (θ)√

2(i |yz, ↓〉+ |xz, ↓〉)− i cos (θ) |xy, ↑〉 ,

|↓〉G =sin (θ)√

2(−i |yz, ↑〉+ |xz, ↑〉)− i cos (θ) |xy, ↓〉 ,

tan (2θ) = 2√

2λ/(λ+ 2∆). (6.3)

The energy difference between the ground and first excited doublets is given by −λ +

(λ+ 2∆)(1 + 1/ cos (2θ))/4, which goes to zero as ∆→ 0, and approaches ∆−λ/2 when

∆ λ. We consider the case where this separation is large enough to focus on the

lowest-energy doublet. This will require the tetragonal crystal field to be significantly

larger than the exchange coupling J , regardless of the relative strength of spin-orbit

coupling. By projecting out the higher-energy states, we obtain a pseudo-spin-12

model.

Within this projection, we consider the form of the interactions in an otherwise cubic

double perovskite, beginning with the quadrupole-quadrupole interaction. Due to the

fixed orbital occupation in (6.3), this interaction is constant and will not contribute to

our models. The orbitally off-diagonal ferromagnetic interactions, of strength J ′, generate

pseudo-spin interactions that are both spatially and spin-anisotropic. For our models,

we will focus on the antiferromagnetic interactions. Nearest-neighbour interactions along

the undistorted x-y, x-z and y-z planes are given by

HAF = J∑

<ij> in x-y

(Si · Sj −

1

4

)ni,xynj,xy + (xy → yz) + (xy → xz), (6.4)

where ni,xy is the occupation operator of the dxy orbital at site i. [103] Upon projection to

the lowest-energy doublet, we obtain a Heisenberg model in the pseudo-spin-12

operators

Pi,

H′ = N

(−J

4

)+

∑<ij> in x-y

cos (θ)4JPi ·Pj

+∑

<ij> in x-z

sin (θ)4J

4Pi ·Pj +

∑<ij> in y-z

sin (θ)4J

4Pi ·Pj. (6.5)

For ∆ λ, this result reduces to the one obtained by Chen et al. in the easy-plane limit

of the cubic perovskite model with J ′ = 0. [103] Without an accurate estimate for the

strength of Hund’s coupling to Coulomb repulsion, the ratio J ′/J is difficult to ascertain.

However, we note that the easy-plane result of Chen et al. is an antiferromagnetic state


for J ′ < J . [103] Consequently, we consider the physical picture of antiferromagnetic

interactions, and as a first-order approximation we ignore the ferromagnetic contributions

to the Hamiltonian.

We note that the introduction of spin-orbit coupling results in a reduction of the

magnetic moment compared to the case of dxy occupation when λ = 0.

6.2.4 Planar-Anisotropy and General-Anisotropy Models

The first, and simpler, of the two models considered in this chapter is concerned primarily

with the effects of the tetragonal crystal field splitting. Without spin-orbit coupling,

we see easily from (6.4) that preferential dxy orbital occupation leads to anisotropic

interactions that are stronger on the x-y planes. In this case, we have a true spin-12

antiferromagnetic Heisenberg model. However, considering spin-orbit coupling and

tetragonal distortion leads to the pseudo-spin-12

antiferromagnetic Heisenberg model in

(6.5), with a similar form of anisotropy. From this, we are motivated to study the

pseudo-spin-12

antiferromagnetic Heisenberg model where coupling along the x-y plane

differs from the coupling along the y-z and x-z planes. The planar anisotropy model is

given in terms of pseudo-spin-12

operators (henceforth referred to as Si) by

HPA = Jin

∑<ij> in x-y

Si · Sj + Jout

∑<ij> in y-z

Si · Sj + Jout

∑<ij> in x-z

Si · Sj. (6.6)

Both Jin and Jout are antiferromagnetic, and one can consider this model as a general-

ization of the antiferromagnetic model in Eq. (6.5). The ratio Jout/Jin depends on the

strengths of the spin-orbit coupling and tetragonal distortion of the octahedra, seen in

∆/λ. In addition, it captures certain geometrical effects of the monoclinic distortion,

such as the global c-axis elongation, contributing to the particular planar anisotropy in

(6.6).

The other model considered in this chapter will include in full the geometrical effects

of the monoclinic distortion. This will generate many other anisotropic interactions,

breaking the symmetry of the x-y plane. Effective pseudo-spin exchange energies will

become intrinsically anisotropic, in addition to the effects of orbital occupation. We

will model these like the spin-dimer calculation does, with different strengths of the

nearest-neighbour couplings shown in Fig. 6.2. Due to spin-orbit coupling, the partic-

ular parameters Jn in Table 6.1 will not be quantitatively correct. Nonetheless, we will

consider them as a starting point to understand the effect of further anisotropy in the

interactions. Estimates for corrections due to spin-orbit coupling are given in Sec. 6.6.2.


The general anisotropy model is given by

HGA =∑i

∑n

JnS(ri) · S(ri + δn). (6.7)

To analyze the model Hamiltonians (6.6) and (6.7), we will use the Sp(N) general-

ization of the Heisenberg model, which offers several advantages. The first is that the

parameter N allows for a controlled expansion, beginning from the saddle-point solution

as N → ∞. This reproduces the Schwinger boson mean-field theory of Sec. 2.2.3. The

second is that quantum fluctuations can be controlled by a parameter κ (where κ = 2S in

the SU(2) case) allowing a transition from a classical-spin limit (large κ) to one dominated

by quantum fluctuations (small κ). This may capture a changing value of (pseudo)-spin.

The gapped Z2 spin liquid, obtained as a disordered state in the Sp(N) generalization,

is often seen as a potential ground state in many Heisenberg models. [140,200]

This framework may be capable of naturally capturing the changing behavior with

S seen in the family of magnetic materials isostructural to La2LiMoO6. The spin-32

La2LiRuO6 is magnetically ordered, while spin-12

La2LiMoO6 shows short-range corre-

lations and suppression of magnetic order. The isostructural spin-1 La2LiReO6 is more

amenable to a multi-orbital model, and falls outside the scope of these calculations. [98]

6.3 Classical Ordering

In this section, we solve both planar anisotropy and general anisotropy models in the

limit of classical spins. The magnetic ordering patterns and wavevectors are determined

by the O(N) model, where we generalize to N → ∞ components of the spin vector, as

explained in Appendix A.1. We will see in Sec. 6.4.4 that this corresponds also to the

classical limit of the Sp(N) model.

6.3.1 Planar-Anisotropy Model

In the planar anisotropy model (6.6), two phases are found with varying Jout/Jin, the

ratio of inter-plane to intra-plane interactions. For Jout < Jin, the intra-plane interactions

create antiferromagnetic Neel order within each x-y plane. For Jout > Jin, the inter-plane

interactions create antiferromagnetic order between planes.

For Jout > Jin, the ordering wavevector q is given by

q =π

a/2(0, 0, 1) or

π

a/2(1, 1, 0) . (6.8)


Spins on each x-y plane are aligned, while spins on neighbouring planes are antiparallel.

Neel ordering is found along the x-z and y-z planes. The antiferromagnetic interactions

between x-y layers are satisfied, as seen in Figure 6.3.

Figure 6.3: View along the z-axis of FCC lattice magnetic ordering of the planaranisotropy model for Jout > Jin. The solid lines indicate an x-y plane of the FCClattice, while the dotted lines indicate a neighbouring plane. Spins are aligned on eachof the x-y planes, but Neel ordered along x-z or y-z planes.

For Jout < Jin, the ordering wavevector q is

q =π

a/2(1, 0, kz) ,

π

a/2(0, 1, kz) (6.9)

for arbitrary kz. Each x-y plane takes on the Neel order for a square lattice. The

degeneracy in kz indicates that spins on neighbouring planes may take any relative overall

orientation. An example of this ordering, with kz = 0, is given in Figure 6.4. We will see

in Sec. 6.4.4 that this degeneracy is broken by the introduction of quantum fluctuations,

choosing kz = 0.

Both of these states show Type I antiferromagnetic ordering on the FCC lattice, where

ordering is antiferromagnetic on two of the x-y, x-z or y-z planes, and ferromagnetic on

the other.

6.3.2 General Anisotropy Model

The two parameter sets in Table 6.1 also yield antiferromagnetic ordering in the x-y

plane, similar to the Jout < Jin case. However, the degeneracy of kz is broken here at the

classical level, where kz = 0 for both parameter sets. Ordering as in Figure 6.4 results.


Figure 6.4: View along the z-axis of FCC lattice magnetic ordering of the planaranisotropy model for Jout < Jin with kz = 0. The solid lines indicate an x-y planeof the FCC lattice, while the dotted line indicates a neighbouring plane. There is Neelordering along each of the x-y and y-z planes, but ferromagnetic ordering along the x-zplane. Also possible is a state where the ferromagnetic ordering is along the y-z planeinstead.

6.4 Sp(N) Mean Field Theory

6.4.1 Sp(N) Generalization of the Spin Models

As discussed in Section 2.3.4, the Sp(N) method is a large-N generalization of the

Schwinger boson spin representation. In the physical case N = 1, Sp(1) is isomor-

phic to SU(2), and we have the standard Schwinger boson representation wherein Sia =12b†iα(σa)αβbiβ and the boson number per site b†iαbiα ≡ nb = 2S determines the spin quan-

tum number. Here, α, β =↑, ↓ label the primitive spin-12

species that comprise the full

spin angular momentum. We generalize to 2N flavors of bosons, where α = (m,σ), la-

belled by m = 1 . . . N , and σ =↑, ↓, transforming under the group Sp(N). [129] κ = nb/N

acts in analogous fashion to 2S in the SU(2) case, controlling the strength of quantum

fluctuations.

When generalized to Sp(N), the Heisenberg Hamiltonian (6.7), up to constants in-

volving nb, is written as

H =−1

2N

∑i

∑n

Jn(J αβb†iαb†i+δn,β

)(Jγνbγi bνi+δn). (6.10)


Here, Jαβ is a 2N × 2N block-diagonal antisymmetric tensor, given by

Jmσ,m′σ′ = δm,m′

(0 1

−1 0

). (6.11)

6.4.2 Mean-Field States

The quartic terms in (6.10) can be quadratically decoupled by the mean field

Qin =1

N

⟨∑m

εσσ′b†imσb

†i+δn,mσ′

⟩. (6.12)

ε is the 2×2 antisymmetric tensor. This is a generalization of the singlet pairing param-

eter η discussed in the N = 1 case in Sec. 2.3.2.

When the boson dispersion becomes gapless, we allow for a condensate, bi1σ =√Nxiσ ∈ C, where σ =↑, ↓, so that 〈bi1σ〉 has a finite expectation value. This will

account for the appearance of long-range magnetic order.

The projective symmetric group analysis may be used to characterize possible mean-

field ground states; for Sp(N) this has been applied to many other Heisenberg models.

[140] However, we will not pursue the PSG classification of bosonic spin liquid states

with the symmetry of the FCC lattice – particularly due to the fact that the interactions

break most of the symmetries already. Instead, we choose our mean-field Hamiltonians

by first noting that qualitatively different states may be distinguished by the value of a

flux quantity for plaquettes of the lattice, as in Sec. 3.3. The flux on a plaquette of sites

a . . . z is defined by the phase Φ in [201]

|Ξ|eiΦ =∑a...z

Qab(−Q∗bc) . . . Qyz(−Q∗za). (6.13)

A nearest-neighbour Heisenberg model will favor the zero-flux states at small κ, particu-

larly for plaquettes of smaller length. [201] On the bipartite cubic lattice, for instance, a

translationally invariant choice of Qij yields zero flux on any plaquette. Since the FCC

lattice is frustrated, a translationally invariant Qij = −Qji, while giving zero flux on

most plaquettes, leaves π flux on a small number of plaquettes. In particular, assuming

all Q to be translationally invariant and positive, the four-site plaquettes with π flux

have sites on both the x-y and y-z planes, such as i, i + δ1, i + y, i + δ3, where i and

i± y are joined by the plaquette. There are eight such plaquettes with π flux, of a total

of thirty-six four-site plaquettes involving site i. This provides motivation to consider

translationally invariant mean-field solutions, which we restrict ourselves to in this work.


6.4.3 Mean-Field Hamiltonian

After decoupling in the site-independent Qn fields, the Hamiltonian (6.10) becomes

H =∑i,n

Jn

[−Qn

2εσσ′

(N∑m=2

bmσi bmσ′

i+δn + xσi xσ′

i+δnN

)+ h.c.+

N

2|Qn|2

]

+∑i

µi

(−nb +

N∑m=2

b†imσbmσi +Nx∗iσx

σi

). (6.14)

Here, the boson number constraint is enforced on average by the inclusion of the Lagrange

multiplier µi. We assume translational invariance, with µi = µ. We have allowed the

m = 1 component to condense, represented by xσi ∈ C.The saddle-point Hamiltonian (for N → ∞) is derived in full in Appendix A.2.

The first step is a Fourier transform defined by bi = 1√Ns

∑k bke

−ik·ri . The second step

is a Bogoliubov transformation diagonalizing the Hamiltonian, yielding a quasiparticle

energy ωk =√µ2 − (

∑n JnQn sin (k · δn))2. The transformation is defined by b = T−1γ,

where the Hamiltonian is diagonal in the γ basis. The condensate enters only via the

total density n =∑

kσ |xσk|2, and ±k1, the wavevectors of the boson dispersion minimum

where the condensate forms.

We then write the diagonalized Hamiltonian as

HNsN

=∑δ

Jδ2|Qδ|2 + µ (−1− κ+ n) + n

∑δ

JδQδ sin (k1 · δ)

+1

Ns

∑k

ωk

(1 + γ†k↑γk↑ + γ†k↓γk↓

). (6.15)

6.4.4 Semi-classical Large-κ Limit

We take advantage of the Sp(N) fluctuation parameter κ to look at the semi-classical

magnetic order from the κ → ∞ limit. This provides a link from the classical order of

Sec. 6.3 to the magnetic order seen at finite κ.

We begin by approximating the Hamiltonian for κ 1. Here, leading-order behavior

in the Hamiltonian is of O(κ2). Corrections, of O(κ), act to split degeneracy of the

classical ordering. [129] We have that Q, µ and n are all O(κ) as κ 1. EC , the largest

contribution to the energy is of O(κ2):

ECNsN

=∑δ

Jδ2|Qδ|2 + µ (−κ+ n)n

∑δ

JδQδ sin (k1 · δ), (6.16)


while the first-order quantum correction E1, of O(κ), is given by

E1

NsN= −µ+

1

Ns

∑k

ωk, (6.17)

where Q, µ and n are given by solutions minimizing the classical energy (6.16). [129] The

mean-field equations for EC are easily solved, yielding n = κ, µ = −∑

n JnQn sin (k1 · δn),

and Qm = −κ sin (k1 · δm). We can then write EC as a function of the minimum wavevec-

tor k1:

ECNsN

= −κ2∑n

Jn2

sin2 (k1 · δn). (6.18)

With the boson dispersion minimum at ±k1, spin ordering occurs at the wavevectors

q = ±2k1. The minimum of EC corresponds to an ordering pattern equivalent to that of

the classical O(N) model (see Appendix A.1 for details). [202] The correction (6.17) can

then easily be computed for all k1 (with corresponding Q, µ, n) in the degenerate set of

minima of (6.18).

6.5 Planar Anisotropy Model Results

In this section we study the planar anisotropy model with in-plane coupling Jin (J1 = J2)

and out-of-plane coupling Jout (J3 = J4 = J5 = J6). We study the effect of quantum

fluctuations, controlled by κ, and coupling anisotropy, controlled by Jout/Jin. In Sec. 6.3,

we saw classical Neel ordering on each x-y plane. The first-order quantum correction E1

in (6.17) breaks the degeneracy. After this “order by disorder”, the ordering wavevectors

are

q =π

a/2(1, 0, 0) or

π

a/2(0, 1, 1) ,

q =π

a/2(0, 1, 0) or

π

a/2(1, 0, 1) . (6.19)

Spins are aligned along either the x-z or y-z planes. Ordering along one such direction

was seen in Figure 6.4.

As κ is reduced from this limit, we wish to see the evolution of the ordering wavevector

and mean-field parameters. For small κ, we investigate the destruction of the ordered

state by quantum fluctuations. We note that the semi-classical solutions, for all values

of Jout/Jin, all feature |Q1| = |Q2|, |Q3| = |Q4|, and |Q5| = |Q6|. Motivated additionally


by the equality of in-plane couplings, J1 = J2, and of between-plane couplings, J4 =

J4 = J5 = J6, we take an ansatz with Q1 = Q2, Q3 = Q4, and Q5 = Q6. The relative

signs, such as between Q1 and Q2, correspond to making a particular gauge choice. With

such an ansatz, the semi-classical solutions remain unchanged, with wavevectors (6.8) or

(6.19) as appropriate. Furthermore, relaxing the ansatz suggests that the equivalence

|Q1| = |Q2|, |Q3| = |Q4|, and |Q5| = |Q6| is retained down to low κ. With this ansatz,

we numerically solve the mean-field equations, given explicitly in Appendix A.2. The

resulting phase diagram is given in Fig. 6.5, in which there are five phases to consider.

0.01

0.1

1

10

0 0.25 0.5 0.75 1

κ

Jout/(Jin+Jout)

(b) k = (1,0,0) x-y Plane Neel Order Q3=0

(a) k = (0,0,1) Inter-Plane AF Order Q1=0

(e) Quasi-2DSpin LiquidQ3=Q5=0

(c) Spin LiquidQ1=0

(d) Spin LiquidQ3=0

Figure 6.5: Heuristic phase diagram for the Q1, Q3, Q5 ansatz of the planar anisotropymodel. Note that the label Qm = 0 indicates that Qm is negligibly small (compared to κand the finite Q) in the condensed phase; Qm is identically zero in the corresponding spinliquid phases. Solid lines indicate second-order transitions, while dashed lines indicatefirst-order transitions.

6.5.1 Inter-plane Antiferromagnetic Order

This state is an extension of the classically ordered state for Jout > Jin, with antiparallel

magnetization on neighbouring x-y planes. Ferromagnetic ordering is seen along the x-y

plane, with Neel ordering along the x-z and y-z planes. In this state, the intra-plane

Q1 = Q2 is significantly smaller than the intra-plane Q3 through Q6. The ordering

wavevector has only small corrections to the classical result (6.8).


6.5.2 x-y Plane Neel Order

This state is an extension of the classically ordered state for Jout < Jin, with Neel order

on the x-y planes. It is characterized by large |Q1| = |Q2| within the x-y plane. Of the

two independent inter-plane Q, one is significantly smaller than the other, depending

on the gauge choice of ferromagnetic order direction (along the x-z or y-z plane). The

ordering wavevector has only small corrections to the semi-classical result (6.19).

6.5.3 Inter-Plane Spin Liquid

This state is a disordered analogue of the inter-plane ordered state (Sec. 6.5.1) for

Jout > Jin. However, the intra-plane Q1 = Q2 are identically zero in this state. While

the direct intra-plane correlations are consequently zero, the finite inter-plane Q prevent

the lattice from decoupling. The minimum wavevector, determining short-range order,

still has only small corrections compared to the ordered minimum (6.8). The transition

into this state from the intra-plane ordered state, as κ is lowered, is second-order.

