karol gregor- aspects of frustrated magnetism and topological order

8/3/2019 Karol Gregor- Aspects of Frustrated Magnetism and Topological Order

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Aspects of Frustrated Magnetism and Topological Order

Karol Gregor

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

November, 2006


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c Copyright 2006 by Karol Gregor.All rights reserved.


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Abstract

In the first part we study O(N) spins on geometrically frustrated pyrochlore and planar

pyrochlore lattices. For classical spins the strong frustration causes a lack of ordering

for N = 2 at all temperatures. Nevertheless at low temperatures the system is stronglycorrelated. We find degrees of freedom that are relevant in this regime and find that the

correlations are dipolar up to distance 1/T and then decay exponentially. We usevarious approaches most notably, we show that this holds to any order in the 1/N

expansion. Then we look at spin evolution under precessional and Monte-Carlo dynamics

and find that in both cases the dynamical correlations decay exponentially in time with a

scale 1/T and that the long distance and long time correlations are described by thesimple Langevin evolution of the degrees of freedom found in the static case. Finally we

look at the effects of quantum fluctuations by studying the quantum rotor model on the

pyrochlore lattice. Using quantum Monte-Carlo simulations and analytical arguments we

find that for every N there is a region in quantum couplingtemperature plane with long

range nematic order.

In the second part we study stability of topological order in Kitaevs model under ar-

bitrary small local perturbations. Using the Fredenhagen-Marcu order parameter [13] we

show in quantum perturbation theory that in two or more dimensions, the topologically

ordered phase is distinct from more conventional phases. Further we show that this distinc-

tion survives to positive temperatures in three or more dimensions and that the topological

phase is also distinct from the high temperature phase. We extend this result to odd Ising

gauge theories and note that the relation between these and dimer models strongly suggests

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that this result also holds for the Ising RVB phases found in the latter [16] and in the related

spin models.

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Acknowledgements

I am very grateful to my advisor Prof. Shivaji Sondhi for his invaluable guidance, patience,

support and for sharing with me his vast knowledge of physics. I learned a lot of physics

through him, he pointed me to many interesting topics and he taught me not only physics

but many other important things that are needed for modern researcher.

I am also very grateful to my advisor Prof. David Huse for the working with me and for

me having chance to experience his vast knowledge of physics and his simple and effective

way to approach problems and understand physics.

I would like to thank Prof. Roderich Moessner for working with me and for many

interesting discussions that we had together. I also thank Joe Bhaseen for the work we did

together.

I would like to thank Prof. Herman Verlinde for being my advanced project advisor and

Prof. Michael Romalis for giving me a good experience of working in experimental physics

during work on my experimental project.

I am very grateful to my dear Yansong Wang for her love, for many wonderful moments,

for all the time we have spent together, for lot of fun and for many interesting conversations

that we have had.

I am very thankful to Vu Hoang Cao for his friendship, for the great time we spent

together and for letting me and teaching me how to experience life.

I am very thankful to Peter Svrcek for sharing with me his motivation, opinions and his

large knowledge of physics and mathematics.

I would like to thank many people whom I had interesting physics discussions or who

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made my life enjoyable here in any other way. These include Gau Pham Anh, Alexandre

Baitine, Fiona Burnell, Kai Chan, Alex Chuvikov, Chunmei Du, Pedro Goldbaum, Christo-

pher Hirata, Shirley Ho, David Hsieh, Minhyea Lee, Alexey Makarov, Subroto Mukerjee,

Vassilios Papathanakos, Kumar Raman, Srinivas Raghu, David Shih, Amol Upadhye, Gary

Vaganek, and the members of the Czech table especially Vaclav Cvicek, Iva Kleinova and

Martina Vondrova.

Finally I am very grateful to my parents for raising me in loving family, for always caring

about me and for teaching me. I would like to dedicate this thesis to them.

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Contents

Abstract iii

Acknowledgements v

Contents vii

1 Introduction 1

2 Classical Statistical Mechanics on Pyrochlore and Planar Pyrochlore lat-

tices 16

2.1 Basic facts about pyrochlore and planar pyrochlore lattices and the ordering

of classical spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Mode counting criterion for spin ordering . . . . . . . . . . . . . . . 17

2.1.2 Single tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3 Pyrochlore lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.4 Planar pyrochlore lattice . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Large N Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 N = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 1/N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Planar pyrochlore lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Physical Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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2.4.2 Exact consequences of the symmetries of the lattice . . . . . . . . . 31

2.4.3 N = correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.4 1/N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Pyrochlore Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.1 Exact consequences of the symmetries of the lattice . . . . . . . . . 43

2.5.2 N = correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.5.3 1/N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Height and Gauge Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7 Large N theory and 1/N expansion in magnetic field . . . . . . . . . . . . . 52

3 Dynamics on Planar Pyrochlore Lattice 54

3.1 Spin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Exact consequences of the symmetry of the lattice . . . . . . . . . . . . . . 55

3.3 Theory of dynamical correlations . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Large N derivation of dynamical correlations for Monte Carlo dynamics . . 57

3.5 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Quantum Rotor Model on Pyrochlore Lattice 66

4.1 Theories of quantum frustrated magnets . . . . . . . . . . . . . . . . . . . . 66

4.2 Single tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.1 Quantum ground state . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2 T > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Pyrochlore lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Quantum ground state . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.2 T > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.3 Scaling argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.4 Spin wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.1 Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2 Order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.3 Phase diagram and the order of transition . . . . . . . . . . . . . . . 82

4.5 Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Stability of Ising topological phase at zero and positive temperatures: An

order parameter argument 89

5.1 Topological order in Ising theories . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Definition of Ising lattice gauge theory . . . . . . . . . . . . . . . . . . . . . 95

5.3 Basic properties of Ising lattice gauge theory. . . . . . . . . . . . . . . . . . 99

5.3.1 Properties of pure Ising lattice gauge theory. . . . . . . . . . . . . . 99

5.3.2 Phase diagrams, properties and difficulty of distinguishing between

the phases of Ising lattice gauge theory with matter. . . . . . . . . . 104

5.4 Order parameter distinguishing between the phases of Ising lattice gauge

theories with matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.1 Order parameter in the Classical theory . . . . . . . . . . . . . . . . 106

5.4.2 Classical system with finite number of time slices (quantum system

at positive temperature) . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.3 Order parameter in the Quantum system. Quantum system at zero

temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.4 Quantum system at positive temperature . . . . . . . . . . . . . . . 116

5.4.5 Additional comments. . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.5 Odd Ising gauge theories and dimer models . . . . . . . . . . . . . . . . . . 122

A Duality of Quantum Ising Lattice Gauge theory 124

B Duality of Classical Ising Lattice Gauge theory 127

References 130

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Chapter 1

Introduction

A very interesting problem and the one that appears in many contexts is the following.