6.5.4 Three-Dimensional Intra-Plane Spin Liquid

This state is a disordered analogue of the inter-plane ordered state (Sec. 6.5.2) for Jout <

Jin. However, one of the intra-plane Q is now identically zero, such as Q3 = Q4. The

other intra-plane Q is nonzero, but still smaller than the in-plane Q1 = Q2, preventing

the lattice from decoupling. As before, the minimum wavevector, determining short-

range order, has only small corrections compared to the ordered minimum (6.19). The

transition into this state from the intra-plane ordered state, as κ is lowered, is second-

order.

6.5.5 Quasi-Two-Dimensional Spin Liquid

In this state, all inter-plane Q vanish: Q3 = Q4 = Q5 = Q6 = 0. The system then consists

of decoupled two-dimensional x-y planes in this mean-field theory. The transitions into

this state, from either the ordered or disordered intra-plane states for Jout < Jin, are

weakly first-order. The minimum (short-range order) wavevector no longer takes the

semi-classical value, instead taking a different value among the classical solutions (6.9),

with kz ∼ 0.15.


6.5.6 Tricritical Point and Destruction of Order

We find a tricritical point at Jout = 0.58Jin separating the intra-plane spin-liquid phases

from the x-y plane Neel ordered phase. For Jin > Jout > Jout, the ordered state first

enters the three-dimensional spin-liquid state as κ is decreased. A first-order transition

to the two-dimensional spin liquid follows as κ decreases further. The κ range of this

three-dimensional spin liquid narrows as Jout reaches tricritical point, as seen in Figure

6.5. For Jout < Jout, in-plane coupling pushes the system to decouple. However, we

expect that the Q = 0 decoupling seen in all three mean-field spin liquid states is an

artifact of the mean-field theory, and that 1N

corrections will restore a small yet non-zero

value to these Q.

The critical κ value of the destruction of magnetic ordering, κc, is fairly small in this

planar anisotropy model. κc ranges from 0.1 for large Jout to 0.4 for small Jout. In the

physical N = 1 case, κ = 1 corresponds to the “most quantum” limit of S = 12. Our

N →∞ solution indicates that ordering is likely to occur, even though mean-field theory

overestimates ordering. While κc will differ in the exact N = 1 theory, the values of

κc ∼ 0.1− 0.4 are too small to account for the behavior of La2LiMoO6.

6.6 General Anisotropy Model Results

6.6.1 Spin Dimer Parameters

We now turn to the particular parameter set in Table 6.1 modelling La2LiMoO6. We

saw that the semi-classical limit led to Type I antiferromagnetic order, with Neel order

on the x-y planes. As for the planar-anisotropy model, we take advantage of coupling

symmetry to simplify the mean-field calculation. We make the ansatz Q3 = Q4 and

Q5 = Q6, since J3 = J4 and J5 = J6. The semi-classical result satisfies this, while

relaxing the ansatz again suggests this structure carries to low κ. Then we numerically

solve the resulting mean-field equations. The mean-field solution finds that ordering

persists down to κc = 0.986. As in the planar anisotropy case, the ordering wavevector

changes little with κ, and Q5 remains significantly smaller than the other Q. At κc, there

is a weakly first-order phase transition into a disordered state with Q1 = Q3 = Q5 = 0.

This highly anisotropic mean-field solution consists of decoupled quasi-one-dimensional

chains, with Q2 contributing the only non-zero correlation. The phase diagram for the

general anisotropy model with parameters modelling La2LiMoO6 is given in Figure 6.6.

As before, we expect 1N

corrections to remove this decoupling.

The parameter set for Sr2CaReO6 in Table 6.1 behaves similarly, although the tran-


κ

0.9860

AF Magnetic OrderAnisotropic Spin Liquid

Figure 6.6: Phase diagram as a function of κ for the general anisotropy model withparameters for La2LiMoO6, from Table 6.1. For κ larger than κc = 0.986, the system isin a three-dimensional magnetically ordered state, as in the semi-classical limit. For κsmaller than κc, the system is in an anisotropic and highly decoupled spin liquid state.

sition occurs at a smaller κc ∼= 0.41, similar to the values from the planar anisotropy

model.

Two comparisons to the planar anisotropy model are relevant. The first is that at large

exchange anisotropy, the mean-field theory continues to predict immediate transitions

from magnetic order into maximally decoupled spin liquid states. Additionally, this

anisotropy stabilizes these decoupled states. For the La2LiMoO6 parameters, we see a

marked increase in κc, which falls quite close to 1. This saddle-point solution suggests

that the S = 12

system must be very close to the transition to a spin-liquid state, even

if magnetic order eventually appears at very low temperature. The effect of further

quantum or thermal fluctuations may be sufficient to destroy the order. This could

explain why no long-range order is observed in La2LiMoO6 down to 2 K, while µSR

shows at most short-ranged order. The distortion of La2LiMoO6 from the cubic perovskite

structure is key in moving beyond the magnetic order predicted by the planar anisotropy

model.

6.6.2 Corrections to In-Plane and Out-of-Plane Anisotropy

While the Table 6.1 parameters give a good picture of the anisotropy of La2LiMoO6, they

will not be quantitatively correct. We wish to look at deviations due to the inclusion

of spin-orbit coupling, from the viewpoint of in-plane and out-of-plane anistropy. The

change in orbital occupation will result in a reduction of dxy-mediated coupling as spin-

orbit coupling increases, along with new contributions, primarily out-of-plane, from dxz

and dyz occupation. From these considerations, we estimate changes to Jn so as to

minimize the resulting anisotropy, thus estimating a lower bound for κc upon inclusion

of spin-orbit coupling. We determine the effective couplings Jn in a manner similar to

model (6.5), but with intrinsically anisotropic exchange modified by orbital occupation.

In general, we have

Jn → cos (θ)4Jxyn +1

4sin (θ)4Jxz,yzn , (6.20)


with θ as defined in (6.3). While the θ = 0 spin-dimer parameters give Jxyn , the Jxz,yzn

are unknown. Since they arise from octahedral tilting, the in-plane Jxz,yzn will be quite

small, similar to how the out-of-plane Jxyn are small. Since 14

sin (θ)4 is also small, we

ignore that term by estimating Jxy1,2 = 0. For the out-of-plane interactions, we will make a

large estimate for Jxz,yzn to minimize the out-of-plane anisotropy, by taking Jxz,yz3,4,5,6 = Jxy2 ,

the largest exchange scale in the problem. In terms of the spin-dimer parameters JSDn ,

we estimate the change in magnitude of Jn due to the change in orbital occupation from

spin-orbit coupling by taking

J1,2 = cos (θ)4JSD1,2 ,

J3,4,5,6 = cos (θ)4JSD3,4,5,6 +1

4sin (θ)4JSD2 . (6.21)

For the case of λ ∆, we find that κc reduces to 0.86. However, for a moderate case of

λ = ∆, we find that there is only a slight reduction in κc to 0.98. For moderate values

of λ∆

, these mean-field results indicate that the system is still close to a disordered state;

however, this will be sensitive to the value of λ∆

.

Exchange anisotropy has shown to be very important, from the results for the spin-

dimer parameters and the spin orbit coupling rescaled values (6.21). To better understand

the combined effect of in-plane and out-of-plane anisotropy, we consider a model with

slightly less than the full anisotropy, where J1 = RIJ2, J3 = J4 = ROJ2, and J5 = J6 =

RORIJ2. This captures the in-plane (RI) and out-of-plane (RO) anisotropy, differing from

the full anisotropy only in the very small exchange parameters J5 and J6. In Figure 6.7

we show κc as a function of RO, for several values of RI . We see that κc decreases fairly

evenly as either RO or RI increases. This confirms that both in-plane and out-of-plane

anisotropy are important in securing a large κc.

6.7 Finite Temperature

Thermal fluctuations of the quasiparticles in (6.15) introduce, beyond quantum fluctua-

tions, another mechanism inducing disorder. At nonzero temperatures, these excitations

have a thermal Bose distribution. The energy 〈H〉 and the mean-field equations, (6.15)

and (A.10), are modified accordingly. Thermal fluctuations will reduce magnetic ordering

and correlations. We see different finite temperature behavior depending on the state

(ordered or spin liquid) seen at T = 0 for a given set of Jn and κ.


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

κc

RO

RI = 0.14

RI = 0.16

RI = 0.19

RI = 0.24

RI = 0.29

RI = 0.34

Figure 6.7: Critical value κc of destruction of magnetic order. Two types of anisotropyare considered. RI is the ratio of the anisotropy within x-y planes, while RO is the ratioof anisotropy between these planes.

6.7.1 Zero-Temperature Disordered Phases

From disordered phases, as T increases, the magnitudes of all Q decrease. The smaller

the value of Q at T = 0, the lower the temperature at which Q reaches zero. At a large

enough temperature, all Q are zero, describing a perfectly paramagnetic state, where

spins are independent and completely uncorrelated. This unphysical behavior at high

temperature is typical of N →∞ solutions of Schwinger boson mean-field theories, and

disappears for smaller values of N . [203]

6.7.2 Zero-Temperature Magnetic Phases

From ordered phases, as T increases, the condensate density n decreases along with the

mean-field parameters |Qn|. It similarly reaches zero at a large enough T . At large κ,

the transition to the perfect paramagnet state is first-order, with the system remaining

in the ordered state until all Q and n discontinuously jump to zero. This occurs even

for moderate values of κ, such as κ ∼ 0.5 in the planar anisotropy model. For instance,

with Jout = 0.54Jin and κ = 0.5, this transition occurs at T = 0.44Jin. With θC = −45

K and S = 12, the transition temperature T = 53 K, an overestimate to be expected of

mean-field theory.

For smaller κ, close to the disordered state boundary, the transition is second order.

Furthermore, the order can be destroyed before the Q become zero; the system has a

second-order transition to a thermally disordered state before entering the perfect para-

magnet state. We show such an example in Figure 6.8. Here, κ = 0.2, just above the

zero-temperature critical κc for Jout = 0.54Jin. At T = 0, the transition with varying κ


went from ordered state directly into a quasi-two dimensional spin liquid. At finite tem-

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.05 0.1 0.15 0.2 0.25 0.3

T (Jin)

Q1Q3Q5

CondensateDensity

Figure 6.8: Mean field and condensate density (closely dotted) destruction with increas-ing temperature, shown for Jout = 0.54Jin and κ = 0.2 in the planar anisotropy model.Above T ∼ 0.1Jin, the magnetic ordering is destroyed, leaving a thermally disorderedstate. As Q5 (dotted) and Q1 (solid) become zero, the system enters a two-dimensionallyor completely decoupled state, respectively.

perature, we see that there is a window, 0.1Jin . T . 0.15Jin, where a three-dimensional

disordered state exists, in contrast with the decoupling behavior of the T = 0 mean-field

theory.

The general anisotropy model with La2LiMoO6 parameters shows similar behavior.

However, at κ = 1, the transition from the ordered state looks weakly first-order, with

the system directly entering a quasi-two-dimensional decoupled state where only Q1 and

Q2, both in the x-y plane, are nonzero. A fully three-dimensional disordered state is

not predicted here by the finite-temperature mean-field theory. Nonetheless, this case

illustrates how fluctuations destroy magnetic order and inhibit coupling in the spin-liquid

states. As before, we expect 1N

corrections to further restore correlations.

6.7.3 Heat Capacity

The presence of the perfect paramagnet state is an artifact of the mean-field theory.

Regardless, the magnetic contribution to the heat capacity is an important physical

quantity, and can be reliably calculated in this approach at low temperatures. CV is found


straightforwardly from d〈H〉dT

. In the magnetically ordered states, we find that CV ∝ T 3 at

low temperatures. This is expected from three-dimensional antiferromagnetic spin wave

contributions. In the disordered states, CV ∝ exp (− ∆G

kBT). ∆G scales roughly with the

spin gap, as expected for gapped states. Unfortunately, the lattice match material for

La2LiMo6 was not useful in subtracting the lattice contribution to the heat capacity. [102]

Without clear data for the magnetic contribution to the specific heat, direct comparison is

not feasible. For a system close to the ordering transition, such as the general anisotropy

model for La2LiMoO6, the T 3 behavior persists only at extremely low temperatures,

further complicating potential comparison.

6.8 Conclusion

We have modelled the effects of monoclinic distortion and spin-orbit coupling in 4d1 or

5d1 double perovskites. Local z-axis distortion of the magnetic ion-oxygen octahedra

changed dxy orbital occupation compared to the other t2g orbitals. Geometrical effects of

monoclinic distortion changed orbital overlaps, introduced multiple exchange pathways

and generated significant anisotropy. Considering spin-orbit coupling in conjunction with

the local z-axis crystal field yielded a lowest-energy doublet of states and a pseudo-spin-12

Heisenberg model from antiferromagnetic interactions. We considered first the general

case where interactions between sites on x-y planes differ in strength from interactions

between these planes. This planar anisotropy model was studied for a general ratio of

these two couplings. Geometrical changes of the monoclinic distortion induce further

anisotropy among the interactions, especially within the x-y plane, leading to the general

anisotropy model, studied for particular parameters modelling La2LiMoO6, estimated

from spin-dimer calculation. [102] We solved both these models in the saddle-point limit

of the Sp(N) generalization of the Heisenberg model. Semi-classical ordering was deter-

mined to be Type I antiferromagnetic, with antiferromagnetic order on two of the x-y,

x-z, y-z planes, and ferromagnetic order on the other. The Sp(N) method connected the

semi-classical results to the limit of large quantum fluctuations. The large interaction

anisotropy of the general anisotropy model predicted disordering at a relatively large

κc = 0.986. The N = 1 pseudo-spin 12

system was determined to be very close to a

disordered state, even if order sets in at a low temperature. This could explain the lack

of long-range order seen down to 2 K in La2LiMoO6. Furthermore, estimates of the effect

of spin-orbit coupling on the spin-dimer calculation parameters of Table 6.1 reduced κc

only to 0.98 for moderate strength of spin-orbit coupling. The system is still close to a

disordered state in this case.


Further experimental and theoretical enquiries follow as natural extensions of our in-

vestigation. Single-crystal experimental results would be useful, primarily in determining

the short-range ordering wavevector of La2LiMoO6. Results at temperatures lower than 2

K could determine specifically how antiferromagnetic order is being suppressed. Finally,

estimates of the strength of the spin-orbit coupling and crystal field splitting would guide

a more precise model of the monoclinic distortion.

Chapter 7

Spinon Fermi Surface and Spin

Density Wave Order in Coupled

Chain Model for Volborthite

7.1 Introduction

As discussed in Sec. 1.4, the isotropic kagome lattice has been the subject of great focus,

both theoretically and experimentally, with anisotropy spoiling many potential candidate

realizations. However, recent studies suggest that such anisotropy in two dimensional

frustrated magnets may lead to a host of interesting effects. This includes effective

dynamic dimensional reduction in the anisotropic triangular lattice compound Cs2CuCl4

[204] and access to unconventional quantum phase transitions. [205, 206] Another such

candidate for interesting behaviour in an anisotropic two-dimensional kagome lattice

system is the powder form of the Mott insulator Volborthite, Cu3V2O7(OH)2 · 2H2O.

[128, 207] Here, the Cu2+ ions sit on the distorted kagome lattice composed of isosceles

triangles. The compound is highly insulating, [208] and the low energy magnetic degrees

of freedom consists of an unpaired S = 12

on the Cu atoms.

As shown in Fig. 7.1, in powder samples of Volborthite, there are two different chem-

ical environments for the Cu atoms [209] which are indicated as Cu(1) and Cu(2). The

magnetic susceptibility data shows a Curie-Weiss behaviour well below the mean field

ordering temperature (ΘCW ∼ −115 K) and does not show any sign of long range mag-

netic order to 1.8 K. While initial results indicated spin-glass behaviour at low tempera-

tures, [126] this is now seen as an impurity effect that can be removed by the application

of a magnetic field. [127] 51V-NMR shows a transition into a magnetically ordered state

82

Chapter 7. Spinon Fermi Surface and Order in Volborthite 83

below 1 K. [127] The NMR line shape is Lorentzian, which indicates spin density wave

(SDW) order of magnetic moments on the Cu sites, as opposed to a rectangular line

shape found in conventional antiferromagnetic ordering. Furthermore, 1T1

is linear in

temperature, indicating dense low-energy excitations uncharacteristic of a conventional

ordered state. Both responses are field-independent within this low field phase (below

4.3 T ). Thermodynamic measurements show, at these low fields and low temperatures

(below 4 K), a linear magnetic specific heat contribution, cM = γ T , with a sudden

decrease in the magnitude of the constant γ around 1 K, which is where NMR sees the

development of SDW order. [208] As the magnetic field is increased, the value of the

specific heat coefficient, γ, decreases. Finally, above 4.3 T, conventional magnetic order

sets in. These experimental findings collectively suggest a rather unconventional ground

state for powder samples of Volborthite since these features are at odds with conventional

magnetically ordered systems.

However, the synthesis of single-crystal samples has unveiled an entirely different

story. A unique structural transition is found at 310 K, wherein the orbital state of Cu(1)

switches from ddz2−r2 at high temperatures to dx2−y2 . [148] Consequently, the structure,

microscopic model, and magnetic properties are all different from the powder case. The

structure is even more anisotropic, and long-range ordering sets in near 1 K. [148] In

this chapter, we will be considering the monoclinic C2/m structure found in powder

samples, although this has not been stabilized in single crystals to date. We note that

the role of disorder in powder samples of Volborthite is thus an open question, though

further experimental effort into the powder samples may help to shed more light onto the

stability of this non-conventional ordered state in comparison with the spin-glass state.

For the rest of the chapter, we mean the powder samples specifically, and the structure

and models relevant to it, when we refer to Volborthite.

In this chapter, we propose an unconventional U(1) spin liquid ground state for Vol-

borthite to account for the above experimental results. This spin liquid has gapless

Fermionic spinon excitations about a spinon Fermi surface, which, we argue, accounts

for the linear specific heat. The sudden decrease in γ below T = 1 K, we show, results

from spin density wave (SDW) instability of the spinon Fermi surface which gaps out

portions of the Fermi surface. However, since parts of the reconstructed Fermi surface

is present, the low energy spectrum still contains gapless fractionalized spinons. Hence,

we distinguish this phase from the conventional SDW phase by denoting it as SDW∗.

We find that application of a magnetic field enhances nesting of the spinon Fermi sur-

face and hence promotes the SDW∗ phase. However, on increasing the magnetic field

further, it completely gaps out the spinons resulting into a gapped U(1) spin liquid. It


J1

J2J¢

A

BC

CuH1LCuH2L

Figure 7.1: Kagome lattice of Cu2+ moments of Volborthite. The two ions marked asCu(1) and Cu(2) have different chemical environments. Indicated are NN interactions J1

(ferromagnetic) and J ′ (antiferromagnetic), and NNN interactions J2 (antiferromagnetic),which comprise our model. Also shown is ferrimagnetic ordering of the moments, whichis the un-frustrated ordering seen for small J2 (also see discussion in the main text).

is well known that in two spatial dimensions, such a gapped U(1) spin liquid is unstable

to confinement, [178] which according to experiments occurs around B ∼ 4 T . Beyond

this magnetic field conventional commensurate ordering sets in. We show that a cou-

pled chain model, suggested from first principle calculations on Volborthite, [147] indeed

realizes the above scenario within slave-Fermion mean-field theory.

The rest of the chapter is organized as follows. In Sec. 7.2, we outline the model

Hamiltonian that has been proposed to be the microscopic spin Hamiltonian from the

first principle calculations. We use this Hamiltonian in the rest of the chapter to perform

our calculations. In Sec. 7.3, we outline the results obtained from the classical treatment

of the model as well as the Schwinger-boson mean-field theory approaches. We show that

both approaches give rise to conventional magnetically ordered ground states and hence

fail to explain the properties of the material. In Sec. 7.4, we present our slave fermion

analysis and show that a U(1) spin liquid with a spinon Fermi surface is stabilized within

self-consistent mean field theory. We study possible spin density wave instabilities of the

spinon Fermi-surface and the effect of magnetic field in Sec. 7.5. The relevance of our

results to the experimental situation is discussed in Sec. 7.6, and we conclude in Sec.

7.7. The details of various calculations are presented in the appendices.


7.2 The coupled-chain Spin Hamiltonian

We start with the discussion of the microscopic Hamiltonian that has been suggested

to capture the low energy properties of Volborthite. The presence of lattice distortions

make the magnetic exchanges anisotropic. To capture this, a distorted nearest neighbour

Heisenberg antiferromagnet on a kagome lattice was studied in Refs. [210] and [211].

However, a recent density functional calculation indicates a different set of magnetic

exchanges from this simple anisotropic nearest neighbour model. [147] These results in-

dicate presence of chain-like structures made out of the Cu(2) atoms.The Cu(2) atoms

belonging to the same chain (as shown in Fig. 7.1) have ferromagnetic (FM) nearest-

neighbour (NN) and significant antiferromagnetic (AFM) next-nearest-neighbour (NNN)

Heisenberg exchanges. These chains further interact with each other via the interstitial

Cu(1) atoms through separate antiferromagnetic exchanges, again shown in Fig. 7.1.

Accordingly, we may view the system as a coupled-chain model, in sharp contrast with

the isotropic model. The competing interactions lead to frustration in this model. The

chains themselves are frustrated since the NN FM and NNN AFM interactions compete

with each other, and the one-dimensional problem has already shown a very rich phase

diagram. [212–214] We expect that the presence of further couplings among the chains

may stabilize a two dimensional quantum paramagnetic phase.

Motivated by these density-functional results, [147] we study an anisotropic spin-12

Heisenberg model on the kagome lattice. The coupled chain Hamiltonian is given by

H = −J1

∑〈ij〉∈A,B

Si · Sj + J2

∑〈〈ij〉〉∈A,B

Si · Sj + J ′∑

〈ij〉,i∈C

Si · Sj, (7.1)

where, Si refers to the spin-12

operators at the Cu2+ sites; 〈ij〉 and 〈〈ij〉〉 denote inter-

actions between NN and NNN sites respectively. (A) and (B) refers to the two sites

belonging to the chains, while the third interstitial site is denoted as (C). Each unit

cell is comprised of these three sites, as shown in Fig. 7.1. The exchanges J1, J2, J′ > 0

indicate that the nearest neighbour exchanges (J1) and the next-nearest neighbour ex-

changes (J2) along the chain are ferromagnetic and antiferromagnetic, respectively, while

the interchain nearest neighbour couplings (J ′) are antiferromagnetic. The model ig-

nores other interactions, such as those of Dzyaloshinsky-Moriya type, which may arise at

smaller energy scales. The terms we consider thus retain SU(2) spin rotation symmetry.

In the rest of this chapter we study the above Hamiltonian using various mean field

techniques to obtain information about its possible ground states, and compare the results

against the experimentally observed low energy properties of Volborthite.


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

J′/J

2

J1/J2

FerrimagneticOrder

Classical Spiral OrderS =1/2 Spiral Order

(b)

Figure 7.2: (a) Classical ferrimagnetic order that is obtained as a small-J2 ground state(see text). (b) Classical and Schwinger Boson mean field theory phase diagram as afunction of J1, J ′. The two magnetically ordered states are ferrimagnet and spiral.There are no disordered phase within Schwinger Boson mean field theory. We show thephase boundaries between the ferrimagnetic and spiral ordered states, for the classicallimit (S →∞), as well as S = 1

2.

7.3 Classical Ground state and Schwinger-boson mean

field theory

In this section, we explore the classical solution of the Heisenberg Hamiltonian (7.1) as

well as results from Schwinger Boson mean-field theory.

7.3.1 Classical ground state

For the classical ground state, we limit ourselves to solutions where the magnetic unit

cell is same as the structural unit cell of the kagome lattice. This can then be calculated

with the Luttinger-Tisza approximations. [215]

For the Hamiltonian in Eq. 7.1, the classical magnetic ordering is found to be planar

throughout the J1-J2-J ′ phase diagram. As expected, when the frustrating interaction

J2 is small compared to J1 or J ′, the system takes on unfrustrated ferrimagnetic order,

satisfying all J1 and J ′ interactions, as seen in 7.2(a). In this state, the chains are

ferromagnetically ordered, while the interstitial spins are anti-parallel to the chains.

As J2 increases, the ordering changes from the uniform Q = 0 ferrimagnetic state to

a Q 6= 0 spiral phase. Interestingly, this spiral order only has Qx 6= 0 (while Qy = 0),

spiral order along the J1-J2 chain direction. Qx increases as the system moves deeper

into the spiral phase. The classical phase diagram can be seen in 7.2(b). This classical

solution agrees with the S → ∞ limit of the Schwinger Boson approach, which we now


discuss.

7.3.2 Schwinger-Boson Mean-field theory

In this section, we analyze the Heisenberg model introduced in Eq. (7.1) using a

Schwinger boson mean-field theory, and we proceed as discussed in Sec. 2.3.2. This

gives us the following mean field parameters (A, B, and C indicate the sublattices in Fig.

7.1).

1. FM NN Interactions, J1 : χAB, χBA

2. AF NNN Interactions, J2 : ηAA, ηBB

3. AF NN Interactions, J ′ : ηBC , ηCB, ηCA, ηAC.

Assuming that the set of ansatze does not break further lattice point group symme-

tries, we get the following constraints

|ηBC | = |ηCB| = |ηCA| = |ηAC |,

|ηAA| = |ηBB|, |χAB| = |χBA|. (7.2)

Following the mean-field decouplings, the quadratic mean-field bosonic hamiltonian

is diagonalized using Fourier transform. The details are given in Appendix B.1.

In the absence of condensation, the mean-field Hamiltonian is given by

HMF = HC +∑k

b†kHkbk, (7.3)

HC = Nc

(2J1|χAB|2 + 2J2|ηAA|2 + 4J ′|ηBC |2

)−Nc(2µa + µc)(2S + 1),

bTk =(bkA↑, bkB↑, bkC↑, b

†−kA↓, b

†−kB↓, b

†−kC↓

), (7.4)

and the form of Hk is given in Appendix. B.2.

The quadratic Hamiltonian in (7.3) is then diagonalized by the Bogoliubov transfor-

mation bk = Rkγk, with γk being the rotated vector given by a relation similar to Eq.

7.4. For the diagonalized bosons in γk to have the proper commutation relations, we

must have (R†kηRk)mn = ηmn, where the matrix η is given by

ηαβ = δαβ

1 1 ≤ α ≤ 3

−1 4 ≤ α ≤ 6(7.5)


(since we have three flavours of ↑ or ↓ bosons, one for each site in the unit cell). This is

accomplished by taking the columns of Rk to be the eigenvectors ymk (with eigenvalues

ωmk) of ηHk. [189]

When the Hamiltonian Hk is positive definite, the corresponding spinon dispersions

|ωmk| are gapped. This leads to a disordered ground state. In the case of a gapless

dispersion, ηHk0ym′k0 = 0 for some m′ and k0, so that both Hk0 and ηHk0 have a zero

eigenvalue for at the dispersion minimum k0. Condensation of such zero-energy bosons

gives rise to a magnetically ordered state. To describe this condensation, we replace the

operators bk0 with macroscopic constant values,∑m′

xm′k0 =∑m′

cm′k0

√Nucym′k0 , (7.6)

where y†m′k0ym′k0 = 1 and Nuc is the number of unit cells. With the condensate contri-

bution, the diagonalized Hamiltonian is

HMF = HC +∑k

6∑m=1

γ†mk|ωmk|γmk +∑k0

∑m′

x†m′k0Hk0xm′k0 , (7.7)

where m′ runs over the set of zero-energy eigenvalues for the minimum wavevector k0.

Spin ordering from the boson condensate is found at the wavevector 2k0.

We self-consistently solve for the mean-field values of χ and η, while finding the values

of the chemical potentials µ and (if applicable) condensate parameters y.

We first determine the magnetic ordering for large value of S, which we find agrees

with the classical ordering found earlier. Next, we look at the evolution of the phase

diagram as S is lowered.

The S = 12

phase diagram can be seen in 7.2(b). The systems remains magnetically

ordered throughout the phase diagram, and we see a shift in the boundary between

ferrimagnetic and spiral phases. The ferrimagnetic phase stabilizes for larger J ′ relative

to the classical order, while the spiral state stabilizes slightly for larger J1. Nonetheless,

we do not see a large qualitative change from the classical result as the number of bosons

per site is lowered.

These results presented in this section suggest a conventional magnetically ordered

ground state both within classical approach and Schwinger-boson mean-field theory. This

is at odds with the many experimental results discussed in Sec. 7.1: these results would

predict a low temperature contribution to the magnetic specific heat, Cm ∝ T 2.


We now consider a slave fermion mean-field theory to see if it can stabilize an uncon-

ventional paramagnet that may explain the experimental results.

7.4 Slave Fermion Approach

Following the Fermionic mean-field theory discussed in Sec. 2.3.2, we can re-write the

Heisenberg terms in singlet and triplet mean-field parameters, as follows:

1. FM NN Interactions, J1 : EAB, EBA

2. AF NNN Interactions, J2 : χAA, χBB

3. AF NN Interactions, J ′ : χBC , χCB, χCA, χAC

For the particle-particle decouplings represented by η and D, we set both of them

equal to zero:

ηij = Dij = 0 ∀ij. (7.8)

As a result, this ansatz has a U(1) gauge redundancy. We shall briefly discuss the

role of such particle-particle pairing terms at the end of this chapter. The mean field

Hamiltonian is invariant under the gauge transformation

fiα → eiθifiα, χij → χijei(θi−θj), Eij → Eije

i(θi−θj). (7.9)

Hence, our ansatz describes a U(1) spin liquid within projective symmetry group classi-

fication of the spin liquids. [176]

We perform the Fourier transform (see Appendix B.1) and include chemical potential

terms as in the bosonic case. We choose a similar ansatz that preserves various lattice

symmetries and get

χ′ : |χBC | = |χCB| = |χCA| = |χAC |,

χ2 : |χAA| = |χBB|, E1 : |EAB| = |EBA|. (7.10)

While the projective symmetry group can determine all the triplet U(1) spin liquids with

the symmetry of our problem, the number of such states makes classification difficult.

The ansatz we have chosen aims to strike a compromise between generality and fruitful

computation.


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

J1/J

2

0.2

0.4

0.6

0.8

1

1.2

J′/

J2

a b c

d e f

Figure 7.3: Phase diagram for the slave fermion method described in Sec. 7.4. Eachphase is a U(1) spin liquid with different symmetries. We label them by which mean-fieldparameters are non-zero, as follows. a. χ2 6= 0. b. χ2,E1 6= 0. c. E1 6= 0. d. χ2, χ

′ 6= 0.e. χ2, χ

′,E1 6= 0. f. χ′,E1 6= 0.

We have the mean-field Hamiltonian

HMF = HC +∑k

f †kHkfk, (7.11)

HC = Nc

(2J1|EAB|2 + 2J2|χAA|2 + 4J ′|χBC |2

),

fTk =(fkA↑, fkA↓, fkB↑, fkB↓, fkC↑fkC↓

). (7.12)

The explicit form of Hk is given in Appendix B.2.

The quadratic Hamiltonian in (7.11) is diagonalized by the Bogoliubov transformation

for Fermions that is given by fk = Zkρk, with ρk being the rotated Fermion operators

defined in the same way as fk in Eq. 7.12. For Fermionic Bogoliubov rotations, the

diagonalized Fermions in ρk have the proper anti-commutation relations if (Z†kZk)mn =

δmn. This is accomplished by taking the columns of Zk to be the eigenvectors (with

eigenvalues ωmk) of Hk. [189]

As with the bosonic case, we now solve the mean-field Hamiltonian self-consistently,

choosing the Lagrange multipliers µ to enforce the constraint.

7.4.1 Results

Here, we discuss the results of our self-consistent mean-field theory. We present the phase

diagram obtained from the slave fermion method in Fig. 7.3 as a function of J ′/J2 and

J1/J2. In each phase, we find a different U(1) spin liquid that is characterized by the set

of non-zero hopping parameters.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(k⋅v)/2π

(k⋅u)/2π

(a)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(k⋅v)/2π

(k⋅u)/2π

(b)

Figure 7.4: Spinon Fermi surfaces at zero field for a. J1 = 0.3, J ′ = 0.8 and J2 = 1.0, andb. J1 = 0.8, J ′ = 0.8 and J2 = 1.0. J ′ = 0.8 and J2 = 1.0. Hole pockets are depicted asforward slashes and spinon pockets as backward slashes, while 1 spinon per wave vectoris depicted as empty. u and v are reciprocal lattice vectors, described in Appendix B.1.In phases where the mean-field solution is partly decoupled (some parameters zero), wesee perfect nesting of the spinon Fermi surface, making instabilities much more likely.

In terms of the ansatz in Eq. 7.10 the six phases are as discussed below. Some

corresponding Fermi-surfaces are shown in Fig. 7.4.

State a: χ2 6= 0. This is a one-dimensional state, in which inter-chain couplings χ′ = 0.

One-dimensionality is likely an artifact of the mean-field theory. This state has Fermi

points, and we expect this state to be unstable to confinement, since (1+1)-dimensional

U(1) gauge theory is always confining.

State b: χ2,E1 6= 0. This state is one-dimensional and gapped, and thus unstable as

well.

State c: E1 6= 0. Like the two previous states, this is also one-dimensional state. This

state is gapped, and immediately unstable to confinement as before.

State d: χ2, χ′ 6= 0. In this state, E1 = 0, and hence J1 plays no role at the mean field

level.The likelihood of a magnetically ordered ground state is still high in this region,

where J1 is small and the frustration is not acute (refer to the bosonic analysis in the

previous section). This is reflected in the spinon Fermi surface, an example of which is

given in Fig. 7.4(a), which has perfect nesting. From our analysis of the classical model

and Schwinger-boson mean field theory in the previous section, we may expect that this

state is unstable to a spiral ordered state.


State f: E1, χ′ 6= 0. In this state, χ2 = 0 and J2 plays no role, leaving the resulting

interactions un-frustrated. Its Fermi surface also contains perfect nesting, pointing to

an instability towards ferrimagnetic order. This is consistent with the bosonic analysis,

although in that case spiral order sets in at smaller J2 than where this state f is found.

State e: χ2, χ′,E1 6= 0. Unlike the aforementioned states, this state has neither low

frustration inherent in J, nor perfect nesting of the Fermi surface, as seen in Fig.

7.4(b). Furthermore, it is two dimensional and gapless, hence stable to confinement. [216]

Thus, this state is an example of a two-dimensional, stable (within mean field analysis)

U(1) spin liquid with a spinon Fermi surface.

Of all the above states obtained in this slave-Fermion mean field theory, only the

state e seems to be stable. All the others have either a perfectly nested Fermi surface

(like states d and f) or are one dimensional (like states a, b and c). Hence all these

other states are unstable to confinement, where the confined state may have magnetic

order as found in our previous classical analysis. The density functional analysis suggests

coupling parameters that are likely to lie within the region of state f. Since we do not

know the precise magnitudes of the coupling constants in our present model that suits

to describe the physics of Volborthite, here we study the properties state e as a possible

candidate ground state for this compound. This in turn puts some bounds (within mean

field theory) on the values of the coupling constants.

Once again, we note that the state e, which does not decouple into one-dimensional

chains and is in a strongly frustrated regime, has a Fermi surface lacking obvious in-

stabilities. These features make the spin liquid slave-Fermion representation a relevant

starting point for analysis of this state. However, the complicated Fermi surface, as seen

in Fig. 7.4(b), still shows some nested segments in the Fermi surface. Thus, this state

may also have a partial nesting instability where the nested segments are gapped out due

to this instability leaving remnant parts of the Fermi surface still intact which then un-

dergoes reconstruction. Only part of the Fermi surface might be gapped out as a result.

In this scenario, the low-energy excitations of the reconstructed Fermi surface remain in

addition to the SDW order. The spin-density-wave term Sq ∝∑

k f†kασαβfk+qβ describes

a magnetically ordered state, but if spinons remain deconfined, this is an exotic state,

the so-called SDW* state. [217] Although time-reversal and spin-rotational symmetries of

the U(1) spin liquid are broken by the magnetic order, the spinon Fermi surface plays an

important role in the low-energy physics, in conjunction with spin waves. This can then

explain the simultaneous presence of magnetic order and Fermi liquid-like specific heat,

as observed in Volborthite experiments. We substantiate this scenario in the upcoming


0 0.2 0.4 0.6 0.8 1

(k ⋅ u)/2 π

0

0.2

0.4

0.6

0.8

1

(k ⋅ v

)/2 π

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

Figure 7.5: Largest eigenvalue of the bare susceptibility matrix for J1 = J ′ = 0.8 andJ2 = 1.0 at zero field. The maximum of the bare susceptibility occurs at k · u = 0,k · v ∼ 0.05π, where we look for the development of SDW order.

section.

7.5 Spinon Spin-Density-Wave Order

In this section, we probe possible SDW instabilities of the spinon Fermi surface of state

e that arises from partial nesting. Such instabilities may lead to incommensurate SDW

order for the spinons.

To locate the wave vector Q of the SDW instability, we compute the bare spin suscep-

tibility for a given set of χ and E mean field parameters. This is given in Fig. 7.5. The

maximum of the bare susceptibility occurs at k · u = 0, k · v ∼ 0.05π for the parameter

values J1 = J ′ = 0.8, J2 = 1.0. channel in the mean field decoupling, along χ and E, to

determine if such a phase is indeed stabilized and also form of the reconstructed Fermi

surface.

We include short ranged repulsion among the spinons, which may arise in the low

energy theory from the single spinon (per site) constraint. This takes the form of an on-

site Hubbard-like term for the spinons U∑

laσ nlaσnlaσ and calculate the static structure

factor, χGRPA(q, ω = 0), within the generalized random phase approximation (RPA),

where

χGRPA(q, ω) = (I + 2Uχ0(q, ω))−1χ0(q, ω). (7.13)

Details of the calculation are given in Appendix B.3. We find (not shown) that the

magnitude of the susceptibility is enhanced at Q, while the value of Q depends very


weakly on the value of U .

These calculations suggest that the state has a tendency to break spin rotation sym-

metry spontaneously by developing SDW instability of the spinon Fermi surface. To

investigate this further, we can decouple the spin Hamiltonian in Eq. (7.1), keeping the

SDW channel in addition to the χij and Eij channels introduced before. Another route

would be to decouple the aforementioned repulsive Hubbard-like interaction between

spinons, U∑

laσ nlaσnlaσ, in the SDW channel. We must emphasize that this SDW order

results from a spinon “spin-density wave”; a spinon density wave, on the other hand, is

incompatible with the single spinon per site constraint. For the SDW channel we look

for ordering at the wave-vector Q, where the peak of the RPA susceptibility occurs for a

particular set of J within the state e. The associated eigenvector also specifies the spin

structure within the unit cell. We also note that unlike in the bosonic case, the value of

Q is not naturally connected to the underlying structural unit cell. Indeed, this seems

to be a feature of the SDW ordering found in Volborthite.