Imagine we have a large number of interacting objects that by themselves are very simple or

can be approximated by very simple ob jects. These objects interact among themselves with

interactions that are also very simple or can be approximated by simple interactions. There

are many ways to arrange the objects and the interactions. We can lay the objects out on a d

dimensional lattice and let only close neighbors interact with each other. The interaction can

be regular or random. Or we can arrange them at random in a d dimensional space and let

them interact with near neighbors in a regular way or at random. Or we can let them move

around. More generally the objects with interactions form a network which changes in time

according to some rule. We would like to understand these systems, which I think means

the following. It means to find patterns in the behavior of the system, that you can reliably

predict. The prediction should require much less computation then the one that would

be necessary to simulate the system exactly (assuming some computable approximation of

the underlying simple systems). The information contained in our understanding of large

systems is necessarily much smaller then the information contained in them and therefore

the understanding of these systems will almost always be approximate.

The number of such systems (networks) is very large. One just have to imagine few

examples to conclude that it is impossible to understand them in general and even that

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there are many systems which cannot be understood at all. One complex example (which is

being understood better and better) is the network of chemical reactions inside the cell. The

richness of these networks represents the richness of life that we see around us. Fortunately

most of the systems around us can be understood by us to an increasing level as we spend

more and more effort to do it.

Condensed matter physics studies some of the simplest and most fundamental such

systems. This research is driven by two basic reasons, written without order of importance

(which is very personal). First, our society has seen great advances in technology which

caused a large expansion of the set of things that we are able to do. At the base of this

technology is an understanding of basic materials which are one of the simplest physical

systems of the form mentioned above. Second reason is that one of the greatest satisfactions

of human mind is to understand things. We find these systems interesting because they are

on one hand very simple as they are build of simple building blocks arranged in a simple

way and interacting via simple interactions and yet they can exhibit many different kinds of

behavior. Furthermore, they are objects of the real world around us, and we have a strongpreference to understand this world.

The building blocks of condensed matter systems are atoms, electrons and light and they

are described by laws of quantum mechanics, which in many cases can be approximated

by classical physics. Examples of condensed matter systems are: Regular lattice of atoms

(Aluminum crystal), almost regular lattice of few types of atoms with several atoms of dif-

ferent type scattered around (La2xBaxCO4), two dimensional layer of electrons (interface

between GaAs and AlGaAs), substances made of polymers (Rubber). They can exhibit

various behaviors such as (corresponding to the above examples) conductivity, high tem-

perature superconductivity and ferromagnetism, quantum hall effect, large compressibility.

Our understanding consists for example of observations that notions such as volume, pres-

sure, temperature consistently make sense for different objects, that the system responds

linearly in volume to the applied pressure, linearly in temperature to the added energy and

knowledge of the values of the corresponding coefficients (compressibility, heat capacity).


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The basic approach to finding what happens in these systems is to approximate them

by some ideal system that we can solve exactly and calculate the behavior of the actual

system in perturbation theory. Other approach (which can usually be formulated in the

form above) is a mean field theory, where we approximate the actual interactions acting

on some subsystem by their expected average. Again the true behavior is then calculated

in perturbation theory. When these approaches dont work the systems is called strongly

correlated. This is not any exact definition. In fact often, using physical intuition, experi-

mental results or numerical simulations we can identify a degrees of freedom that are more

relevant for the given problem, but are different then the ones one would think of at first, in

terms of which the mean field theory and perturbation around it gives a good description

of the system. There are many approaches one can take and it is beyond the scope of this

introduction to review them all here.

Now, let me introduce an example which is relevant to my work and where these ideas

and many of the ideas of my work can easily be illustrated. Consider a set of spins of

magnitude S arranged on a cubic lattice and interacting via exchange interactions, that is,described by the Hamiltonian

H =

ij

Jij Si Sj (1.1)

where J is a matrix that specifies what is the interaction between every pair of spins. This

is called Heisenberg model. Usually only the interactions between close neighbors need to

be considered. One would like to understand the behavior of this system as a function of

the couplings, magnitude of the spins and temperature. Note that at large S, the system

becomes classical: The spins become classical vectors.

Let us discuss what is the ground state and the lowest excited states of this system.

First, imagine that only the nearest neighbors interact. If the interaction J is negative

(ferromagnet) the ground state is the direct product of the same state for each spin. If

it is positive (antiferromagnet) the ground state for classical spins is also simple: spins on

one sublattice point in one direction and the spins on the other in the opposite direction.

Quantum ground state is no longer simple:

|S

on one and

| S

on other sublattice is


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no longer an eigenstate. However the ground state can be obtained in perturbation theory

about the classical state in 1/S expansion and hence is close to this state. The lowest

energy excitation of the system of noninteracting spins is to excite one of the spins. When

they interact, the excitations are to the lowest order just a linear combinations of single

spin excited, propagating in some direction: spin waves. They carry the same quantum

numbers as the excitation of the underlying objects (spins).

Now consider the nearest neighbor antiferromagnet and turn on the next nearest neigh-

bor interaction which is also antiferromagnetic. This interaction wants the next nearest

neighbors to point in the opposite direction, whereas nearest neighbor interaction wants

them to point in the same direction. When these interaction are of comparable strength,

they are competing with one another. In that case some of the bonds will not have spins

close to to the state that minimizes the bond energy. This property is called (bond) frus-

tration.

Competing interactions and frustration are the main themes in my work. When the

interactions are competing one can expect a lot of exotic phases to appear near one another.Typical is also a large density of states near the ground states compared to the nonfrustrated

systems. As the interactions compete, the system is far away from simple states, which

makes the perturbative approach often impossible. One then has to find clever methods to

analyze these problems. In the system that we have considered so far various exotic ground

states have been proposed. States of one kind are valence bond states where the spins pair

up into singlets that are arranged in various crystalline patterns. This is well established for

some range of parameters. Other (not well established) is a resonating valence bond state

[2] where these bonds form a liquid and whose excitations are particles that carry spin 1/2.