The spin-density wave order parameter for the wave vector q and sublattice m is

given by the expectation value of the operator

Sqm =1

Nc

∑k

f †kmασαβfk+qmβ, (7.14)

where Nc is the number of unit cells. To probe for instabilities we keep only the channel

q = Q. Here we note that, while it is possible to look for SDW instability for a general q

by varying it, numerical implementation of this approach is tedious, since the folding of

the Brillouin zone depends on the value of the wave vector at which the SDW instability

occurs. We avoid such complications in the present mean field treatment, which at any

rate can only find local minima. Also, since we are interested in development of SDW

order in the state e, we assume that the this state is stable to the development of such

order. Hence, to the leading order, we neglect the renormalization of the parameters χij

and Eij due to the development of SDW order, 〈SQ〉.

The mean field decoupling of the Hubbard-like interaction of the Hubbard-like inter-

action in the SDW channel is given by

U∑m

(− 〈SQm〉S−Qm − 〈S−Qm〉SQm + | 〈SQm〉 |2

), (7.15)

where 〈SQm〉 is determined self-consistently for a fixed Q. Since SQ couples wave-vectors

k to k + Q (see Eq. 7.14), we need to extend unit cell (the basis in Eq. (7.12)) to include


the terms up to

fk+Q mα, fk+2Q mα, . . . , fk+pQ (7.16)

where m = A,B,C, α =↑, ↓ and p is an integer such that pQ · r (r being the Bravias

lattice vector) is an integer multiple of 2π.

7.5.1 Results

Here, we consider the magnetic instabilities described above for a representative point in

state e of Fig. 7.3, with J1 = J ′ = 0.8, J2 = 1.0. The bare susceptibility, as shown in

Fig. 7.5 and discussed before, shows a maximum at k · u = 0, k · v ∼ 0.05π.

On incorporating the SDW channel, we indeed stabilize a state with non-zero SDW

order at wave-vector Q, to have lower energy. Within the mean-field theory, this requires

a finite value of U ∼ J2/2, and we find that SDW order increases gradually with U , with

more of the spinon Fermi surface becoming gapped out. We take the value U = J2 to

show the effect of the SDW instability upon the spinon Fermi surface. We repeat that the

value of Q is not tied, in any obvious way, to the underlying lattice structure as in the case

of bosonic analysis. After folding the Fermi surface due to SDW order, we find gapless

portions of the reconstructed Fermi surface remain after reconstruction. Working with a

finite lattice, we choose the system size to obtain the reciprocal lattice vector Q closest

to the RPA result. With incommensurate wavevectors being approximated, the Brillouin

zone must be folded to an area reduced on the order of the system size. The reconstructed

Fermi surface thus does not connect in a useful way to the original one, so we do not

present it here. The gapless spinon excitations about the reconstructed Fermi surface

still protect this state from confinement. [216] Additionally, the spin rotation symmetry

is broken and this SDW order, in principle, can explain the NMR experimental data on

Volborthite. We shall discuss the experimental implications in greater detail in the next

section.

It is difficult to compare the Fermi-surfaces with and without SDW order, because

their shapes are complicated and they are not gauge invariant. Nevertheless, it is possible

to compare their effects in thermodynamic quantities like heat capacity, which depends

on the density of states at the Fermi-surface as well as the effective mass of the spinons.

In Fig. 7.6 we show the low-temperature specific heat contributions from the spinons.

It shows T -linear behaviour that results from low-energy excitations around the spinon

Fermi surface. The T -linear behaviour is seen both below and above the kink (where

higher-lying bands become important). At the mean-field level, this is a direct conse-


0

0.01

0.02

0.03

0.04

0.05

4.0⋅10-4

8.0⋅10-4

1.2⋅10-3

1.6⋅10-3

2.0⋅10-3

C/k

BJ2

kB

T/J2

Figure 7.6: Heat capacity C(T ) per unit cell for J1 = J ′ = 0.8, J2 = 1.0 and no magneticfield. It exhibits low-temperature T -linear behaviour above and below the kink.

quence of free spinons with the reconstructed band structure, which will have a number

of bands proportional to the size of the system.

Gapless gauge field fluctuations in a U(1) spin liquid are an integral part of the

state. Beyond being responsible for the confinement transition, they may provide im-

portant corrections to the standard behaviour of the mean-field state. For instance, the

low-temperature specific heat for free Fermions coupled to a U(1) gauge field in two di-

mensions scales at C ∝ T23 . [218] How the spinon band structure changes this behaviour

remains an intriguing open question.

Given the interesting parallels to the low-field phase of Volborthite, (discussed in

greater detail in Sec. 7.6) we will look more closely at this phase by including the effect

of a non-zero external magnetic field.

7.5.2 Zeeman Field

Here, we consider the effect of an external field upon the state e. We add an external

Zeeman term to the Hamiltonian (Eq. 7.1),

HZeeman = −∑i

h · Si, (7.17)

where h = gµbH and we set the parameters to J1 = J ′ = 0.8 J2 as a representative point

to explore the effect of Zeeman field. We will take h ∝ z.

An increase in h brings about a change in the spinon Fermi surface, where the smaller

pockets in Fig. 7.4(b) shrink and eventually vanish. Meanwhile, the two large pockets


Figure 7.7: Magnitude of the spin-ordering wavevector giving the largest contributionto the spin susceptibility. We show the smallest such magnitude as a function of h, forJ1 = 0.8, J ′ = 0.8 and J2 = 1.0. For reference, |Q| = 0.65 A, the relevant short-rangeorder at low temperatures in Volborthite, is also shown.

lose their curvature and gain more obvious nesting, as seen in Fig. 7.8. Thus the value of

the wave-vector Q for SDW instability, calculated from the RPA susceptibility, changes.

We show the increase in the magnitude |Q| with field h in Fig. 7.7.

With an increase in nesting, the spin-density-wave instability lowers the density of

states around the Fermi level and hence decreases the specific heat. We probe this by

systematically looking at the effect of h upon the low-temperature specific heat. For fields

h ∼ 0.8 and higher, the nesting instability completely gaps out the spinon Fermi surface.

For larger fields, the gap is present even before instabilities are taken into account.

Such a gapped U(1) spin liquid immediately becomes unstable to confinement, [178]

and hence a conventional magnetically ordered phase results at high fields once the spinon

Fermi surface is completely gapped out. Since the linear specific heat is sensitive to the

presence of spinon Fermi surface, the decrease in the gapless portions of the Fermi surface

results in decrease in the constant of proportionality, γ. This is shown in Fig. 7.9 where

the low-temperature specific heat for several values of h is plotted. We see the general

trend of a decrease in γ with h.

Having described the general framework of our slave fermion mean field theory, we

now consider its applicability to the current experiments on Volborthite.

7.6 Comparison to Experiment

In this section, we discuss how the experimental results on Volborthite may be explained

within our slave-Fermion analysis.

In the SDW∗ phase discussed above, the Fermi surface of spinons, which survives even


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(k⋅v)/2π

(k⋅u)/2π

Figure 7.8: Spinon Fermi surface for J1 = 0.8, J ′ = 0.8 and J2 = 1.0, and h = 0.08. Thehole pocket is depicted as forward slashes and the spinon pocket as backward slashes,while 1 spinon per wavevector is depicted as empty. u and v are reciprocal lattice vectors,described in Appendix B.1. As the Zeeman field increases, so too does the nesting of thespinon Fermi surface (compare to Fig. 7.4 (a)), making instabilities much more likely.

0

0.005

0.01

0.015

0.02

0.025

2⋅10-4

4⋅10-4

6⋅10-4

8⋅10-4

1⋅10-3

C/k

BJ2

kB

T/J2

h/J2 = 0.00

h/J2 = 0.02

h/J2 = 0.04

Figure 7.9: Heat capacity C(T ) per unit cell for J1 = J ′ = 0.8, J2 = 1.0 and h/J2 =0, 0.02, 0.04. There is a general decrease with h of the low-temperature slope γ. Thisbehaviour indicates a lowered density of states at the Fermi level as field increases.


after the introduction of spin-density-wave order, dominates the low-energy thermody-

namic properties of this phase. The T -linear heat capacity and 1T1

spin-lattice relaxation

rate follow from the low-energy excitations around the Fermi surface which is in close

parallel with a regular Fermi-liquid metal.

The coexisting spin-density order at incommensurate wavevectors gives rise to the

distribution of magnetic moments seen in the Lorentzian NMR line shape. According

to the experiments, such an SDW instability of the spinon Fermi surface seems to set

in around T ∼ 1 K. A T 2 contribution to the specific heat coming from the Goldstone

modes of broken spin rotation symmetry gives a sub-leading effect.

A further advantage of the SDW* state is that the ordering wavevector is an emergent

result of the spinon Fermi surface, which may allow a greater flexibility in comparison

with neutron scattering. Here, the results show short-range order, with wavenumber

Q2 = 1.15 A (corresponding to q = 0 order for the antiferromagnetic kagome model)

being overtaken by wavenumber Q1 = 0.65 A as temperature decreases. [219] However,

this cannot be accounted for by the traditional order-by-disorder seen in the Heisenberg

kagome antiferromagnet, [219] nor can it be obtained from classical or Schwinger boson

magnetic order in our coupled-chain model. In both cases, spiral order is along the x-axis,

since q ·(v+u/2) = 2π. As |q| is too large, this range of wavevectors cannot describe this

Q1 ordering. In contrast, the wavevector of the SDW* state results from nesting details

of the spinon Fermi surface. As we’ve seen, this is sensitive to the couplings J , and |Q|varies significantly with h. This may be consistent with Q1, though comparison in finite

field (absence of spin-glass behaviour) is difficult. Besides the behaviour of ordering, this

SDW* state can capture other qualitative features of the low-field phase diagram as h

increases. In particular, it finds the decreasing coefficient of specific heat γ, and provides

the transition to conventional order as the entire spinon Fermi surface becomes gapped.

The scenario described in this chapter may be a finite-temperature story, since the

relevant specific heat data in the low-field phase has been found above 0.8 K, up to 1

K. [208] At lower temperatures, the nature of specific heat may be different, and pairing

may play a prominent role. However, for the field-induced phase transition scenario

studied in this chapter, a state with pairing is a less natural explanation, as we now

briefly sketch. With additional pairing terms, we then have a Z2 spin liquid, which may

have a similar SDW* instability. When this Z2 spin liquid is gapless, the low-energy

excitations will give similar behaviour to the U(1) SDW* state. In contrast with the

U(1) case, gapped Z2 spin liquids (and corresponding SDW* states) are stable, and

do not undergo the confinement transition. As in the conventional ordered state, the

SDW order will have a T 2 contribution to the heat capacity, which is now the leading


contribution. Without a confinement transition, the gap must open at the same time that

the SDW order becomes commensurate to capture the transition to the high-field phase.

Achieving this commensurate order from nesting will be difficult without first opening

a gap. Conversely, the conventional order after the U(1) confinement transition will

not be directly connected to the Fermi surface structure before the transition, allowing

for a different route to commensurate order. We present the U(1) SDW* state as an

exotic state whose confinement transition provides a possible picture for the low-field to

high-field phase transition.

Beyond the discussion above, there remain other interesting results left unaccounted

for. Beside the transition at 4.3 T, there are seen step-like magnetization jumps at

25.5 and 46 T, at 16

and 13

of saturation, respectively. [220] Recent NMR measurements

indicate another transition between magnetically ordered states at 26 T, coinciding with

the magnetization jump. [221] The destabilization of states with fractional magnetization

runs counter to the usual stabilization, and subsequent magnetization plateau, of such

states. Furthermore, an understanding of the effect of disorder on possible spin liquid

states remains an interesting problem, due to the disorder-induced spin glass state.

7.7 Conclusion

We have investigated the Heisenberg Hamiltonian (7.1) of coupled frustrated chains as

a model for the distorted spin-12

kagome compound Volborthite. We have used three

different approaches (classical ordering, Schwinger boson and slave fermion) to probe

both the semi-classical and quantum regimes. The Schwinger boson approach, along with

the classical analysis, found conventional ferrimagnetic (for small J2) or spiral order. The

slave fermion approach yielded different magnetic instabilities of the spinon Fermi surface

in different regions of the phase diagram. While conventional order correspondingly

resulted in the relatively unfrustrated cases of large or small J1, we also found an exotic

SDW* state for the relatively frustrated case of J22

. J1 . J2, J ′ & J22

. In this state,

spin-density-wave order coexists with a reconstructed spinon Fermi surface, and spinons

may remain deconfined.

This SDW* state may explain several properties of the low-field phase of Volborthite.

This phase features a spin-density-wave type of magnetic moment distribution, as found

in the incommensurate wavevectors of the SDW order parameter in the SDW* state.

The Fermi-liquid-like specific heat and 1T1

spin-lattice relaxation rate (both are linear in

temperature) coincide with the reconstructed Fermi surface of spinons in the SDW* state.

The T -linear heat capacity coefficient γ decreases with applied field, which is mirrored by


the spinon Fermi surface becoming increasingly gapped out. These results are inconsistent

with the conventional magnetic order found in the classical or Schwinger boson analyses of

this model. Furthermore, the spiral was found along the chain direction, disagreeing with

the low-temperature neutron scattering wavevector. Our mean-field treatment suggests

that the exotic SDW* state may be a relevant part of the description of the low-field

phase, even as models are refined beyond the current Heisenberg coupled frustrated chain

model.

As mentioned earlier, a two-dimensional U(1) spin liquid with a Fermi surface is

expected to have C ∝ T23 at low temperatures. It is unclear how the band-structure of the

spinons changes this behaviour, and at what temperature. A systematic understanding

of gauge-field fluctuations on heat capacity, or extremely low-temperature magnetic heat

capacity data, could help to reconcile the observed T -linear behaviour to date with the

T23 expectation.

This work serves as a basis for considering the relevance of the SDW* state, and

coupled-chain model, to the low-field phase of Volborthite. While exotic, the SDW*

state can explain the anomalous features of the low-field phase. Further corrections to

the coupled-chain model could yield quantitative corrections to compare directly with

experiment, particularly the behaviour of the heat capacity. Experimentally, cleaner

samples are instrumental in suppressing the spurious spin-glass formation at zero and very

low fields. Along with single crystals of this structure, neutron scattering could determine

the precise nature of the magnetic order in the low-field state. Finally, thermal transport

measurements may be able to detect deconfined spinons, [222] further pinpointing the

unconventional nature of this low-field phase.

Chapter 8

Quantum Spin Liquids in the

Absence of Spin Rotation Symmetry

8.1 Introduction

In Sec. 1.4, we saw that the isotropic kagome lattice material Herbertsmithite is a

candidate spin liquid. The nearest-neighbour kagome antiferromagnet was also seen to

lie close to quantum phase transitions upon tuning second-neighbour or Dzyaloshinsky-

Moriya interactions.

For Herbertsmithite, this means that, due to the potential proximity to a quantum

phase transition, small perturbing interactions may play an important role, particularly

at low energies. [144] Hence, it may be important to account for these smaller perturbing

energy scales in order to understand the experiments on Herbertsmithite. Electron spin

resonance (ESR) and magnetic susceptibility measurements suggest that perturbations

break spin-rotation symmetry, with both a Dzyaloshinsky-Moriya (DM) interaction and

an Ising-like easy-axis exchange, each of strengths around 10 percent of the NN antifer-

romagnetic exchange interaction. [115,120,223,224]

In this chapter, we study the effect of these further perturbations on various QSL

states that potentially offer an explanation of the unusual phenomenology of the non-

magnetic ground state in Herbertsmithite. We contrast these states with respect to their

signature in spin-spin correlations, which is measured in inelastic neutron scattering

measurements, by calculating the dynamical spin-structure factor for a host of candidate

Z2 and associated U(1) spin liquid states that are allowed by the projective symmetry

group analysis on the Kagome lattice. [117] In addition, we calculate the ESR absorption

spectra that provide useful information in systems without spin-rotation invariance, and

102

Chapter 8. Spin Liquids Without Spin Rotation Symmetry 103

hence can help us to identify the nature of the QSL. [225,226] ESR absorption resulting

from DM interactions has been probed in, for instance, the quasi-two-dimensional system

Cs2CuCl4. [227] In this compound, the ESR absorption can be interpreted in terms of

deconfined spin-12

spinons, in contrast with the spin chain Cu Benzoate, where bound

states dominate the ESR spectra. [228,229]

Due to the involved nature of the results presented here, before discussing further de-

tails, we briefly summarize first the current situation of the spin liquid physics on Kagome

antiferromagnets in view of Herbertsmithite, and second the main results obtained in this

work.

8.1.1 Summary of results: Spin liquid physics in Herbertsmithite

The current theoretical framework of constructing QSL states is largely based on slave-

particle mean-field theories, as discussed in Chapter 2. In particular, we shall focus on the

slave-fermion mean-field theory, whose gauge redundancy has been discussed in Chapter

3. If the Hamitonian is spin-rotation symmetric, the hopping and the pairing channels

can be completely separated into a spin independent (singlet) part and a spin dependent

(triplet) part (Section 8.3.1). In this work we refer to them as singlet and triplet channels

(or terms) respectively. In usual studies of models with frustrated antiferromagnetic

spin-rotation invariant exchanges, the singlet ansatze give rise to stable QSL mean field

solutions and hence the triplet channels are absent. Thus, we obtain symmetric QSLs

that do not spontaneously break spin-rotation symmetry. However, it is known that in

the presence of competing ferromagnetic and antiferromagnetic interactions, both singlet

and triplet channels can exist (and may give rise to exotic three dimensional gapped U(1)

spin liquid states with topologically protected gapless surface spinon modes [168,230]).

A PSG analysis [117] of the singlet ansatze on Kagome lattice shows that there are

eight Z2 QSL states with gapped or gapless fermionic spinons that can be stabilized by

NN and NNN hopping and pairing terms for the spinons. All these states are found

in the neighbourhood of four parent U(1) states, one of which, with Dirac-like spinon

excitations, was seemingly realized as the ground state of the NN-HKAF model in recent

VMC studies. [111] Hence, we start by looking at these parent QSLs, and note that they

comprise only a small number of the possible U(1) spin liquid ansatze on the kagome

lattice. Since the above spin liquids are time reversal symmetric, there exists a gauge

where the singlet U(1) QSL mean-field ansatze can be written employing NN real, singlet

hopping terms. These spin liquids are conveniently labelled using the following notation:

U(1)[a, b], [117] where a (b) denotes the magnetic flux of the emergent gauge field through


the triangular (hexagonal) plaquette of the Kagome lattice. Time reversal symmetry

constrains these fluxes to be either 0 or π. Accordingly, we have U(1)[0, 0], U(1)[0, π],

U(1)[π, π], and U(1)[π, 0] QSLs. [117, 136] From these four U(1) QSLs, the eight Z2

QSL are obtained by tuning in appropriate spinon-pairing terms (up to second nearest

neighbour) allowed by PSG. These pairing terms necessarily break the gauge structure

from U(1) down to Z2, hence the name. The nomenclature for the Z2 spin liquids is

derived straightforwardly from their parent U(1) states. [117] The added Latin/Greek

letter distinguishes between two or more Z2 QSL obtained from the same U(1) parent.

[117] Both the U(1) and Z2 QSLs have characteristic spinon band structure. We note

that, while this spinon dispersion is not a gauge invariant quantity, it is still useful to

understand its general features in order to capture various properties of these states

like the nature of the experimentally observable spin-structure factor. [28] Hence, we

summarize the general characteristic features of the spinon spectra (discussed in detail

in Section 8.4.1) in the second column of Table 8.1.

More recent work, however, indicates that the NN-HKAF model likely sits close to a

quantum phase transition between a Z2 spin liquid and a valence-bond-solid state upon

tuning the second-neighbour interaction, or a magnetically ordered state upon tuning the

Dzyaloshinsky-Moriya interaction. [108,112–114,118,119,144] It is this proximity to the

quantum phase transition that may make the system very sensitive to small perturbations

which then can have a sizable effect. [144] The perturbations to the HKAF seen in

Herbertsmithite are then likely to be relevant to a proper description of the material,

and we endeavour to gain an understanding of how the aforementioned symmetric spin

liquid states are affected under these perturbations, particularly the breaking of spin-

rotation symmetry induced by DM interaction and Ising anisotropies.

Since the spin-rotation symmetry is broken, the singlet and the triplet spinon de-

coupling channels (both hopping and pairing) can no longer be separated, and a more

general QSL ansatze incorporating both singlet and triplet decomposition, and allowing

their intermixing, needs to be introduced. [177] While it is possible to use such a general

four-component spinor representation, as in the case of a triplet BCS Hamiltonian, here,

we find that since the DM vectors and the Ising anisotropies both are perpendicular to

the Kagome plane, we can use a two-component spinor representation. The effect of

such spin-rotation symmetry breaking terms on the low-energy excitations of these spin

liquids is also summarized in the Table 8.1 (third column). Generally, the triplet terms

decrease the density of states at the spinon Fermi-level in most of the U(1) and Z2 QSLs.

Beyond mean field theory, apart from the gapless/gapped spinon excitations, there

exist additional low-energy excitations, depending on the gauge structure. For example,


QSL Label Singlet Ansatz [117] “Singlet + Triplet” Ansatz

U(1)[0, 0] Fermi surface (F.S.) F.S. is alteredbut not gapped out

U(1)[0, π] Dirac point (D.Pt.) D. Pt. is gapped out (∗)U(1)[π, π] Flat bands Bands acquire dispersion,

& D.Pt. D.Pt. remains intactU(1)[π, 0] Flat bands Bands acquire dispersion

Z2[0, 0]A F.S. gapped out to Band touching pointsBand touching points remain

Z2[0, π]β F. S. becomes Gap isfully gapped altered

Z2[0, 0]B F.S. shrinks compared F.S. gapped out toto parent U(1) band touching points

Z2[0, π]α D.Pt. changes to Band touching pointsband touching points remain

Z2[0, 0]D F.S. shifted compared F.S. gapped out toto parent U(1) band touching points

Z2[0, π]γ D.Pt. changes to Band touching pointsband touching points remain

Z2[π, π]B Negligible change Bands gapped to D.Pt.compared to and band touching pointsparent U(1)

Z2[π, 0]B Flat Bands are gapped F.S. gapped toto form a F. S. band touching point

Table 8.1: (* This state is unstable to confinement). Structure of low-energy spinonexcitations in the U(1) and Z2 under consideration, in the mean-field parameter regimedescribed in Sec. 8.3.1. These behaviours persist within the neighbourhood of the re-spective parameter sets; it gives a general understanding of these states. The singlet caseshows the evolution from U(1) to Z2 spin liquids upon addition of the bolded terms in Ta-ble 8.2. The “singlet + triplet” case shows the evolution of all these states upon additionof nearest-neighbour triplet terms. For Z2 spin liquids, F.S. indicates a Fermi surface ofFermionic quasi-particles, obtained upon diagonalization of the mean-field HamiltonianHQ. The low energy excitations lie about the spinon Fermi-level which is set by the halffilling constraint on the spinons (Eq. 2.11).


in gapless U(1) QSL in two spatial dimensions, there is an emergent gapless photon. For

Z2 spin liquids, the “magnetic flux” associated with the Z2 gauge field leads to gapped

topological excitations called visons. [171] In this work, we shall be mostly concerned with

the question of the nature of the ground state and spinful excitations, such as spinons

and spin-1 excitations.

Having identified the spin liquids, we turn to their dynamical spin-structure factor,

which is just the Fourier transform of spin-spin correlation function. The dynamical spin-

structure factor has been recently measured in inelastic neutron scattering experiments on

a single crystal sample of Herbertsmithite. It shows diffuse scattering with no signature

of the spin gap down to at least 1.5 meV. [116]

In our computations, almost all of the above spin-liquid ansatze show patterns of

diffuse intensity in the dynamical structure factor throughout the Brillouin zone, the

overall shape of which is largely influenced by the parent U(1) spin liquid. The structure

factor of the U(1) states have characteristic flat and dispersive features within the diffuse

continuum. In the Z2 spin liquids, the dispersive features are broadened and become

significantly more complex. A central result is that, for some Z2 spin liquids, these

dispersive features are only slightly stronger than the diffuse background, so that the

dynamical structure factor looks mostly diffuse. This is particularly true for the states

labelled Z2[0, π]β and Z2[0, π]α (the dynamical structure factor is shown later in Figs.

8.10 (e) and (f), respectively). This may be consistent with present neutron scattering

results where similar diffuse scattering is seen over a large energy window in large parts

of the Brillouin zone. [116] It is found that the spin-rotation symmetry breaking terms

primarily introduce non-zero intensity at the centre of the Brillouin zone. They also

split the spinon bands, so that diffuse scattering is found more evenly throughout all of

the Brillouin zone in the presence of DM and Ising interactions. The inelastic neutron

scattering indeed shows some intensity at the centre of the Brillouin zone, with a broad

maximum near 0.4J .

To better understand the effect of triplet terms on the ansatze we consider the ESR

absorption spectra that show a non-trivial response solely from spin-rotation breaking

terms. The ESR signal shows the largest absorption intensity at a δ-function peak at

the Zeeman field energy scale. Triplet terms create additional “satellite” absorption

peaks, offset by their energy scale. The number, position, and broadening of these

extra absorption peaks differentiate between the different ansatze. Particularly for the

aforementioned Z2[0, π]β and Z2[0, π]α states, the absorption spectrum provides a distinct

qualitative features (as in Figs.8.14 (e) and (f)) that can be verified experimentally.

The rest of this chapter is organized as follows. We begin in Sec. 8.2 by discussing the


123

45 6

789

1011 12

45 6

1

23

1011 12

7

89

Figure 8.1: Bond labels and directions for nearest-neighbour (solid) and next-nearest-neighbour (dashed) bonds on the Kagome lattice. The nearest-neighbour directions alsoindicate the directions of the Dzyaloshinsky-Moriya terms. The bonds labelled 1 are theones upon which mean-field parameters are defined as in Table 8.2; parameters on otherbonds are determined from these through the application of symmetry operations.

spin model used to capture the behaviour of Herbertsmithite. Following the introduction

of the spin Hamiltonian, in Sec. 8.3 we describe the fermionic slave-particle construction

of mean-field states. In Sec. 8.4 we describe the projective symmetry group analysis of

the Z2 ansatze, describing their respective U(1) parent QSL states as well as the effect

of the spin-rotation symmetry breaking terms. The spin correlations of these states are

characterized in in Sec. 8.5. Finally, in Sec. 8.5.5 we discuss the implications of these

correlations in light of the recent neutron scattering results. We summarize our results

in Sec. 8.6. The details of different calculations are given in various appendices.

8.2 Spin-12 Model on the Kagome Lattice

The starting point for our analysis is the spin-12

model with antiferromagnetic nearest-

neighbour Heisenberg interactions on the Kagome lattice,

HNN−KHAF = J∑〈ij〉

Si · Sj. (8.1)

where J ∼ 17 meV ∼ 200 K denotes strong antiferromagnetic interactions, and forms the

largest energy scale in Herbertsmithite. [121, 122] Experiments also indicate that there

may be three kinds of interactions acting as perturbations to the above Hamiltonian:

Dzyaloshinsky-Moriya (DM) interaction:

HDM = D∑〈ij〉

Dij · Si × Sj. (8.2)


C6σ

T1

T2

b

a

w u

v

Figure 8.2: Symmetry transformations of the Kagome lattice, including translations T1

and T2, a rotation C6 and reflection σ.

This has been estimated to have energy scale of |D| ∼ 15 K (perpendicular to the

Kagome planes) from ESR experiments [120] and fitting of the anisotropy of the magnetic

susceptibility data. [115] In Eq. (8.2), the orientation of Dij = z is out of the plane when

the bonds are counted in a counter-clock wise way around each triangle (Fig. 8.1).

The in-plane components of the DM interactions (∼ 2 K) appear to be negligible in

comparison with the out-of-plane component, and we will neglect it in our calculations.

XXZ anisotropy: The susceptibility anisotropy also indicates a sizable easy-axis spin-

spin interaction along an axis perpendicular to the Kagome plane,

HIsing = ∆∑〈ij〉

Szi Szj , (8.3)

where ∆ ∼ J/10 ∼ 20K.

Next-nearest-neighbour coupling: As discussed in the introduction, even small J2

can drastically affect the stability of spin liquids of the HKAF, as found in recent numer-

ical calculations. Accordingly, we consider adding an isotropic next-nearest-neighbour

antiferromagnetic Heisenberg interaction J2.

HNNN = J2

∑〈〈ij〉〉

Si · Sj. (8.4)

The complete Hamiltonian is then obtained by adding Eqns. (8.1) – (8.4). All these

terms have full symmetries of the Kagome lattice, shown in Fig. 8.2, and enumerated

below. These are:

• Translations T1 and T2 in the Kagome plane.

• A sixfold rotation C6 around the centre of a hexagon.


• A reflection σ through a hexagon.

With this Hamiltonian, we now construct possible Z2 spin liquid states with fermionic

spinons, both gapped and gapless, and then contrast their properties in context of ex-

periments on Herbertsmithite. We shall limit ourselves to Z2 spin liquid ansatze that

can be realized with just first and second-neighbour terms (both hopping and pairing).

Since mean-field theory only gives an order of magnitude estimate of the microscopic

parameters, we shall investigate the existence of the above spin liquids in somewhat a

broad range of the parameters, especially J2, whose effect is known to be quite sensitive

in stabilizing Z2 spin liquids. [114]

8.3 Slave-Fermion Construction of Quantum Spin Liq-

uid States

8.3.1 Mean-Field Decoupling

We use the slave-fermion representation as discussed in Chapter 2. The projective sym-

metry group classification follows as in Chapter 3. However, here we will use a spinor

basis, which will be more convenient for this case, where only the z-components of triplet

terms are found.

In the singlet case, HQ is conveniently written using a two-component Nambu basis

νi, which most clearly reveals the underlying gauge redundancy of HQ. The Hamiltonian

is given by [24,175,176,231]

HsingQ =

∑ij

ν†iUsingij νj, (8.5)

where the Uij matrix satisfies Uji = U †ij, and

νi =

(fi↑

f †i↓

), U sing

ij =

(χ∗ij −ηij−η∗ij −χij

). (8.6)

The exact constraint of single spinon per site (Eq (2.11)) is then relaxed to an average

constraint and imposed by Lagrange multipliers, given by [24,175,176,231]

µa∑a

ν†i τaνi. (8.7)


where µa (a = 0, 1, 2, 3) are the Lagrange multipliers. We note that the Nambu (νi) basis

components involve operators of the same spin type. This form clearly demonstrates the

explicit SU(2) gauge redundancy [24,175,176,231]

νi → Wiνi

Uij → WiUijW†j ∀Wi ∈ SU(2). (8.8)

The introduction of triplet terms necessitates, in principle, a four-component basis.

[168,177] However, since in the following we will use only the z-components of the triplet

terms, we may use the same basis νi as the singlet case, writing terms with the matrix

form U tripij , given as

U tripij =

(Ez∗ij Y z

ij

−Y z∗ij Ez

ij

). (8.9)

The quadratic form for the triplet part is now given by ν†iUtripij νj. Importantly, this form

has the same gauge redundancy as the singlet case, so the projective symmetry group

classification will be similar, as to be discussed in Sec. 8.4. These terms combine in the

quadratic Hamiltonian to give

HQ =∑ij

ν†i (Usingij + U trip

ij )νj + h.c. (8.10)

As pointed out earlier, with the introduction of the DM and Ising anisotropy terms on

the NN bonds, the singlet and triplet channels mix and the saddle point is characterized

by a combination of the two kinds of channels. For example, in the hopping sector, we

can re-diagonalize the interactions to write (for NN bonds):

JSi · Sj + ∆Szi Szj +Dz · Si × Sj ⇒

ω−χ†−χ− + ω+χ

†+χ+ +

J + ∆

8Ex†Ex +

J + ∆

8Ey†Ey, (8.11)

where we withhold the i, j subscripts on all operators for the rest of this section, and

ω± = −J + ∆

8± 1

4

√J2 +D2,

χ± =

√J2 +D2 ∓ J

ξ±χ∓ i D

ξ±Ez,

|ξ±|2 = 2√J2 +D2

(√J2 +D2 ∓ J

). (8.12)


For the pairing channels, the same is true with η in place of χ, and Y in place of E.

Both x- and y-components of the triplet terms lead to unstable mean-field states, so

we will only consider ansatze with singlet (χ, η) and z-component triplet (Ez, Y z) terms.

We can write the mean-field Hamiltonian in the basis described by (8.6) and (8.9). On

a NN bond, we have

HNNij = ω−

(χ∗−χ− + η∗−η−

)+ ω+

(χ∗+χ+ + η∗+η+

)+h.c.− ω−|χ−|2 + |η−|2 − ω+|χ+|2 + |η+|2. (8.13)

On a NNN bond (since there is only antiferromagnetic Heisenberg term, and no D or ∆)

we decouple only in the singlet channels:

HNNNij = −3J

8(χ∗χ+ η∗η) + h.c.+

3J

8

(|χ|2 + |η|2

). (8.14)

The quadratic Hamiltonian Hij = HNNij +HNNN

ij can now be solved to obtain the spinon

wave-function, |ψ〉spinon.

To gain insight on the actual values of different mean-field parameters that charac-

terize the different QSL ansatz, we perform self-consistent mean-field calculations, the

details of which are outlined in Appendix C.3. We note that in the self-consistent cal-

culation, we find that the next-nearest-neighbour exchange, J2, necessary to stabilize

six out of the eight Z2 spin liquids in Table 8.2 is: J2 > 0.4J . This large value of J2,

required to stabilize non-zero next nearest neighbour hopping and pairing interactions

for the spinons, is certainly an artifact of our mean field treatment.

These mean-field results leading to stable solutions act as a guide for choosing a

representative mean-field parameter set for calculation of the dynamical spin-structure

factor. The general qualitative features of the structure factor so obtained are expected

to be less sensitive to the details of the exact parameter values. In any case it is likely that

the parameter values obtained from a mean-field theory are renormalized by quantum

fluctuations. Hence, we take mean-field parameters of singlet Z2 terms wherein pairing

and hopping terms are of similar magnitudes, to demonstrate the qualitative features of

these Z2 states in comparison to the U(1) ones. We will also take next-nearest-neighbour

terms on the order of 0.4 times the magnitude of the nearest-neighbour terms, and a

triplet-singlet ratio of 0.1 ∼ D/J when considering a spin-rotation symmetry-breaking

ansatz.


8.4 Projective Symmetry Group and Mean-Field Ansatze

The projective symmetry group classification follows as in Chapter 8.4, although in this

section we will use the basis in (8.10), where both singlet and triplet parameters are

described by the same 2×2 Uij matrix. The singlet Z2 spin liquids have been determined

by Lu et. al., [117] and the same PSGs apply in the triplet case, since we use the same

basis as the singlet one (8.6). However, the resulting ansatze differ between these cases,

primarily in the existence and structure of the triplet terms. We will leave these details

to Appendix C.2, and summarize the results here.

8.4.1 Singlet Mean-Field Ansatze

In this section, we characterize the singlet spin liquid ansatze on the Kagome lattice that

are of interest to us. [117] These include the eight Z2 spin liquids that occur in the vicinity

of one of four different parent U(1) spin liquids and can be stabilised by tuning spinon

hopping and pairing terms up to the second nearest neighbour sites. We shall begin by

briefly describing the spinon excitation spectra of the four parent U(1) spin liquids at

the nearest-neighbour level. The Z2 states are obtained upon appropriate introduction

of additional hopping and pairing terms that break the IGG to Z2, as are shown in Table

8.2.

The nomenclature of the four U(1) QSLs have already been introduced in Section

8.1.1. These are identified as U(1)[0, 0], U(1)[0, π], U(1)[π, π], and U(1)[π, 0] phases. In

Appendix C.1 we show the signs of the hopping parameters used (in the gauge that we

have chosen). Here we summarize the features of their spinon spectrum in the above

gauge. We note once again that though gauge dependent, the spinon spectra provides

valuable clues to the nature of the experimentally measurable and gauge invariant spin-

spin correlation functions. We plot the dispersion along the high symmetry direction

Γ→M → K → Γ of the original Brillouin zone (Fig. 8.3).

U(1)[0, 0]: The uniform U(1)[0, 0] state has a Fermi surface of spinons. Since this state

has uniform hopping, its dispersion is simply the band structure for the Kagome lattice,

and is seen in Fig. 8.4 (a). It has a flat band at the maximum of the dispersion.

U(1)[0, π]: This ansatz breaks translational symmetry of the Kagome lattice, and has

Dirac points in its dispersion, as seen in Fig. 8.4 (b). It has a doubly-degenerate flat

band at the maximum of the dispersion, similar to the U(1)[0, 0] case.


K

Γ

M

M' K'

Figure 8.3: The cut along the high symmetry direction of the primitive and extended firstBrillouin zone is shown respectively. The boundary of the primitive (extended) Brillouinzone is shown in light (dark) gray. While the spinon dispersion is periodic within theprimitive Brillouin zone, the spin-structure factor is periodic in the extended Brillouinzone (see text for details).

-0.6

-0.4

-0.2

0

0.2

Γ M K ΓU(1)[0, 0]

-0.6

-0.4

-0.2

0

0.2

Γ M K ΓU(1)[0, π]

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[π, π]

Spin

ondis

per

sion

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[π, 0]

Figure 8.4: Spinon dispersion for U(1) singlet spin liquid ansatze. These dispersions gofrom the centre of the Brillouin zone of the Kagome lattice, Γ, to the edge M , to thecorner K, and back to the center Γ. The Fermi level is at zero energy.


SL Label UNN UNNN

Z2[0, 0]A τ 2, τ 3 τ 3

Z2[0, π]β τ 2, τ 3 τ 3

Z2[0, 0]B τ 2, τ 3 τ 3

Z2[0, π]α τ 2, τ 3 τ 3

Z2[0, 0]D τ 3 τ 2, τ 3

Z2[0, π]γ τ 3 τ 2, τ 3

Z2[π, π]B τ 2 τ 3

Z2[π, 0]B τ 2 τ 3

Table 8.2: The general structure of the singlet QSL ansatz for the Z2 spin liquids. Theparticular bonds chosen for UNN and UNNN are shown in Fig. 8.1. The bold termsshown in UNN and UNNN give the pairing and hopping terms whose existence is requiredto break the U(1) parent spin liquid’s gauge group down to Z2. We note that we havechosen a gauge in which τ 2 terms for Z2[0, 0]A and Z2[0, π]β ansatze are zero. [117]

U(1)[π, π]: This state has a Dirac point at the centre of the Brillouin zone, but also

has double-degenerate flat bands at the Fermi level, as seen in Fig. 8.4 (c).

U(1)[π, 0]: We find that there are at least two distinct U(1) states with a [π, 0] hopping

flux pattern. Only one of them can stabilize a Z2 spin liquid (Z2[π, 0]B) upon addition

of the appropriate next-nearest-neighbour hopping and pairing terms; therefore, this is

the state that we will consider. Like the U(1)[π, π] state, there are also flat bands at zero

energy, as seen in Fig. 8.4 (d). However, there is a gap to the subsequent single-spinon

excitations throughout the entire Brillouin zone.