These excitations are unlike the excitations of the original building blocks of this system

(spins) for which spin can only be changed by an integer. These particles are examples of

patterns that we have observed, they form a part of our understanding of the given system.

We can predict their existence and their behavior - they are described by the same laws

as the particles in elementary particle physics - this description is approximate, but very


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precise.

My work belongs to the area of geometrical frustration which is a special type of frus-

tration induced by geometry. The basic building block of most of these systems is a set of

three spins on triangle interacting antiferromagnetically. For now let us restrict ourselves

to the classical spins. On a triangle there is no configuration with spins pointing in the

opposite directions on every bond, Figure 1.1a. There are various lattices that are made of

triangles and tetrahedra (which are made of triangles) Figure 1.1b,c,d. The most common

one and the one I will focus on is the pyrochlore lattice, Figure 1.1c, a three dimensional

lattice made of corner sharing tetrahedra. It was found that the the number of degrees

of freedom that are in the ground states grows linearly with the total number of degrees

of freedom [1]. This large ground state degeneracy results in the strong suppression of or-

dering. In fact for three and higher dimensional spins and also for Ising spins (spins with

values 1) the system stays paramagnetic at all temperatures. For XY spins (points ona circle), there is finite temperature phase transition into an ordered state, that is due to

different amount of phase space around different ground state (termed order by disorder).This system is similar to the one considered above when the nearest and the next nearest

neighbor couplings are tuned appropriately. What is different here is that no fine tuning is

need because the frustration results from the geometry.

There are many compounds with pyrochlore structure. However in a physical system

there are always interactions other then the nearest neighbor ones and the spins are quan-

tum. These can often be considered as a perturbations. They will lift the ground state

degeneracy and there are many ways to do it, depending on interactions, that is depending

on the compound and the conditions that it is in. Therefore, we expect pyrochlores to ex-

hibit various kinds of behavior. This is confirmed by experiments. If the perturbations are

weak, the system will stay paramagnetic to much lower temperatures than the temperature

given by the strength of the interaction. The parameter characterizing this is the frustration

parameter f that equals the ratio of the would be transition temperature if the crystal was

not frustrated to the actual transition temperature. Now I will review some characteristic


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?

(a) (b)

(c) (d)

Figure 1.1: (a) Frustration: Impossibility to satisfy all bonds (b) Kagome lattice (c) Py-rochlore lattice (d) Planar Pyrochlore lattice. The squares with crossing are tetrahedra -there is no spin at the crossing.

compounds and experimental observations.

The most common compounds with magnetic pyrochlore structure are spinels and py-

rochlores. The spinels have structure A2+B3+2 O4 where the B sites form pyrochlore lattice.

The pyrochlores have structure A3+B4+O7 where both A and B lie on pyrochlore lattice

and both can in principle be magnetic.

An example of compound that orders into an long range antiferromagnetic state is

the pyrochlore Gd2T i2O7. Its Curie-Weiss temperature is CW 9.6K whereas theordering occurs at Tc 1K [3]. This can be observed for example from the magneticsusceptibility data. At high temperature it obeys the Curie-Weiss law which states that


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the inverse susceptibility is linear in temperature and becomes zero (by extrapolation) at

the Curie-Weiss temperature which approximately equals the minus of the transition (Neel)

temperature of the ordering into a long range antiferromagnetic state, if the spins were

on non-frustrated lattice with the each spin having the same number of nearest neighbors.

The Figure 1.2, taken from [3], illustrates this and also shows the absence of ordering to

temperatures much lower then the minus of Curie-Weiss temperature.

0 100 200 3000

10

20

0 5 10 15 20 250.0

0.5

1.0

1.5

2.0

2.5

Gd2Ti2O7

-1(

mol/emu)

T (K)

Figure 1.2: Measurement of the inverse susceptibility of Gd2T i2O7. The graph taken from

[3].

Another type behavior occurs in Y2M o2O7 [4] [5]. As usual, it stays paramagnetic to

temperatures much lower then the Curie-Weiss temperature, but then it enters a spin glass

state. A spin glass, which occurs in many other non-geometrically frustrated compounds,

arises from randomness of the spin distribution and/or randomness of the interactions, due

to impurities in the sample. In the spin glass due to randomness not all bonds can be

satisfied and so the spin glass in general is by itself an example of a frustrated system.


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Characteristic features are: Large density of states at low energies, history dependent be-

havior of the sample and a very slow relaxation as the systems lowers its energy by tunneling

between many local ground states. What is surprising in this pyrochlore case is that the

spin glass occurs at immeasurably small amount of disorder. The spin glass behavior can

be observed for example in difference between spin susceptibility when cooling the sample

in zero field and non zero field (history dependence) as illustrated in Figure 1.3 which is

taken from [4].

Figure 1.3: Zero field and nonzero field cooled susceptibility in Y2Mo2O7. The graphs istaken from [4].

In the next example, the spins in Ho2Ti2O7 can be approximated as Ising. The Ti4+ ions

are nonmagnetic and Ho3+ are magnetic, with J = 8. In the centers of the Ho tetrahedra

lies an oxygen ion. Two oxygen ions in nearby tetrahedra are close to one another and

produce a strong anisotropy for the Ho ion lying in between. This splits the 5I8 state,

producing two fold degenerate ground state spanned by |8,8. The first excited state is

several hundreds of Kelvin above the ground state. Hence the spins at low temperatures

are to a very good approximation Ising. The interaction between the physical spins is

ferromagnetic. However, because of the spin orientations, the interaction between the Ising


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variables, taken as

1 if they point in/out, is antiferromagnetic. The system is in a ground

state if two spins point in and two out. This is called ice rule because this problem is

equivalent to the water ice as follows: The oxygen atoms of ice lie on the diamond lattice

that is the lattice of the centers of tetrahedra of pyrochlore lattice. Between every oxygen

atoms there is one hydrogen atom. However the hydrogen atom is always closer to one

of the oxygen atoms. Furthermore there is the ice rule which states that always, two

hydrogen atoms are closer and two further from a given oxygen atom. This maps in the

above compound to two spins pointing in and two out. Therefore such compound is called

spin ice. One difference from the spin ice is that once the ice freezes, the position of hydrogen

atoms is also frozen, in one of the many possible configurations. In spin ice however the

spin interaction energy is much lower then the crystal freezing temperature, and hence the

spins are free to move before the temperature is lowered enough that this spins freeze.