On adding appropriate spinon hopping and pairing terms up to second nearest neigh-

bour, the IGG of the above spin liquids is broken down from U(1) to Z2. [117] There are

eight such Z2 spin liquids as given in Table 8.2. The addition of these terms changes

the structure of the low-energy quasiparticle excitations, with parameters as discussed

in Sec. 8.3.1. In some cases, a gap is opened (Z2[0, π]β). In others, the line degeneracy

of zero-energy excitations from the U(1) state’s Fermi surface remains (Z2[0, 0]B and

Z2[0, 0]D). For some other states, only band touching points remain (Z2[0, 0]A, Z2[0, π]α

and Z2[0, π]γ). The flat bands at zero energy of the U(1)[π, π] and U(1)[π, 0] states either

remain (Z2[π, π]B) or are lifted to a line degeneracy (Z2[π, 0]B). Table 8.1 summarizes

the structure of the low-energy quasiparticles for the above U(1) and Z2 spin liquids.

The full form of GS(i) is given in Appendix C.2.


-0.6

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[0, 0]

-0.6

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[0, π]

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[π, π]

Spin

ondis

per

sion

-0.4

-0.2

0

0.2

0.4

Γ M K ΓU(1)[π, 0]

Figure 8.5: Spinon dispersion for U(1) spin liquid ansatze with “singlet + triplet”channels for NN bonds. These dispersion go from the centre of the Brillouin zone ofthe Kagome lattice, Γ, to the edge M , to the corner K, and back to the center Γ. Thefermi level always lies at the zero of the energy scale. We note that since the “singlet+ triplet” U(1)[0, π] spin liquid state is gapped, it is unstable to confinement due toinstanton tunnelling events. [178]

8.4.2 “Singlet + Triplet” Mean-Field Ansatze

All of the above spin liquids on the Kagome lattice are also realized even in absence of

spin-rotational symmetry, if time-reversal and the symmetries of the Kagome lattice are

preserved. The allowed Uij now has both real singlet terms and imaginary z-component

triplet terms for the nearest neighbour hopping and pairing, and real singlet terms for

next-nearest-neighbour as well.

Again, we first consider the effect of the triplet terms on the four U(1) spin liquids.

In all cases, the spectra show additional features in their dispersions, coming from the

triplet terms that removes the flat bands previously present throughout the Brillouin

zone. The U(1)[0, 0] state has an altered Fermi surface, which does not split the f↑ and

f↓ spinons (Fig. 8.5 (a)). The Dirac cone in the U(1)[0, π] state, (Fig. 8.4 (b)) however, is

gapped out (Fig. 8.5 (b)). Further, the bands are also spin split throughout the Brillouin

zone. We note that such a two-dimensional gapped U(1) spin liquid is unstable toward

a confinement transition. [136, 178] So we infer that triplet decoupling may indirectly


render the U(1)[0, π] state unstable. The U(1)[π, π] and U(1)[π, 0] states both have their

zero-energy flat bands mostly gapped out, leaving a filled band with a line degeneracy at

zero energy. The Dirac cone in the U(1)[π, π] state remains intact (8.5 (c)), while newer

Dirac nodes are created for the U(1)[π, 0] state on including the triplet terms (Fig. 8.5

(d)).

Allowing for small triplet terms in the singlet Z2 ansatze also changes the low-energy

structure of the quasiparticles. The Z2[0, π]β state has its gap altered. For states with

line degeneracies, we find that only band touching points remain (Z2[0, 0]B, Z2[0, 0]D and

Z2[π, 0]B). Also, the band touching points are found to be robust to spin-rotation sym-

metry breaking perturbations (Z2[0, 0]A, Z2[0, π]α) and Z2[0, π]γ). In the state Z2[π, π]B,

the flat bands at zero energy are lifted to band touching points, while the Dirac node

persists. These changes are summarized in Table 8.1.

With this, we have completed the description of the candidate QSL states. We shall

now calculate the dynamical spin-structure factor and study their broad features that

can help us identifying the nature of the spin liquid, possibly realized in Herbertsmithite,

from inelastic neutron scattering and ESR experiments.

8.5 Spin Correlations in U(1) and Z2 Spin Liquids

8.5.1 Structure Factor and Experimental Probes

To determine the spin correlations, we will calculate the dynamical structure factor; the

matrix is given by

Sαβ(q, ω) =

∫ ∞−∞

dt

2πeiωt

∑ij

eiq·(ri−rj)⟨Sαi (t)Sβj (0)

⟩, (8.15)

where α, β ∈ 1, 2, 3. It characterizes spin-1 magnetic excitations of energy ω and

wavevector q. Up to a magnetic form factor, Sαβ(q, ω) can be measured directly by

inelastic neutron scattering. [116] We note that Eq. (8.15) is not periodic in the reciprocal

lattice vectors of the Kagome lattice. Since ri − rj can be half of the primitive lattice

vectors a or b in Fig. 8.2, we must extend the Brillouin zone to double its size. We will

plot the dynamical structure factor along the path Γ → M ′ → K ′ → Γ in the extended

Brillouin zone, where Γ is the centre, M ′ is the midpoint of an edge, and K ′ is the

corner of the edge of the extended Brillouin zone (EBZ) (Fig. 8.3). We found that the

equal-time (ω-integrated) structure factor shows little qualitative differences among our

ansatze, so we concentrate on the ω-resolved features for different QSL states. Further,


for the singlet ansatze, since the quadratic Hamiltonian HQ commutes with the total

spin operator and the ground state is an eigenstate of the total spin with eigenvalue 0.

Thus, the dynamical structure factor of singlet ansatze is zero at q = 0.

The dynamical structure factor can also be probed by ESR, under an external Zeeman

field HZ

∑i Si · z. Absorption of a transverse microwave field (along axis α) of frequency

ω probes the long-wavelength q ≈ 0 dynamical structure factor Sαα(0, ω). [225] The

absorption intensity is given by

I(ω) =H2mω

2

(1− e−βω

)Sαα(0, ω), (8.16)

where the field is applied in the α direction, with amplitude Hm and frequency ω. [226]

The SU(2)-invariant terms affect the intensity trivially; that is, I(ω) ∝ δ(ω−HZ), where

HZ is the Zeeman field strength. Hence, ESR line-shape is sensitive to spin anisotropy,

and so the triplet terms. Thus, ESR experiments can reveal important information about

the effect of the spin-rotation symmetry breaking perturbations.

8.5.2 Dynamical Structure Factor of Singlet U(1) Spin Liquid

Ansatze

We will begin by considering the singlet U(1) spinon ansatze. Our choice of color scale

is made to accentuate the low-intensity scattering compared to the zero-intensity back-

ground.

Fig. 8.6 shows the dynamical spin-structure factor for the U(1)[0, 0] state. The low-

energy domes of scattering anchored around the Γ and M ′ points are contributed by the

excitations near the Fermi surface. Above these domes, we see additional flat scattering

coming from contribution of the the higher energy, flat bands (refer to Fig. 8.4 (a)).

In Fig. 8.7 we plot the dynamical spin-structure factor for the U(1)[0, π] state. In

contrast with the U(1)[0, 0] case, low-energy cones of scattering are seen at the Γ, M ′,

and intermediate points in the extended Brillouin zone, a consequence of the linearly-

dispersing low-energy single-spinon excitations of the Dirac spin liquid. The flat, intense

band of scattering seen at the highest energies are manifestations of the flat spinon bands

(Fig. 8.4 (b)).

Fig. 8.8 shows the structure factor for the U(1)[π, π] state. In this case, we can see

intensity all the way down to zero energy across the extended Brillouin zone. Within

this diffuse scattering, there are two major variations of intensity. The first is dispersive,

rising from the Γ point, due to the Dirac point in the spinon dispersion. The second is


Γ M ′ K ′ Γ

ω/J

S(q,ω

)

Figure 8.6: Dynamical structure factorof the NN singlet U(1)[0, 0] spin liquidalong the high symmetry direction ofthe extended Brillouin zone. The char-acteristic low energy dome-like structureresults from the spinon excitations nearthe fermi surface (Fig. 8.4 (a)).

Γ M ′ K ′ Γ

ω/J

S(q,ω

)

Figure 8.7: Dynamical structure fac-tor of the nearest-neighbour singletU(1)[0, π] spin liquid along high sym-metry directions of EBZ. Compared toU(1)[0, 0] state, the low energy contin-uum near Γ and M ′ points are replacedwith cones of scattering which is a con-sequence of the Dirac node at the fermilevel for the spinons (Fig. 8.4 (b)).

Γ M ′ K ′ Γ

ω/J

S(q,ω

)

Figure 8.8: Dynamical structure fac-tor of the nearest-neighbour singletU(1)[π, π] spin liquid along the highsymmetry direction of the EBZ. The ex-tensive continuum of low energy scatter-ing is a contribution of the flat bands atthe fermi-level (Fig. 8.4 (c)).

Γ M ′ K ′ Γ

ω/J

S(q,ω

)

Figure 8.9: Dynamical structure fac-tor of the nearest-neighbour singletU(1)[π, 0] spin liquid along high symme-try directions of the EBZ. There is fi-nite scattering intensity exactly at zeroenergy because of the flat bands in thespinon dispersion (Fig. 8.4 (d)), whichis followed by a lack of scattering due ato gap in the spinon spectrum.


the flat band of intensity in the middle of the energy range, again due to zero-energy flat

bands of the dispersion.

Finally, Fig. 8.9 shows the structure factor for the U(1)[π, 0] state. While the flat

bands in the dispersion contributes to the scattering exactly at zero-energy, the gap to

subsequent excitations is seen in the absence of scattering. The highest intensity is seen

at a particular point of scattering at M ′.

We see that the low-energy spin correlations are an effective way to distinguish be-

tween these U(1) spin liquids. Other dispersive scattering features characteristic to the

states also show up at comparatively higher energies.

Next, we will look at the effect of the additional pairing terms of the Z2 spin liquid

ansatze on the dynamical structure factor, particularly as a means to distinguish between

Z2 states with the same parent U(1) spin liquid, in spite of their similarities.

8.5.3 Dynamical Structure Factor of Singlet Z2 Spin Liquid

Ansatze

We see that the dynamical structure factor of Z2 states are similar in overall shape to

that of their parent U(1) states. To illustrate this point, we will consider the Z2[0, 0]A, B

and D states, in Fig. 8.10 (a,b,c). The Z2[0, 0]D state has a structure factor that is most

similar to the parent U(1)[0, 0] case (Fig. 8.6), with zero-energy scattering along much

of the M ′ −K ′ line. Above that, the scattering is diffuse, with dispersive features that

are only slightly stronger than the diffuse background. For the Z2[0, 0]A state, the low-

energy domes of scattering persist over small energies, and the flat features have become

much more dispersive. There is a large intensity at the M ′ point at higher energies. In

the Z2[0, 0]B case, there is low-energy intensity only near the Γ and M ′ points. Many

dispersive features are seen, particularly near the K ′ point.

The Z2[π, π]B state has a dynamical structure factor (Fig. 8.10 (d)) which is strikingly

similar to its parent U(1)[π, π] state (Fig. 8.8). Only the intensity of the dispersive bands

increases, and scattering close to Γ is found up to all energies, but there are no obvious

qualitative differences between the two.

We also see similar resemblance in case of the the Z2[0, π] QSL states with their

parent U(1)[0, π] state (Fig. 8.10 (e,f,g)). The Z2[0, π] states retain flat intensity at

the highest energy, and relatively constant diffuse scattering across their energy range

from their parent state. The Z2[0, π]β state has a fairly significant spin-gap, where the

low-energy cones of scattering of the parent U(1) state (Fig. 8.7) have been rounded off.

This spin-gap is found to be rather momentum independent. Also the flat features of the


Γ M ′ K ′ Γ

ω/J

Z2[0, 0]A

Γ M ′ K ′ Γ

S(q,ω

)

Z2[0, 0]B

Γ M ′ K ′ Γ

ω/J

Z2[0, 0]D

Γ M ′ K ′ Γ

S(q,ω

)

Z2[π, π]B

Γ M ′ K ′ Γ

ω/J

Z2[0, π]β

Γ M ′ K ′ ΓS

(q,ω

)Z2[0, π]α

Γ M ′ K ′ Γ

ω/J

Z2[0, π]γ

Γ M ′ K ′ Γ

S(q,ω

)

Z2[π, 0]B

Figure 8.10: Dynamical structure factor for Z2 singlet spin liquid ansatze. These struc-ture factor are plotted from the centre of the extended Brillouin zone of the Kagomelattice, Γ, to the edge M ′, to the corner K ′, and back to the center Γ.


U(1)[0, π] state are found to be broadened, and the scattering is fairly diffuse, although

some dispersive intensity can be seen coming from the Γ point, due to the remnant of the

Dirac cone. At low energies, the intensity for the Z2[0, π]α state has many broad domes

of scattering, and zero-energy excitations exist almost throughout. At larger energies,

the intensity is mostly diffuse, where the U(1)[0, π] state’s high-energy intensity has been

split into two neighbouring broad, diffuse, yet still almost flat bands. The low energy

features of the Z2[0, π]γ state is similar to the U(1) case. Above this, the intensity is also

separated into two broad, diffuse bands, more prominently than the Z2[0, π]α state.

The Z2[π, 0]B state (Fig. 8.10 (h)) shows many separated broad and diffuse bands,

including the band near zero energy, which is the remnant of the strictly zero energy

scattering coming from the flat bands of the U(1)[π, 0] state (see Fig. 8.9). Furthermore,

the intensity is relatively constant across the cut, only diminishing as the zone centre (Γ

point) is approached.

With the exception of the [π, π] states, the low-energy intensity and dispersive features

allow the twelve spin-liquid ansatze in Table 8.2 to be distinguished from each other

qualitatively. In the next subsection, we will see the effect of adding triplet terms to the

QSL ansatze to the dynamical structure factor.

8.5.4 Dynamical Structure Factor of “Singlet + Triplet” U(1)

and Z2 Spin Liquid Ansatze

The introduction of the triplet terms Ezij and Y z

ij breaks the spin-rotational symmetry of

the spin-polarized dynamical structure factor Sαβ(q, ω), where Sxx = Syy 6= Szz. While

the singlet structure factors showed a vanishing intensity approaching q = 0, these triplet

terms generate a non-zero intensity in Sxx and Syy. Also, the scattering intensity at low

energies changes, due to changes in the spinon dispersion (summarized in Table. 8.1).

Since the spinon bands are, in general, split by triplet terms, the scattering becomes

increasingly diffuse.

We will showcase these changes by calculating the trace of the dynamical structure

factor matrix, i.e., (Sxx + Syy + Szz)/3, and then plotting the difference from the singlet

structure factor. In Appendix. C.4, we show the actual structure factor for all these

“singlet + triplet” states.

The “singlet + triplet” U(1)[0, 0] structure factor looks very similar to the singlet case,

particularly in the low-energy scattering. However, the diffuse scattering at moderate

energies is no longer flat. While this is also true for the “singlet + triplet” U(1)[0, π]

structure factor, it also has two qualitative differences. The first is that the cones of


Γ M ′ K ′ Γ

ω/J

U(1)[0, 0]

Γ M ′ K ′ Γ

∆S

(q,ω

)

U(1)[0, π]

Γ M ′ K ′ Γ

ω/J

U(1)[π, π]

Γ M ′ K ′ Γ

∆S

(q,ω

)

U(1)[π, 0]

Figure 8.11: Difference between the dynamical structure factor for U(1) “singlet +triplet” spin liquid ansatze. These structure factor are plotted from the centre of theextended Brillouin zone of the Kagome lattice, Γ, to the edge M ′, to the corner K ′, andback to the center Γ. We note that since the “singlet + triplet” U(1)[0, π] spin liquidstate is gapped, it is unstable to instanton tunnelling events. [178]


scattering at low energy are gapped out; this can be seen more clearly in the full “singlet

+ triplet” structure factor in Appendix C.4, Fig. C.3. The second is that the flat band at

the highest energy is split into two. For the U(1)[0, π] state, though we plot the structure

factor for completeness, we note that in this state the spinon spectrum is gapped and

hence the state is unstable to confinement transition due to instanton events. [178] The

“singlet + triplet” U(1)[π, π] structure factor is very similar to the singlet case, where

the low-energy scattering sees a band of intensity between Γ and M ′, as well as Γ and

K ′. This is absent in the singlet case, which can distinguish between these states. Such

a band is also seen in the “singlet + triplet” U(1)[π, 0] state, which otherwise is again

similar to the corresponding singlet case.

The Z2 (S+T) spin liquids are shown in Fig. 8.12. Most of the structure factors closely

resemble their singlet counterparts with important general differences. Here, we shall limit

ourselves to pointing out these differences only. Generally, due to the triplet decoupling

channels, the spinon bands are spin-split. Hence, the sharp dispersing structures are

somewhat lost and the scattering intensity becomes more diffuse compared to the singlet

only ansatze, with increasing strength of the triplet terms. As with the U(1) states, the

intensity of low energy scattering does not go to zero at q = 0. Hence there is a slight

enhancement of scattering at the EBZ centre compared to the singlet states. Also, for

most of the states where there is a spin-gap in the structure factor, the gap magnitude

becomes increasingly independent of the momentum.

Finally, we end this section by noting that the spin-liquid state Z2[0, π]β seems to

be the best candidate in the context of present inelastic neutron scattering experiments

and DMRG calculations. We shall discuss this in some detail, along with the signatures

of the other spin liquids, in the next two sub-section in the context of inelastic neutron

scattering and ESR experiments.

8.5.5 Implications for Inelastic Neutron Scattering Experiments

It is useful to compare the spin-structure factor obtained above with the results of the

recent inelastic neutron scattering experiments on single crystals of Herbertsmithite. [116]

These experiments show almost uniform diffuse scattering over a large energy window.

Furthermore, no spin-gap is observed down to 1.5 meV. It also shows no obvious signature

of broken lattice symmetries.

With parameters from the mean-field solution, (as discussed later in Appendix C.3)

excitations are found for ω . 0.35J . In contrast, the inelastic neutron scattering shows

strong diffuse intensity up to the largest measured energy 11 meV ∼ 0.65J . However,


Γ M ′ K ′ Γ

ω/J

Z2[0, 0]A

Γ M ′ K ′ Γ

∆S

(q,ω

)

Z2[0, 0]B

Γ M ′ K ′ Γ

ω/J

Z2[0, 0]D

Γ M ′ K ′ Γ

∆S

(q,ω

)

Z2[π, π]B

Γ M ′ K ′ Γ

ω/J

Z2[0, π]β

Γ M ′ K ′ Γ∆S

(q,ω

)Z2[0, π]α

Γ M ′ K ′ Γ

ω/J

Z2[0, π]γ

Γ M ′ K ′ Γ

∆S

(q,ω

)

Z2[π, 0]B

Figure 8.12: Difference between the dynamical structure factor for Z2 “singlet + triplet”spin liquid ansatze. These structure factor are plotted from the centre of the extendedBrillouin zone of the Kagome lattice, Γ, to the edge M ′, to the corner K ′, and back tothe center Γ.


the numerical values from our mean-field calculation can, at best, serve as a consistency

check.

Comparing with our calculated dynamical spin-structure factor with the above fea-

tures of the experiment, we may infer that the U(1)[π, π], U(1)[π, 0], Z2[π, π]B, and

Z2[π, 0]B states clearly appear to be inconsistent with the neutron scattering data. Fur-

ther comparison of their ESR line shapes and peak distribution (see next section) may

confirm/ invalidate this conclusion.