It was found theoretically that the ground state degeneracy of Ising spins on pyrochlore

lattice is extensive: approximately (2/3)3/2 of states are in the ground state. Experimentally

this can be verified by integrating experimentally measured (magnetic part of) C(T)/T toobtain the low temperature entropy. At high temperature the entropy is given by equipar-

tition theorem and hence the (almost) zero temperature entropy can be obtained, which

is the logarithm of the ground state degeneracy. This measurements confirms the entropy

mentioned above. Thus it would seem that this explains the basic physics. However the

leading interactions are not exchange interactions but dipolar ones, which are long ranged,

decaying as the inverse third power of the distance. This has been explained through a lot

of work but recently in quite a simple way [6] briefly as follows. There are more interactions

that have the same states as ground states, and in their work, one such has been found

that is very close to the dipolar interaction but is much easier to analyze. The physical ex-

planation of why this works is motivated by work described in this thesis: The correlations

of nearest neighbor interaction model are dipolar and hence they are compatible with the

dipolar interactions.

Finally I would like to mention that there are many other systems with competing


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interactions then magnetic insulators that display rich behavior. One simple one is Frenkel-

Kontorowa model [7]. It is a system of particles attached by a string that has an equilibrium

distance l and elastic constant k. These particles sit in a periodic potential with period

d and stiffness K. If these two distances are not simple multiples of each other they are

incompatible and the corresponding interactions are competing. In one limit, K k, theparticles will not care about the potential but sit at the distance l from one another. In

the other limit, K k, they will not care about the string between them but will sit inthe minimum of the potential. When K and k are comparable however, they will transition

between many phases as we vary l/r. There is a critical value of K/k for which there are

infinitely many phases of different constant density which together form a Cantor set of

non-integral fractal dimension. Plotting the density versus l/k will give us a curve called

devils staircase.

The system with perhaps richest phase diagram known is a system of electrons in two

dimensions in a magnetic field in the presence of disorder. As we lower the temperature and

disorder and vary the magnetic field there appear more and more phases (whose numberseems to increase without a bound). The phases appearing are insulators and integer

and fractional quantum hall phases. This seems to resemble the Frenkel-Kontorowa model

mentioned above but is more complex. Further the fractional quantum hall phases are very

exotic, having excitations with fractional charge and fractional statistics. This is another

example of phenomena (a pattern that we managed to observe) that arises only when many

particles are put together.

After this introduction I give an outline of my thesis. The thesis is divided into four

parts. In the first two we consider classical O(n) spins (n-dimensional vectors) on pyrochlore

and planar pyrochlore lattices. As mentioned for n = 2 the system stays paramagneticat all temperatures. However at low temperatures the spins are much more restricted

and correlated and therefore the nature of this state is different from the one at high

temperatures. Our goal is to find the properties of this state.

In the first part we focus on static properties. Part of this work is published in [8]. At


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zero temperature we find that the long distance degrees of freedom that are relevant are

transverse fields (fields with zero divergence) with gaussian probability distribution. The

transverse fields arise from the ground state condition on the spins: Spins have to add up

to zero in every tetrahedron, which translates into zero surface integral of the field and

therefore its zero divergence. In the action, this term corresponds to the usual mass term

(quadratic term with no derivatives) - with the difference that it gives nontrivial correlations.

Since the mass term is the most relevant term in an action in any dimension (in the sense

of renormalization group), other terms in the action are irrelevant, and therefore at long

distances the action for this field is Gaussian, despite being in two or three dimensions. The

correlations following from this action are dipolar: They form is that of the electric field of

an electric dipole and they decay as the third power of distance in three dimensions. At

positive temperature the longitudinal modes appear. The resulting correlations are dipolar

up to distance 1/T and then decay exponentially. This dependence doesnt acquireany nontrivial exponents for the reasons above.

We approach this problem in the following ways. First is the physical argument outlinedin the previous paragraph. The second is the large N (where N is the dimension of the

spins - N in O(N)) argument which is equivalent to a self-consistent gaussian approximation.

They both predict the form of correlations above. The third is consideration of fluctuations

around the large N result in the 1/N expansion. It predicts that the correlations stay

dipolar to all orders in 1/N expansion and that the correction at the order 1/N is very

small. The fourth is the numerical simulations. We find that for pyrochlore lattice the

agreement between the numerical simulations and the large N prediction is excellent for

Ising spins and very good Heisenberg spins.

In the second part we focus on dynamical properties. Since the spins are dynamical

they dont have dynamics by themselves as opposed to quantum spins that are evolved by

Schrodinger equation. The dynamics that we consider is the dynamics of the quantum spin

operators in Heisenberg picture - the precessional dynamics - every spin rotates around

the instantaneous sum of its neighbors. We also consider simpler dynamics: The Langevin


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12

dynamics where each spin is acted upon by a random force.

Again, we are interested in the degrees of freedom that are important at low tempera-

tures and the space-time correlations. We find that the important degrees of freedom are

the same and their long distance properties are described by the same theory as if they

were evolving under a Langevin dynamics. We use the large N method for the Langevin

and numerical simulations for both dynamics to verify this.

These results should relevant for the real materials for the temperatures smaller then

the exchange interaction but larger then the ordering temperature. As we discussed this is

often a large range of temperatures. These results are also important as they are results for

the ideal system which is often a starting point for understanding the effects of additional

interactions or quantum fluctuations.

In the third part we consider the effects of quantum fluctuations. This work is published

in [9]. There has been a lot of work done on finding the behavior of quantum spins, and

it will be briefly reviewed in the chapter. In this work we consider a different quantum

fluctuations then those of the regular spins by considering a set of quantum rotors onpyrochlore lattice. Quantum rotor is equivalent to a quantum mechanical particle on a

sphere. We find that for any N the quantum fluctuations induce a long range nematic

order (all spins point along a common axis but with different orientations along it) in a

region of quantum coupling/temperature plane. The transition is first order and happens

at temperatures lower then 0.015 (in terms of exchange coupling) and quantum coupling

lower then 0.045. These low values are manifestations of frustration in the system. We

havent been able to determine if there is any further ordering determining which spins

order in which direction along the axis or if there is a transition to such states at even lower

values of temperature and quantum coupling.

The topic of the last section is topological order. Generally, a system is called topolog-

ically ordered if its low energy excitations are described by gauge theories. The simplest

one is the Chern-Simons theory in two dimensions. This theory is purely topological having

finite number of states whose number depends only on the topology of underlying manifold


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13

- which is the origin of the term topological order. Theories with continuum of excitations

are Abelian (U(1)) or various other non-Abelian (e.g. SU(2)) gauge theories. The topo-

logical order is increasingly popular topic as it seems to describe many strongly correlated

systems.

Here we study a particular model exhibiting topological order, Kitaevs model [10]. It

is a system of spin 1/2s residing on the links of a lattice with special types of interactions.

One interaction prefers even number of spins that are on links emanating from a given side

to point down. This implies that the basic variables are close strings (loops) of flipped spins

or strings with ends. The other interaction mixes the strings. If only these two interactions

are present the ground state is the equal amplitude superposition of all such loops - a loop

condensate. It has a special property that the ground state is degenerate and its degeneracy

is two to the number of noncontractible loops (e.g. if the system is square lattice with

periodic boundary conditions - a torus - there are 2 such loops and hence 4 ground states).

In addition if we add an arbitrary local perturbation to the system, these ground states

will only get split by factor ecL

where L is the size of the system and c is a constant,as has been shown in [11][12]. Thus, this system is topologically ordered. For many other

interactions (such as nearest neighbor ferromagnetic) this system has no topological order.

Since this is a sharp distinction the topologically ordered phase is a separate phase with

phase boundaries separating it from other ordered phases. Topological order and string

condensation have been of interest in the area of strongly correlated systems in recent years

as they seems to characterize many strongly correlated systems.

This system, with the two interactions described and the two simplest possible addi-

tional interactions, is the well studied Ising lattice gauge theory. It has two phases - the

deconfined/string condensed phase and the confining/Higgs phase. It is believed that there

is no local order parameter distinguishing between the two phases. However there is a non-

local order parameter distinguishing the two as found in [13][14]. In this chapter we will

do the following. We give an argument in quantum perturbation theory that this order

parameter distinguishes the two phases (the original argument was somewhat different).


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14

We show that the argument is valid under arbitrary local perturbation to the Hamiltonian

mentioned above, thus giving another argument for the stability of the topological phase.

Then we extend the argument to positive temperatures and show that the topologically

ordered phase survives to positive temperatures in 3 + 1 dimensions - that it is separated

from a conventional phase and from the high temperature phase by phase transition. We

will correct the statement in [15] that gives a numerical evidence for this happening in 2 + 1

dimensions. The failure is due to the presence of topological excitations.

We expect that this result is applicable not only to the system of spins on links with such

peculiar interactions but also to much more physical system of triangular and fcc lattice

spin 1/2 anti-ferromagnet with frustrating next nearest neighbor interaction (with spins

sitting on sites). The reason for this is following. As mentioned above there has been a lot

of work done on frustrated spins systems. One approach is to guess that the low energy

states are approximately spanned by states where the spins combine into spin singlets. The

ground state can be for example a particular crystalline pattern of these valence bonds or

a resonating valence bond state. This is still a hard problem to solve and one simplifiesit by replacing such bond by a hard core dimer (link connecting two sites), assume that

dimer states are orthogonal and writes the Hamiltonian for them (this procedure can be

made more rigorous). It has been found that on triangular and fcc lattices [16][17] the

dimer model has a resonating valence bond (or rather dimer) phase. This state and the

topologically ordered state in the model discussed above are somewhat similar. The first is

a condensate of bonds and the second a condensate of strings. By changing the later model

(forcing an odd number of flipped spins on links emanating from a given site) these look

even more similar and it is reasonable that they are continuously connected which means

that they belong to the same phase. Our work would then imply that this phase extends

to finite temperatures in 3 + 1 dimensions.

Our work can probably be extended to more general theories that have string conden-

sates as their ground state. In fact, string (net) condensation has been promoted by Wen

[19] as a rather general framework for strongly correlated systems. Using this picture he


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15

derives a classification of possible phases. However many of these phases dont survive to

positive temperatures, because if the variables are continuous there are typically topologi-

cal excitations that would destroy such phases even in 3 + 1 dimensions (they wouldnt in

higher dimensions, but that is un-physical, at least in condensed matter physics). On the

other hand, the extension of our argument would show how this happens. The simplest

example of continuous theory is the U(1) gauge theory. It can arise from both the Ising

degrees of freedom and continuous degrees of freedom. In [17] it was shown that the U(1)

gauge theory arises in a dimer model (Ising like degrees of freedom) on cubic lattice which

might be realizable in magnets. A spin model realizing this was given in [18]. A model of

boson giving rise to U(1) gauge theory was studied in [20]. A way to realize such phase in

experimental system of cold atomic gases was proposed in [21].


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Chapter 2

Classical Statistical Mechanics onPyrochlore and Planar Pyrochlore

lattices

2.1 Basic facts about pyrochlore and planar pyrochlore lat-

tices and the ordering of classical spins

In this section we introduce pyrochlore and planar pyrochlore lattices, model that we study

and some basic known results (mostly from [1]). This introductory section sets up the

framework and notation for the first three chapters.

The pyrochlore lattice is a lattice made of corner sharing tetrahedra, Fig. 1.1c. The

tetrahedra are of two different orientations each residing on one of the sublattices of a

bipartite diamond lattice. There are four spins per unit cell. We will study N dimensional

classical spins on this lattice with nearest neighbor anti-ferromagnetic coupling between

them, that is, described by Hamiltonian

H = Ji,j

Si Sj (2.1)

where J > 0. N dimensional spin is a classical vector of fixed length in N dimensional

16


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17

space. In the third chapter, we will study quantum rotors on this lattice, which will be

introduced then.

On a nearest neighbor bipartite lattice such as cubic lattice, the ground state ordering is

simple: The spins point in the same direction on one sublattice and in the opposite direction

on the other. The lowest energy excitations are spin waves and there is a finite temperature

phase transition into a disordered phase.