In contrast, several of the [0, 0] and [0, π] (both U(1) and Z2) QSLs do have mostly-

featureless diffuse scattering within some energy window [ωmin, ωmax]. However, in all but

a few cases this window is narrow, and these states have well-defined features at lower

or higher energies and hence are inconsistent with experiments.

The four states, namely Z2[0, 0]D, Z2[0, π]α, Z2[0, π]γ and Z2[0, π]β, have dispersive

features with a very weak intensity, leading to an almost diffuse structure factor. The

dispersive features are broadened and are in relatively poor contrast with the generally

diffuse background. However, we see some modulation in intensity for Z2[0, π]γ state,

and slightly better-defined dispersive features in the Z2[0, 0]D state. However, only the

Z2[0, π]β state shows evidence of a spin gap – Z2[0, π]α is gapless. As already noted, the

inclusion of the spin-rotation symmetry breaking perturbations makes spin-gap increas-

ingly momentum independent in Z2[0, π]β. Further these perturbations also enhances

the scattering near the Brillouin zone centre somewhat in almost all cases (refer to Fig.

8.12). Noting that the DMRG calculations indeed stabilize a gapped QSL state (most

likely Z2), in light of the inelastic neutron scattering measurements, it is tempting to

suggest the Z2[0, π]β state as a consistent choice for a candidate ground state for Her-

bertsmithite. However, we should note that the issue of existence of spin-gap is still not

clear in Herbertsmithite experiments, and except for the lack of spin-gap, Z2[0, π]α is

also a possible candidate.

Next, we describe the ESR absorption spectra for the QSLs that break spin-rotation

symmetry.

8.5.6 ESR Absorption Intensity for different spin-rotation sym-

metry breaking QSLs

To observe the ESR spectra we couple the system with a Zeeman field via HZ

∑i z ·

Si. The ESR absorption intensity given in (8.16). We note that I(ω) ∝ ω at low

temperatures, and peaks occur around the value of the Zeeman field, (ω = HZ) so

absorption is most easily seen at the large values of ω and HZ . We align the Zeeman


HZ − ω

I(ω

)/H

2 m U(1)[0, 0]

HZ − ω

U(1)[0, π]

HZ − ω

I(ω

)/H

2 m U(1)[π, π]

HZ − ω

U(1)[π, 0]

Figure 8.13: Electron spin resonance absorption for U(1) “singlet + triplet” spin liquidansatze, as a function of the Zeeman field strength HZ and microwave field frequencyω. The viewpoint is along the HZ = ω axis. We note that since the “singlet + triplet”U(1)[0, π] spin liquid state is gapped, it is unstable to instanton tunnelling events. [178]

field along the z-axis, the axis along which the spin-rotation is broken. For energy scales

ω near HZ , the Zeeman term breaks SU(2) symmetry, with Uij along with the triplet

terms. For this axis of the magnetic field, Ixx(ω) = Iyy(ω) due to the remaining U(1)

spin-rotational symmetry around the z-axis.

We now discuss the ESR absorption intensity for the four U(1) and eight Z2 QSL

in presence of the triplet channels. We focus on the small-intensity region to show

contributions from the triplet terms; these occur as satellite peaks with smaller intensities,

while the peak at ω = HZ has a much larger intensity. These spectra are shown in Fig.

8.13.

Both the U(1)[0, 0] and U(1)[0, π] spin liquids have an additional ‘satellite’ peak on

either side of the main peak at ω = HZ . The peaks of the U(1)[0, π] state have both

higher intensity and are spaced more closely compared to the U(1)[0, 0] state.

However, the U(1)[π, π] and U(1)[π, 0] states have wildly different ESR absorption

spectra. There is absorption over a broad range of HZ −ω, with many subsequent satel-


lite peaks almost forming a continuum. The intensity of the main peak is significantly

diminished compared to the U(1)[0, 0] and U(1)[0, π] spin liquids. As HZ − ω increases

from zero, in the U(1)[π, π] state, the primary satellite peaks have a monotonically de-

creasing intensity, while the U(1)[π, 0] state features a single pair of prominent satellite

peaks at a finite distance away from zero.

The Z2 spin liquids generally lead to a broadening of the absorption peaks. These

spectra are shown in Fig. 8.14. The Z2[0, 0]A state shows slight broadening of the all

peaks around the base, with diminished intensity compared to the the U(1) case. The

Z2[0, 0]D state shows a similar response, though the satellite peaks are higher compared

to the broadening of the main peak. The Z2[0, 0]B state has very broad absorption

around the now significantly smaller satellite peaks. The largest satellite peaks of the

Z2[π, π]B are broadened at low absorption, but the intensity around the main peak drops

dramatically and is almost zero.

The Z2[0, π]β state shows significant broadening of the main peak around the base,

and nearly complete reduction of satellite peak intensity. The Z2[0, π]α and Z2[0, π]γ

states have smaller amount of broadening around the main peak, where the satellite

peaks are diminished in intensity, but still visible. The intensity of the main peak of the

Z2[π, 0]B is almost zero and the satellite peaks are replaced by small, broad domes of

absorption.

The shape and structure of the satellite peaks provide another qualitative clue to

distinguish between the spin liquid ansatze considered here, and immediate evidence of

spin-rotation symmetry breaking in the kagome planes.

Current ESR measurements down to as low as 5 K, however, find that the absorption

intensity is dominated by the impurity spins below 20 K. [120] These Cu2+ ions contribute

in a nearly-paramagnetic fashion, and the intensity displays a Curie-like response ∝1/T . The line shape remains broad. In the regime where these impurity spins display

a paramagnetic response, isolating the kagome-layer ESR response may prove difficult.

An increase in sample purity can curtail this effect. However, at very low temperatures,

these impurity spins may no longer behave in a paramagnetic fashion, and interact with

the kagome planes with a strength ∼ 10 K. [120, 232] One needs to consider the effect

of the coupling of these impurities to the kagome spin liquid, and the resultant ESR

spectrum. Such analysis is beyond the scope of this present chapter, so we present here

the intrinsic absorption of the spin liquid layers.


HZ − ω

I(ω

)/H

2 m Z2[0, 0]A

HZ − ω

Z2[0, 0]B

HZ − ω

I(ω

)/H

2 m Z2[0, 0]D

HZ − ω

Z2[π, π]B

HZ − ω

I(ω

)/H

2 m Z2[0, π]β

HZ − ω

Z2[0, π]α

HZ − ω

I(ω

)/H

2 m Z2[0, π]γ

HZ − ω

Z2[π, 0]B

Figure 8.14: Electron spin resonance absorption for Z2 “singlet + triplet” spin liquidansatze, as a function of the Zeeman field strength HZ and microwave field frequency ω.The viewpoint is along the HZ = ω axis.


8.6 Discussion

We now summarize our results. In this work, we have attempted to address two important

questions to account for the unusual phenomenology of the non-magnetic ground state of

Herbertsmithite. Firstly, we have calculated the dynamical spin-structure factor for four

U(1) and eight Z2 symmetric spin liquids derived from the former for the NN and NNN

antiferromagnetic Heisenberg model on an isotropic Kagome lattice. These spin liquids

are allowed by spin-rotation, time-reversal and lattice symmetries of a Kagome lattice.

We then consider the effect of small spin-rotation symmetry breaking perturbations (DM

and Ising anisotropy) on the above spin liquids, and on the corresponding spin-structure

factor. Furthermore, we calculate the ESR absorption spectra, which show non-trivial

structures in presence of spin-rotation symmetry breaking. Since recent numerical studies

suggest that the ground state of strictly NN antiferromagnetic Heisenberg model is very

sensitive to small perturbations of second neighbour exchange and Dzyaloshinsky-Moriya

interactions, we expect that the the above perturbations may have important effects in

the low energy features of the experimentally measurable spin-structure factor.

Indeed, we find that the addition of the perturbations make the structure factor

largely diffuse over an extended energy window throughout the Brillouin zone. Similar

scattering has been observed in the recent inelastic neutron scattering experiment on Her-

bertsmithite. We particularly find two Z2 spin liquid states, the so-called Z2[0, π]β and

Z2[0, π]α states, whose spin-structure factor features are qualitatively in conformity with

the experiments. Only the Z2[0, π]β state, however, exhibits a spin-gap in the structure

factor. In the presence of the perturbations, this gap becomes increasingly momentum-

independent. Noting that a similar gapped QSL was obtained as the ground-state of the

NN antiferromagnetic Heisenberg Hamiltonian in recent DMRG calculations, it is tempt-

ing to infer that the Z2[0, π]β state may be adiabatically connected to the ground state

obtained in DMRG and also the ground state of Herbertsmithite. This prediction can

be checked further by measuring the ESR absorption spectrum. Our present calculations

suggest that the the Z2[0, π]β phase shows a characteristic broadening of lines in ESR

absorption spectra.

Chapter 9

Conclusion

In this thesis, we have studied a variety of interesting phases exemplified by quantum

magnets appearing in condensed matter systems. In doing so, we have uncovered some of

the rich behaviour displayed by spin-dimer systems, and a breadth of spin liquid states,

where fractionalization may even persist beyond the breaking of spin-rotation symmetry

and presence of long-range order. To close this thesis, we summarize these results, and

identify future directions arising from this work.

9.1 Summary

In Chapter 4, we studied the quantum phase transition in the spin-dimer system Ba3Cr3O8.

It is formulated as the Bose-Einstein condensation of S = 1 excitations on top of a spin-

singlet background. The dispersion for these excitations was computed within bond-

operator theory, and the condensate was analyzed within a Hartree-Fock-Popov approach.

For moderate condensate densities, we were able to match the experimental magnetiza-

tion profile.

The novel ferromagnetically-coupled spin-dimer system (CuCl)LaNb2O7 was studied

in Chapter 5, using a Schwinger boson approach, rather than a dimer-centered approach.

The classical phase diagram was produced, showing ferromagnetic, stripe, and spiral

ordered phases. Quantum fluctuations breaks a degeneracy between different-direction

spiral order phases, while shrinking the region where it is stabilized. A disordered singlet

phase is indicated when inter-dimer coupling is small. These findings corroborate exact-

diagonalization and strong-coupling results, [158] hinting at the additional possibility of

the spiral phase.

The double perovskite La2LiMoO6 is a geometrically frustrated face-centered-cubic

antiferromagnet, and is studied in Chapter 6, also with a Schwinger boson approach.

130

Chapter 9. Conclusion 131

Spin-orbit coupling of Mo5+ ions competes with a tetragonal distortion of local oxygen

octehedra in shaping the nature of the low-energy manifold. The anisotropic interactions

of the pseudo-spin model resulting from this distortion play a key role in stabilizing a spin-

disordered ground state, which may explain the lack of long-range order in La2LiMoO6.

The kagome compound Volborthite features a distorted lattice, relieving some of the

geometrical frustration but powder samples show an unusual ordered phase with a specific

heat linear in temperature. In Chapter 7, we try to understand the transition from this

exotic phase into a more conventional ordered phase as external field is increased. To

do this, we consider the destruction of the Fermi surface of a U(1) spin liquid within

a fermionic slave-particle analysis. A spin-density-wave excitation of these fermionic

excitations provides magnetic order, until it destroys the spin liquid completely at higher

fields.

Finally, in Chapter 8, we investigate the effects of spin-rotation-breaking interac-

tions in the isotropic kagome lattice, as seen in Herbertsmithite. We identify potential

fermionic spin-liquid states that are consistent with the broad, diffuse inelastic scatter-

ing results. When spin-rotational symmetry is broken, small intensity is seen at the

wavevector q = 0. This may be most easily probed through electron spin resonance, and

we present absorption line shapes that distinguish can further between these proposed

states.

9.2 The Path Ahead

The work considered so far suggests a few possible directions for future research, which

will help to complete the picture being sketched so far, and extend it further.

9.2.1 Disorder in Spin Systems

Improvements in crystal growth have resulted in increasingly pure samples over the years.

Nonetheless, even state-of-the art single-crystal growth techniques leave some disorder

in the samples. Especially for spin-liquid candidates Herbertsmithite and Kapellasite,

disorder plays an important role. Out-of-plane magnetic impurities can mask low-energy

behaviour, while in-plane non-magnetic impurities can significantly alter the spin model.

Both of these effects should be carefully considered, in order to properly compare against

theoretical models, and understand the effect of disorder on spin-liquid states.

Chapter 9. Conclusion 132

9.2.2 Fractionalized Excitations in Non-Symmetric States

The ferromagnetic interactions in Volborthite and the Dzyaloshinsky-Moriya interac-

tions in Herbertsmithite were seen to lead to slave-particle states that naturally break

spin-rotation symmetry. These provide two examples of how deconfined S = 12

spinon

excitations may remain in states that break some kind of symmetry. Spin-nematic states

in other frustrated ferromagnetic models have also been studied, [168] providing other

examples of such states with broken spin-rotation symmetry. Recent work has pointed to

the possibility of broken lattice-rotation symmetry in spin liquids. [233] Broken symmetry

allows a much wider search for fractionalized excitations in frustrated spin systems.

Appendix A

Classical and Saddle-Point Solutions

of the Schwinger Boson Model of

La2LiMoO6

A.1 Classical O(N) Model

We begin by writing the real-space partition function for the Heisenberg Hamiltonian on

the FCC lattice,

Z =

∫DφDµ exp (−S(φ, µ)), where S(φ, µ) =

β∑ij

[Jij2φi · φj +

µi2δij(φi · φi −N)

]. (A.1)

Here, the O(N) model generalizes the spin φ from a three-component vector to an N-

component vector. The first step is to take the Fourier transform defined by φi =1√Ns

∑kφk exp (−ik · ri), where Ns is the number of sites of the lattice. After the Fourier

transform, we have

S

βNsN= −µ

2+

1

Ns

∑k

|φk|2(µ

2+

6∑n=1

Jδn cos (k · δn)

)(A.2)

Z ∝∫dµ∏k

dφkdφ∗k exp (−S). (A.3)

133

Appendix A. Classical and Saddle-Point Solutions for La2LiMoO6 134

We perform the Gaussian integral over φk and φ∗k, giving

Z ∝∫dµ exp

(βµ

2NsN −

∑k

ln

(D(k, µ)Nβ

π

)),

D(k, µ) =µ

2+

6∑n=1

Jn cos (k · δn). (A.4)

The corresponding saddle-point solution gives µ from

1 =1

Ns

∑k

1

NβD(k, µ). (A.5)

The spin-spin correlation function scales as

〈φk · φk′〉 ∝ δk′,−k1

βD(k, µ). (A.6)

As β → ∞, the minimum of D(k, µ) will become the dominant contribution; magnetic

ordering will occur with the wavevector q that minimizes∑6

n=1 Jn cos (q · δn).

A.2 Saddle-Point Solution

To find the saddle-point solution, we first look at the Fourier transform, defined as

bi = 1√Ns

∑k bke

−ik·ri . After taking this transform, the Hamiltonian (6.14) becomes

HNsN

=∑n

Jδn2|Qδn|2 + µ

(−1− κ+

1

Ns

∑k

x∗kσxσk

)

+1

Ns

∑kn

(−JδnQδn

2εσσ′x

σkx

σ′

−keik·δ + h.c.

)

+1

NsN

∑mk

(b†km↑ b−km↓

)( µ Bk

−Bk µ

)(bkm↑

b†−km↓

);

Bk = i∑n

JδnQδn sin (k · δn) (A.7)

where Ns is the number of sites in the system.

The quadratic part of the mean-field Hamiltonian in (A.7) is diagonalized by a stan-

dard Bogoliubov transformation. [189] With the quasiparticle energy ωk, the diagonalized

Appendix A. Classical and Saddle-Point Solutions for La2LiMoO6 135

quadratic terms are

1

Ns

∑k

ωk

(1 + γ†k↑γk↑ + γ†k↓γk↓

), ωk =

√µ2 − (

∑n

JnQn sin (k · δn))2. (A.8)

Here, the transformation is defined by b = T−1γ, where the columns of T−1 are the

eigenvectors of ηM , M is the quadratic Hamiltonian matrix in (A.7), and the 2N × 2N

η is given by

ηαβ =

δαβ α ≤ N

−δαβ α > N.

The structure of the condensate can be determined from the associated mean-field

equation: ∂〈H〉/∂xσk = 0. The solution to the disordered case (x = 0) has a gapped

dispersion. We can track when the gap vanishes and bosons begin to condense. We

find that x↑k is a linear combination of condensates at the minimum wavevectors ±k1:

x↑k = c1δk−k1 + c2δk+k1 . We then rewrite the part of the mean-field energy depending on

x↓ and obtain the mean-field equation

0 =1

NsN

∂E↓∂xk↓

=µ

Ns

x∗k↓ +1

Ns

[∑δ

JδQδ sin (k · δ)

](−c1δk,−k1 − ic2δk,k1) . (A.9)

In the condensed phase, to ensure a gapless dispersion, µ = −∑

n JnQn sin (k1 · δn) > 0.

The form of x↓k follows as x↓k = −ic∗2δk−k1 + ic∗1δk+k1 .

We arrive at the diagonalized Hamiltonian (6.15). From this follow the mean-field

equations

1

NsN

∂E

∂µ= 0 = −1− κ+ n+

1

Ns

∑k

µ

ωk,

1

NsN

∂E

∂Qm

= 0 = JmQm + nJ∆ sin (k1 · δm)− 1

Ns

∑k

∑n JnQn sin (k · δn)

ωk(Jm sin (k · δm)) ,

1

NsN

∂E

∂n= 0 = µ+

∑n

JnQn sin (k1 · δn) (if n > 0). (A.10)

Appendix B

Mean-Field Hamiltonians, Fourier

Transform, and RPA Theory of

Volborthite

B.1 Fourier transform

In this appendix we indicate the formulae involved in the Fourier transform. The kagome

lattice is described by a triangular Bravais lattice with a three point unit cell (see Fig.

7.1). We describe the direct lattice using lattice vectors:

a = (2a, 0) b = (−a,√

3

2a) (B.1)

where the x-axis runs parallel to the J1-J2 chains, as seen in Fig. B.1.

Figure B.1: Lattice vectors a and b of the Kagome lattice.

136

Appendix B. Mean-Field and RPA Theory for Volborthite 137

We now denote every site i with the labels l and m, where l labels the unit cell and

m ∈ A,B,C labels the sub-lattice. The lattice sites are then given by:

rlm = xla + ylb + em (B.2)

where em gives the position within the unit cell. The reciprocal lattice vectors are then

given by

u =π

a

(1,

1√3

), v =

π

a

(0,

2√3

), (B.3)

and general lattice momenta are of the form

k = kuu + kvv. (B.4)

We then have

blmµ =1√Nc

∑k

eik·rlmbkmµ, (B.5)

where Nc is the number of unit cells.

B.2 Explicit Forms of Mean-Field Hamiltonians

In the bosonic case, the matrix Hk is given by

Hk =

(Ck Dk

Ek Fk

), (B.6)

Ck =

µA −J12

(χABe

ik·u2 + χ∗BAe

−ik·u2

)0

−J12

(χ∗ABe

−ik·u2 + χBAe

ik·u2

)µB 0

0 0 µC

, (B.7)

Dk =

−J2

2ηAA (2i sin(k · u)) 0 −J ′

2

(η∗CAe

ik·u+v2 + η∗ACe

−ik·u+v2

)0 −J2

2ηBB (2i sin(k · u)) −J ′

2

(η∗CBe

−ik·v2 + η∗BCe

ik·v2

)−J ′

2

(η∗ACe

−ik·u+v2 + η∗CAe

ik·u+v2

)−J ′

2

(η∗BCe

−ik·v2 + η∗CBe

ik·v2

)0

, (B.8)

with Ek = D†k and Fk = Ck.