Pyrochlore lattice on the other hand is not bipartite. The basic unit is a single tetrahe-

dron which in turn is made of triangles. Spins on triangle are frustrated - they cannot all

point in opposite directions. Tetrahedron in addition has a nontrivial manifold of ground

states (meaning different ground states are not obtained from each other by rotations of the

whole system only) as we shall see below. Therefore finding ordering on the pyrochlore lat-

tice in nontrivial. In the next subsection we describe a general theory of ordering obtained

from counting the degrees of freedom, then describe the ordering on single tetrahedron and

finally turn to the full lattice. Most of these results are from [1].

2.1.1 Mode counting criterion for spin ordering

Classical spins on a lattice are equivalent to a single particle in many dimensional space

(whose degrees of freedom are the degrees of freedom of the spins). As temperature is

lowered the particle is more and more likely to be near one of the minima of the poten-

tial. If there is a single minimum then the behavior of particle is simple: as we lower the

temperature the particle will be more and more likely found near that minimum. However

on pyrochlore lattice as we will see below, there is a continuum manifold of ground states.

Around each point there can be a different amount of phase space having the potential in

a given energy interval. It is possible that this phase space will select a submanifold of the

ground state around which the spins will be predominantly found and which will be selected

in the limit of zero temperature.

Let us find an approximate probability of finding a system near a particular ground

state. First imagine that the ground state consist of isolated points. Approximate the


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neighborhood of each such point by quadratic potential (this approximation will be exact

in the limit T 0) and let 1, . . . , D be the eigenvalues of the normal modes y1, . . . , yD(the eigenvalues of the Hessian matrix). The probability of finding the system around such

ground state is proportional to

Dy exp((1y21 + + Dy2D)/2T) 11D .Now consider a system with a ground state manifold rather then with a ground state

consisting of isolated points. If we look at the degrees of freedom perpendicular to the

ground state manifold and find the eigenvalues of these modes 1, . . . , D then the proba-

bility obtained above will hold approximately (and become exact in the limit T

0).

If at some points of the manifold some of the s go to zero the probability density

at and around those points becomes very large. To find how probable it is to find the

system near these points we need to consider the phase space volume around these points.

Therefore consider the following situation. Let D be the dimension of the ground state

manifold. Let S be the dimension of a special sub-manifold of this ground state where

M eigenvalues become zero. Let be the distance from this sub-manifold in the manifold

of ground states. Then the amount of phase space at distance per unit volume (length,area...) of the special sub-manifold is DS1 (1 because we fixed ). The eigenvaluesthat go to zero are proportional to 2 and therefore the probability density at such point

is M. The probability of finding the system with < is proportional to0

D1SM.

Thus the system localizes around that sub-manifold if and only if D S M 0.Note that the above result seems to be independent of temperature. However this

an artefact of the approximation that probability of finding a system around a particular

ground state is proportional to1

1D . As mentioned this approximation becomes exactin the T 0 limit.

2.1.2 Single tetrahedron

In this subsection we describe the degrees of freedom and ordering of spins on a tetrahedron.

The system is described by the Hamiltonian

H = Ji,j

Si

Sj =

J

2

i

Si2

2J (2.2)


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19

The total number of degrees of freedom is 4(N

1). The spins are in the ground state

whenever they add up to zero. Let us count the number of degrees of freedom in the ground

state. The first two vectors can be pointed anywhere (2(N 1) degrees of freedom). Theother two vectors need to add up to the minus the first two. This means that the third

vector can be pointed anywhere except its projection onto the axis defined by the sum of

the first two vectors is fixed ((N 2) additional degrees of freedom). The fourth vector isthen determined. Thus there are 3N 4 degrees of freedom in the ground state manifold.

r1

r1

r r2 2

Figure 2.1: Left, parametrization of the classical ground states of a single tetrahedron,showing the orientations of the four spins and the reference axis. All the spins are at angle from the reference axis. The short arrows indicate the mode that goes soft as the spinsbecome collinear. The line with no arrows is the reference axis in spin space that we measure from. Right, when the spins are nearly collinear, the space of low energy configurationsthat includes the soft mode may be parameterized by the two small displacements awayfrom collinearity, r1 and r2.

Some of these degrees of freedom are rotations of the whole system and will not select

any ground state. Others are not and selection might occur. In three dimensions, Figure

2.1a, there are two degrees of freedom that are not overall rotations: and . The potential

around a given ground state depends on them. In two dimensions is absent. In more

the three dimensions the four spins that add up to zero always lie in a three dimensional

subspace. In that subspace we can define and as above. The configurations with

spins lying in a different subspace are overall rotations of a configuration with spins in this

subspace. Thus the potential depends only on and .


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20

Evaluating explicitly the eigenvalues of the modes (the eigenvalues of the Hessian matrix

around a given point) we find:

1 = sin2

2 = 1 sin2 cos2 /2

3 = 1 sin2 sin2 /2

4 = . . . = N = 1

N+1 = . . . 4(N1)

= 0

The states where two spins point in the same direction and the other two in the opposite

direction (e.g. = 0) are special: One of the eigenvalues goes to zero. Thus we have precisely

the case we discussed in the previous section with S = 0 (factoring out the overall rotations

that keep the axis of ordering fixed) and M = 1. To find whether the spins localize we

need in addition the amount of phase space around a collinear configuration, given axis of

ordering and small. Pick one of the spins. It has N 2 degrees of freedom for this .The spin that is nearly parallel to it is determined (because we have fixed the axis). The

third spin has also N 2 degrees of freedom and the fourth one is then determined. Thusthe amount of phases space is 2(N2) or D = 2(N 2) + 1 (the 1 is from ). The spins arelocalized if D S M 0 that is if 2(N 2) 0. Thus they are localized for N = 2 andspread out over the full manifold of the ground states for N > 2.

2.1.3 Pyrochlore lattice

In this subsection we describe the degrees of freedom and the systematics of ordering on

the pyrochlore lattice.

The Hamiltonian for the spins on pyrochlore lattice can be rewritten as

H =J

2

tet

(

itetSi)

2 2 (2.3)

Let Ns be the number of spins in the lattice so there are Ns(N 1) degrees of freedom.

The system is in a ground state if the spins in each tetrahedron add up to zero. Each


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21

tetrahedron imposes N constraints and there are four spins per two tetrahedra. Thus the

number of degrees of freedom in the ground state is Ns(N 1) N Ns/2 = NsN/2 Ns.There could be many candidates for the special states where some of the eigenvalues go

to zero. Since in the single tetrahedron spins prefer to align collinearly and this ordering is

consistent with the lattice we will consider collinear configurations (where the spins point

along a given axis some up and some down). This still leaves extensive, but discrete set

of configurations (that of Ising spins). We would like to find whether the system localizes

around a collinear configuration. The dimension of this special state is S = N

1, that

of the rotations of the whole system. As mentioned the dimension of the full ground

state is D = Ns(N 2)/2. We need to find out how many modes become zero as weapproach a collinear configuration. For isolated tetrahedron we found that this number is

one (per tetrahedron) and we assume that this stays true for the whole lattice. A better

argument is presented at the end of this subsection. Thus M = Ns/2. The condition

for ordering is D S M 0. However because of the large number of ground states,

because the determination of M is very approximate and because different eigenvalues goto zero at different rate, we can trust this condition only if it is true extensively that is if

D S M = cNs with c < 0 so for example S 0. We find D S M = Ns(N 3)/2.Thus we conclude that XY spins (N = 2) order collinearly, spins with N > 3 dont order

and the analysis for the Heisenberg spins (N = 3) is inconclusive. Numerical simulations

([1], and as we also verify in the section on quantum rotors) show that the Heisenberg spins

remain disordered.

Now we give more evidence that M = Ns/2 and discuss the distribution of modes

in one particular direction away from a collinear configuration. Consider one particular

collinear configuration: There are two types of tetrahedra on pyrochlore lattice depending

on their orientation (ones sit on one sublattice and the other on other sublattice of the

diamond lattice). Define the collinear ground state by letting spins on all tetrahedra of one

type be in the same configuration with = 0 in Figure 2.1. Now consider one particular

configuration away from the ground state: that by letting the spins on each tetrahedron be


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22

in a configuration of Figure 2.1 with

= 0 and = 0. It is straightforward to calculate

modes around this new configuration. They are those of Figure 2.2b with N 2 copiesof Figure 2.2a. We see that Ns modes go nonzero except along special directions in the

Brillouin zone, that however form a set of measure zero. Each of these modes goes as

i = ki2 and the ks of the lowest two bands ordered are plotted in Figure 2.2c. We see

that while Ns modes go nonzero, some of them have very small coefficient and those along

the special directions stay zero. Thus, if the counting argument is inconclusive, as in the

case of Heisenberg spins, this suggests that the system is disordered at all temperatures -

which is what happens as confirmed by numerical simulations.

0

0.2

0.4

0.6

0.8

1

a)0

0.2

0.4

0.6

0.8

1

b)

0

0.5

1

c)

Figure 2.2: The eigenvalues of the hessian matrix of the potential evaluated at the stateswith a) = 0 and b) = 0.3 and plotted along certain chosen directions of the Brillouinzone. For small the eigenvalues of the two lowest bands go as i = ki

2. The kis, orderedby their magnitude, are plotted in c).

Note that the modes that go nonzero go as 2 as we have seen in the example above.This is simple to understand. If we just consider two such zero modes, say z1 and z2, there

are no terms in the potential of the form z21 and z22. The next lowest term is z

21z

22. If we

move in one of the zero directions say z1 to a point distance away, that is to a point with

z1,0 = , the other mode becomes stiff but its stiffness is z21,0 =

2. In the full lattice there

are of course many modes. However for example we define the as a the root mean squared

displacement of all spins from collinear configurations, it is reasonable that this result stays


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23

valid.

Another way to see that M = Ns is from numerical simulations. One can calculate

the ground state entropy by starting with the high temperature entropy, measuring heat

capacity in the simulation and integrating C(T)/T to get the ground state entropy. This

was done in [22] and confirmed this prediction.

2.1.4 Planar pyrochlore lattice

Planar pyrochlore lattice is a two dimensional lattice of corner sharing tetrahedra Figure

1.1d. All the analysis above goes through in this case except for the fact the the ordering

of XY spins happens only in the limit of zero temperature due to two dimensionality of

the lattice as opposed to the case of pyrochlore lattice which has finite temperature phase

transition.

2.2 Thermodynamics

Moessner and Berlinsky [23] made a prediction for the energy and susceptibility of Heisen-

berg spins on pyrochlore lattice that is exact at zero temperature, asymptotically exact at

large temperatures and a very good approximation in between. Considering the fact that

the correlations are short ranged, they approximate the energy and susceptibility of the full

lattice by that of a single tetrahedron. As one can see on the Figure 2.3, the approximation

is excellent. The approximation is excellent even when the spin are diluted as they checked

for dilutions up to 20%.

2.3 Large N Method

In this section we outline the standard large N method for deriving properties of classical

spins on a lattice. The basic difficulty in analyzing classical spins is their fixed length

constraint. At N = this constraint is turned into a single Lagrange multiplier, makingaction quadratic in the unconstrained spin variables and hence trivial to analyze. The


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24

0 1 2 3 4 5 6 70

1

0 1 2 3 4 5

T/J

0.00

0.02

0.04

0.06

0.08

0.10

0.12

J

Figure 2.3: From [23]. Simulated susceptibility for Pyrochlore lattice (circles) and suscep-tibility of single tetrahedron (line). Inset shows the same for the energy.

method allows calculation of phase transition, critical exponents and spin correlations. To

approach the physical case of finite N correlations and some of the critical exponents can

be calculated order by order in 1/N expansion.

We consider system of classical spins of length

N described by Hamiltonian H =

12

ij SiJij Sj where the spins are normalized to have length N. The partition function

is an integral over all configurations of the Boltzmann factor and can be manipulated as

follows

Z =

DSe

12T

ijSiJijSj

i

(S2i N)

DSDe 1

2T

ijSiJijSj+

1

2Tii(S2iN) (2.4)

DeN

2

(T r log(J

i)+ 1

T

ii

i)

where J is the matrix with indices Jij and is the matrix with indices ij = iij. In the

first line, the delta function imposes the fixed length constraint on every spin. In the second

line we replaced the delta function by the integral over which turned the exponent into a

quadratic function of S. Then we integrated out the S and obtained an effective action for

the :

Seff =N

2T r log(J

i) +1

T

ii

i . (2.5)


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25

The correlation function is obtained by analogous manipulations resulting in:

Sai Sbj = abZ1T

D(J i)1ij

eN2

(T r log(Ji)+ 1T i

ii) (2.6)

where a and b are the vector indices. Let us define the correlation matrix C to be the matrix

with indices Cij = Sai Saj .