In the Fermionic case, we have the matrix Hk given by, in 2× 2 blocks,

Hk =

µAI −J24χ∗AAe

ik·uI −J14

E∗AB · σ −J ′

4χ∗ACe

−ik·(u+v)I

−J14

E∗BA · σeik·u µBI − J24χ∗BBe

ik·uI −J ′

4χ∗BCI

−J ′

4χ∗CAI −J ′

4χ∗CBe

ik·vI µCI

. (B.9)

In the Fermionic case, we replace J4→ 3J

8, so that the coefficients match with those

of the Feynman variational principle. [234] This will be important to the decoupling of

the spin-density wave parameters below.

With the introduction of these spin-density-wave parameters, the quadratic part of

the Hamiltonian is

HQuad =∑

k∈RBZ

f †SDWkHQuadk fSDWk,

where fTSDWk = (fTk , fTk+Q, . . . f

Tk−Q) and we sum over the appropriate reduced Brillouin

zone. In block form,

HQuadk =

HDk V SDW

Q 0 0 . . . V †SDWQ

V †SDWQ HD

k+Q V SDWQ 0 . . .

0 V †SDWQ HD

k+2Q V SDWQ 0 . . .. . .

V SDWQ 0 . . . V †SDW

Q HDk−Q

.

HDk is the Hamiltonian in the absence of SDW terms, as given in (B.9).

We will decouple the residual Hubbard-like interaction between spinons, U∑

laσ nlaσnlaσ,

in the SDW channel. Then, V SDWQ is diagonal in 2× 2 block form:

−U

〈S−QA〉 · SQA 0 0

0 〈S−QB〉 · SQB 0

0 0 〈S−QC〉 · SQC

.

The strength of the coupling U is not known, so we will explore the SDW instability for

a range of U on the order of the energy scale of the problem.

One may also decouple the original Heisenberg interactions in the direct channel. The

Fourier transform of the direct channel spin order parameters are just the SDW order


parameters at different q. We can write the Fourier transform of Slm as

Sqm =1

Nc

∑k

f †kmασαβ fk+qmβ. (B.10)

The Heisenberg interaction can be written as follows:

∑〈ij〉

Si · Sj =∑k

SkA

SkB

SkC

†

Mk

SkA

SkB

SkC

=∑k

3∑n=1

ωknT†knTkn, where Mk =

2J2 cos (k · u) J1(1 + e−ik·u) J ′(1 + eik·(u+v))

J1(1 + eik·u) 2J2 cos (k · u) J ′(1 + e−ik·v)

J ′(1 + e−ik·(u+v)) J ′(1 + eik·v) 0

, (B.11)

upon diagonalization. We take Tkn as our order parameters, which are simply linear

combinations of the Ska SDW order parameters. We then rewrite the direct terms in

terms of the T expectation values, and decouple in the Tkn channel (hence the Sqm

channel). Since not all ωkn are negative, we must treat this mean-field theory in a

variational sense, according to the Feynman variational principle, rather than the usual

construction of a saddle-point solution to a Hubbard-Stratonovich transformation.

B.3 Generalized RPA Theory

We begin by considering the Hamiltonian H0, in our case, the mean-field Hamiltonian,

with weak external perturbations parametrized by ja,

H(t) = H0 −1

Nc

∑qa

jr(q, t)O†r(q), (B.12)

Leading order in time-dependent perturbation theory finds the shift in expectation value

δ⟨O†r

⟩(q, t) =

∫ ∞−∞

dt′χ0rs(q, t− t′)js(q, t′), (B.13)

with the bare susceptibility

χ0rs(q, t− t′) = iΘ(t− t′)

⟨[Or(q, t), O

†s(q, t

′)]⟩. (B.14)


After spectral decomposition into the frequency domain, we have

χ0rs(q, ω) =

∑m

Okr,0mOks,m0

ω + E0 − Em + iδ− Okr,m0Oks,0m

ω + Em − E0 + iδ, (B.15)

whereOkr,nm =⟨n|Or(k)|m

⟩. Here, we are concerned with Or(q) = nqr where r = (a, σ).

From this form, and the diagonalization transformation f = Zγ, we have

χ0rs(q, ω) =

1

Nc

∑k

Z∗krnZk+qrn′Z∗k+qsn′Zksn

nf (ωkn)− nf (ωk+qn′)

ωkn − ωk+qn′ + ω + iδ. (B.16)

From this, we consider the eigenvalues of the static susceptibility χ0rs(q, 0).

We consider also the effect of the diagonal Hubbard interaction

U∑laσ

nlaσnlaσ = U∑qaσ

nqaσnqaσ, (B.17)

which we decouple in the density channels nqaσ that were used to derive (B.16). Upon de-

coupling, we have U∑

qaσ 2 〈nqaσ〉 nqaσ−| 〈nqaσ〉 |2. As a result, nqaσ is coupling not only

to jaσ, but also to 〈nqaσ〉, and we can rewrite the perturbative effect of the expectation

value as

δ 〈nqaσ〉 (q, ω) = χ0aσ,a′σ′ [ja′σ′(q, ω)− 2Uδ 〈nqa′σ′〉 (q, ω)]

δ 〈nqaσ〉 (q, ω) =[(I + 2Uχ0

)−1χ0]aσ,a′σ′

(q, ω)ja′σ′ , (B.18)

and we have the generalized RPA susceptibility matrix

χGRPA(q, ω) = (I + 2Uχ0(q, ω))−1χ0(q, ω). (B.19)

We check for the divergence of its eigenvalues as U increases. We note that the eigenvalues

of χ0 are negative, due to the choice of sign in (B.12).

Appendix C

U(1) Ansatze, Projective Symmetry

Group, Mean-Field and Dynamical

Structure Factors Results for

Herbertsmithite

C.1 U(1) Spin Liquid Ansatze

The flux patterns for the nearest-neighbour U(1) spin liquids under consideration are

shown in Figure C.1. They consist of positive and negative real hopping terms that

retain the translational symmetry of the Kagome lattice (U(1)[0, 0] and U(1)[π, π]) or

break it (U(1)[0, π] and U(1)[π, 0]).

C.2 Projective Symmetry Group

Determining the allowed PSGs GSS, and thereafter the ansatze HQ, comes from con-

straints offered by the group multiplication table of the symmetry group SG. With group

multiplication rules like AB = C that can be rewritten as ABC−1 = I, we can place

constraints on the corresponding expression (GAA)(GBB)(GCC)−1. First, there is no

net physical transformation, so the expression must reduce to a gauge transformation.

Second, each of the operations leaves HQ invariant, and so does the final expression.

Thus, (GAA)(GBB)(GCC)−1 ∈ IGG.

141

Appendix C. Mean-Field Ansatze and Results for Herbertsmithite 142

U(1)[0, 0] U(1)[0, π]

U(1)[π, π] U(1)[π, 0]

Figure C.1: Flux pattern for the real hopping terms of the U(1) singlet spin liquidansatze. Hoppings are either positive (solid) or negative (dashed) on nearest-neighbourbonds.

As derived by Lu et. al., GS are given in the singlet case by the following: [117]

GT1(x, y, s) = ηy12I (C.1)

GT2(x, y, s) = I (C.2)

Gσ(x, y, s) = ηxy12 gσ(s) (C.3)

GC6(x, y, s′) = η

xy+x(x+1)/212 gC6(s

′), s′ ∈ u, v (C.4)

GC6(x, y, w) = ηxy+x+y+x(x+1)/212 gC6(w) (C.5)

GT = iτ 1, (C.6)

where we specify the lattice site i by the position of the unit cell to which is belongs

R = xa + yb, and the sublattice index s = u, v, w, as indicated in Fig. 8.2.

The PSG can also be used to generate terms on symmetry-related bonds of an ansatze.

The action of a lattice transformation S and an SU(2) gauge transformation W on the

quadratic terms of bond, given by Uij, is

SUij → US−1(i),S−1(j)

WUij → W (i)UijW†(j). (C.7)

If GSS leaves the Uij invariant, as in the singlet case, then we can combine the above


relations to generate Uij on symmetry-related bonds:

US(i),S(j) = GS(S(i))UijG†S(S(j)). (C.8)

We will cover the case of the “singlet + triplet” PSG next.

C.2.1 Projective Symmetry Group for “Singlet + Triplet” Ansatze

Now, we discuss how time reversal and (improper) rotations affect “singlet + triplet”

Uij, as in (8.9). We will see that the PSG GSS are the same as in the singlet case, and

that the ansatze allow imaginary triplet terms as well.

We will begin by determining how the standard fermionic time-reversal operator Tf

changes the ansatz Uij, to properly capture the effect of time-reversal on the triplet terms.

Tf = θK, where K is the complex conjugation operator and

θ

(f↑

f↓

)=

(−f↓f↑

). (C.9)

Consider the action on a generic Uij:

U =

(A B

C D

)Tf−→

(−D C

B −A

).

One can instead define a modified time-reversal operator, with an additional iτ 2 gauge

transformation acting on Uij, T = iτ 2Tf , giving

Uiτ2Tf−−−→

(−A −B−C −D

)= −U.

Since the C3 rotations are performed around the z-axis in both real space and spin

space, neither the singlet nor the Ez, Y z triplet terms in our ansatze are affected by them.

As all ansatz will projectively obey time-reversal symmetry, we can use the form of

GT to simplify the ansatz before considering the effect of reflection. The action of T

upon Uij is −Uij in both singlet and triplet cases, thus they affect the ansatze in the

same manner. Thus, T2 = +1 acting on the mean-field states, as before, and we have

the same condition ±GT(i)2 ∈ IGG.

The action in spin space of the reflection, σ, is to flip the sign of our triplet compo-

nents, which are the z-components of a pseudo-vector. Since a gauge transformation, on

physical grounds, can only mix singlet and triplet terms among themselves, this action


commutes with the SU(2) gauge transformations. Only the lattice part of σ enters in

the commutation relation between σ and the gauge transformations GS. Solving for the

time-reversal gauge transformation GT as in the singlet case, the non-zero mean-field

anatze will all have the same GT = iτ 1. [117] For an ansatz to be time-reversal invariant

we must have

τ 1Uijτ1 = −Uij.

With this restriction, Uij can be parametrized by Uij = γ2τ2 + γ3τ

3, where γ2,3 ∈ C.

We note that the real components of γ2,3 are coefficients of the singlet pairing and hop-

ping, respectively, while the imaginary components are coefficients of triplet pairing and

hopping. Reversal of the sign of the triplet coefficients can be performed by taking

Uij → U †ij = Uji.

With this, we have the same projective symmetry group that was derived for singlet

spin liquids with the symmetry of the Kagome lattice. [117] However, the ansatz that

can be generated will differ, due to the effect of σ upon the triplet terms.

We may generate all nearest and next-nearest-neighbour bonds from just one by using

the symmetry operations of the Kagome lattice. For each, we may impose a constraint on

the allowed ansatz for these bonds by devising a non-trivial lattice symmetry operation

that takes Uij back to itself. With the effect of lattice transformations (C.8) along with the

transformation under σ, we can derive the following constraint on the nearest-neighbour

Uij:

gσ(u)gC6(u)gC6(w)UNNg†C6

(v)g†C6(w)g†σ(v) = UNN, (C.10)

and a similar constraint on the next-nearest-neighbour Uij:

gσ(u)gC6(u)UNNNg†C6

(v)g†σ(w) = UNNN. (C.11)

The particular bonds for UNN and UNNN are shown in Fig. 8.1. For a given Z2 ansatz in

Table C.1, these may restrict the structure of Uij to disallow hopping (τ 3) or pairing (τ 2)

terms, but does not place any conditions on whether these terms are real or imaginary.

This differs from the singlet case, where the effect of σ upon the triplet terms was not

considered, and the ansatze become restricted to only singlet terms. Thus, spin-rotational

symmetry-breaking triplet terms are allowed within the same PSG for the Kagome lattice.

Since Uij is now no longer Hermitian, Uji 6= Uij, and the direction of the bonds matter.

They are shown in Fig. 8.1.


SL Label η12 gC6 gσ

Z2[0, 0]A +1 u, v, w : I u, v, w : IZ2[0, π]β −1 u, v, w : I u, v, w : IZ2[0, 0]B +1 u, v, w : iτ 3 u, v, w : IZ2[0, π]α −1 u, v, w : iτ 3 u, v, w : IZ2[0, 0]D +1 u, v, w : iτ 3 u, v, w : iτ 3

Z2[0, π]γ −1 u, v, w : iτ 3 u, v, w : iτ 3

Z2[π, π]B +1 u, v : I w : iτ 1 u, v, w : iτ 3

Z2[π, 0]B −1 u, v : I w : iτ 1 u, v, w : iτ 3

Table C.1: Parameters η12, gC6 and gσ characterizing the PSG GSS of spin liquid(SL) states, as given in (C.6).

C.3 Mean-Field Results

As mentioned in Sec. 8.2, we consider the mean-field solution for each of the Z2 spin liquid

ansatze, with nearest-neighbour perturbations D, ∆ and the next-nearest-neighbour J2.

While we do not expect the self-consistent mean-field theory to give a quantitatively

correct phase diagram, we can still gain some insight from the results. In particular, we

would like to understand the representative mean-field parameter values Uij when the

states in Table 8.2 are stabilized.

We focus on the parameter regime D/J,∆/J ∈ [0, 0.5). Furthermore, we will fix ω−

as the overall nearest-neighbour energy scale, taking the regime J2/ω− ∈ [0, 2). In this

way, we separate the effects of D and ∆ from J2 as much as possible. We begin by

considering general trends across all ansatze.

C.3.1 General Results

DM interactions: Here, we consider the contributions from the χ+ and η+ channels in

Eq. (8.12), in comparison to contributions from χ− and η− channels. The self-consistent

theory suggests that for U(1) and Z2 states labelled as [0, 0] and [0, π] states, the χ− and

η− channels have the dominant contributions, and the ratio of triplet to singlet terms is a

nearly linear function of D/J within our parameter window 0 < D,∆ < J/2. The [π, π]

and [π, 0] states, however, have stable mean-field states where χ+ and η+ are significant,

and dramatically small triplet values, particularly for D/J . 0.2.

Ising Interaction ∆: This term has a negligible effect on the Z2[0, 0] and Z2[0, π]

states, since χ+ and η+ are not relevant. For Z2[π, π] and Z2[π, 0] spin liquids, however,

small values of ∆/J make χ+ and η+ channels less relevant. Since ω+ = (J − ∆)/8 +


O(D), increasing ∆ from zero will actually decrease ω+, so these channels make little

contribution to HQ in the mean-field theory. Within the mean-field theory, these small

values of ∆ yield no qualitatively new behaviour.

Z2 Spin Liquids and Next-Nearest-Neighbour Terms: For the six (all except

Z2[0, 0]B and Z2[0, π]α) Z2 spin liquids in Table 8.2 that are stabilized only in presence of

next-nearest-neighbor terms, within mean-field theory, we must have 0.5 . J2/ω− . 1.2.

The Z2[0, 0] and Z2[0, π] states have a next-nearest-neighbour χ2 hopping parameter that

does not stabilize the Z2 spin liquids from their U(1) parents. Generally, |χ2| increases

with J2 in a monotonic and nearly-linear fashion. However, at a large value of J2 ∼ 1.2ω−,

the Z2[0, π]γ state is stabilized with small |χ1|, small |η2|, and large |χ2|. We expect that

such a state is unlikely to be realized in models where J2/J is small. The Z2[π, π]B state

is stabilized with a similar jump at J2 ∼ 0.9ω− with a large |χ2| value, and a small |η1|.

Out of the other four, in three Z2 spin liquids (Z2[0, 0]A, Z2[0, 0]D and Z2[0, π]β),

η and χ2 increases monotonically with J2 beyond a critical value of J2/J which itself

depends on the type of spin liquid in consideration. While the Z2[π, 0]B sees this be-

haviour for smaller values of J2, it also undergoes a jump around J2 ∼ 0.9ω−, similar to

the Z2[π, π]B state.

C.3.2 Mean-Field Phase Boundaries

Here we present the phase boundaries in the J2-D plane where these Z2 spin liquid phases

can be stabilized within mean-field theory. The Z2[0, 0]B phase is stabilized throughout

the phase diagram, while the Z2[0, π]α state is never stabilized within mean field theory

of the current Hamiltonian. The other Z2 spin liquids are stabilized for sufficiently large

J2.

Fig. C.2 (a) shows the U(1)-Z2 phase boundary in the J2-D plane for the Z2[0, 0] spin

liquids, Fig. C.2 (b) for the Z2[0, π] spin liquids, and Fig. C.2 (c) for the Z2[π, π]B and

Z2[π, 0]B spin liquids. The next-nearest-neighbour interaction J2 stabilizes all other six

Z2 spin liquids. We see that the Z2[0, 0]A, Z2[0, π]β and Z2[π, 0]B phases are stabilized

for smaller J2 than the others, while being destabilized by the Dzyaloshinsky-Moriya

interaction D. However, the Z2[0, 0]D, Z2[0, π]γ and Z2[π, π]B phases are stabilized by

D, despite requiring larger J2.


0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Z2[0,0]AZ2[0,0]D

J2/ω−

D

J

(a)

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Z2[0,π]βZ2[0,π]γ

J2/ω−

D

J

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Z2[π,0]BZ2[π,π]B

J2/ω−

D

J

(c)

Figure C.2: Heuristic mean-field phase boundaries in the J2-D plane of (a) Z2[0, 0],(b) Z2[0, π], and (c) Z2[π, π]B and Z2[π, 0]B states, where J2 is next-nearest-neighbourHeisenberg coupling, D the nearest-neighbour DM interaction in the z direction, and ω−is defined in (8.12).


C.4 Dynamical Structure Factor of Singlet+Triplet

Ansatze

In the main text, we showed the changes in the dynamical structure factor from the

singlet ansatze to when small triplet terms are added. Here we present the trace of the

dynamical structure factor matrix, i.e., (Sxx+Syy +Szz)/3, for these triplet ansatze. We

show the dynamical structure factor for the U(1) and Z2 spin liquids in Fig. C.3.


Γ M ′ K ′ Γ

ω/J

U(1)[0, 0]Γ M ′ K ′ Γ

U(1)[0, π]Γ M ′ K ′ Γ

S(q,ω

)

U(1)[π, π]

Γ M ′ K ′ Γ

ω/J

U(1)[π, 0]Γ M ′ K ′ Γ

Z2[0, 0]AΓ M ′ K ′ Γ

S(q,ω

)

Z2[0, 0]B

Γ M ′ K ′ Γ

ω/J

Z2[0, 0]DΓ M ′ K ′ Γ

Z2[π, π]BΓ M ′ K ′ Γ

S(q,ω

)

Z2[0, π]β

Γ M ′ K ′ Γ

ω/J

Z2[0, π]αΓ M ′ K ′ Γ

Z2[0, π]γΓ M ′ K ′ Γ

S(q,ω

)

Z2[π, 0]B

Figure C.3: Dynamical structure factor for U(1) and Z2 “singlet + triplet” spin liquidansatze. These structure factor are plotted from the centre of the extended Brillouinzone of the Kagome lattice, Γ, to the edge M ′, to the corner K ′, and back to the centerΓ. We note that since the “singlet + triplet” U(1)[0, π] spin liquid state is gapped, it isunstable to instanton tunnelling events. [178]

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