2.3.1 N =

Due to the factor N in the front of the effective action, the effective action is a sharply

peaked function at large N and can be approximated by its saddle point value. This is

obtained by differentiating it with i and equating the result to zero. Because all the points

are equivalent there is a solution with all is the same, ii = , which we will assume. The

J can be Fourier transformed into Jq. For a system with n0 atoms per unit cell, Jq is an

n0 n0 matrix. Its eigenvalues are bands q , = 1, . . . , n0. Then the saddle point value is given by

1

Ns

q,

T

q = 1. (2.7)

where Ns is the number of spins in the system.

For an unfrustrated magnet there is one or more, but finitely many, values ofq at which

the bands have global minimum. Around these points the dispersion is quadratic. For

D > 2 this results in a finite temperature phase transition at temperature Tc because below

this temperature the saddle point equation cannot be satisfied.

For frustrated lattices that we study in this paper however, there is at least one band

that is flat (independent of q) and is at minimum. The saddle point equation in this case

can always be solved: as T 0 it becomes

fT

min(q ) = 1 (2.8)

where f is the fraction of the points that are in the lowest flat bands (the higher bands can

be neglected as T 0). Therefore there is no positive temperature phase transition.


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The correlations at N =

become

Sai S

bj

= ab(C0)ij(q) where the matrix C0(q) is

given by

C0(q) = T(Jq )1 (2.9)

2.3.2 1/N expansion

Let us now turn to the 1/N expansion. In this case we start with the exact effective action

in (2.5) and expand it around the saddle point by writing i = + i. The linear term will

be absent because the action is expanded around the minimum, the quadratic term will give

the propagator for and the higher order terms will give vertices. Rescaling /Nwe get

Seff =N

2T r log(J + + /

N) + N n

+ N

j

j/

N = const +1

4T r(C0)

2

i 16

NT r(C0)

3 18N

T r(C0)4 + (2.10)

The first two vertices are shown on Figure 2.4 where the dotted line is a propagator.

Similar expansion can be derived from (2.6) for the correlation function and to the order

1/N it is given by the diagrams shown on Fig. 2.5.

Let us make one important observation about this diagramatic expansion. In this theory,

there is only one variable - the - and hence only one propagator and infinitely many

vertices. However the structure of the expressions is exactly the same as that of a theory with

two variables, and say S, the corresponding propagators P and C0 (already represented in

the diagrams by dotted and full lines), only one vertex of the form SS and with the loops

consisting of exactly two propagators C0 absent. This observation, which we will use when

discussing the diagrams, can be formally derived as follows. Start with the formula for the

effective action (2.4), add a propagator for , write the diagrams, sum over all two-C0 loops

to modify the propagator for and then set the original propagator for to zero.

The diagramatic expansion can be written in the momentum space. The C0(q) prop-

agator was derived above. The inverse of the propagator, which in the real space is


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Figure 2.4: Vertices.

+ +

Figure 2.5: Diagrams for two point correlation functions up to order 1/N.

P1ij =12C0,ijC0,ij, becomes the convolution

(P1

)

(p) =

1

2

dqC

0 (q)C

0 (p q) (2.11)The first of the order 1/N diagrams becomes

C1,1(p) = C0 (p)

dqC0 (q)P

(p q)C0 (p) (2.12)

where the multiple indices are summed over. The other diagrams are given by the analogous

formulas.

2.4 Planar pyrochlore lattice

In this section we describe low temperature properties of the nearest neighbor anti-ferromagnet

on planar pyrochlore lattice Figure 2.6. Planar pyrochlore lattice is a two dimensional lat-

tice of corner sharing tetrahedra, or simply a square lattice with crossings. While it is not

common experimentally, we will discuss it in detail because it is simple and in the later

section on more complicated pyrochlore lattice we will refer to this section for the details

that are the same between them.


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We find that the correlations are dipolar up to a distance

1/

T as T

0 and

then decay exponentially. We first give a simple physical argument. Then we derive what

the symmetries of the lattice only and the conditions in the ground state imply for the

correlations, without doing any approximations. Then we derive correlations in the large N

limit, find that they are dipolar up to a distance 1/T and then decay exponentially andshow that this stays true at large distances at any order in 1/N, with only a renormalization

of coefficients. Then we present numerical simulations confirming this picture.

y

x

(0,0)

(0,1)

(1,0)

1

2

2

1

1

2

2

1

1

2

Figure 2.6: Planar pyrochlore lattice.

The planar pyrochlore lattice has two spins per unit cell. Let us define the axes as in

the Figure 2.6 so the spins are S1x+1/2,y, S2x,y+1/2 where x, y are integers (we will use this

convention through rest of this section including dropping the spins indices a = 1, . . . , N ).

The nearest neighbor Hamiltonian on any lattice made of corner sharing tetrahedra can be

written as

H =i,j

SiSj =1

2tet

(

itetSi)2 + const

Thus the system is in the ground state iff the spins add up to zero in every tetrahedron.

2.4.1 Physical Argument

We will find the long distance action for coarse grained fields from which the correlations

follow.


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Define the staggered spins B1, B2

B1x+1/2,y = (1)x+yS1x+1/2,yB2x,y+1/2 = (1)x+yS2x,y+1/2

Under coarse graining, for each component of B this defines a two dimensional vector field

B = (B1, B2). At zero temperature B = 0. This can be seen as follows. The spinson each tetrahedron add up to zero. If we draw a circle around a square with crossing

(tetrahedron flattened to two dimensions), two Bs are pointing in and two out of it. If

we take some other large loop containing some number of tetrahedra completely (i.e. each

tetrahedron is either in or out) there will the same number of Bs pointing in as those

pointing out. Under a coarse graining of the Bs and of the loop (one dimensional surface),

this turns into

B dS = 0 which implies B = 0.Now we would like to know the probability density P( B) that we are in some configu-

ration B. For simplicity let us discuss the case of Ising spins. Consider a loop of nearest

neighbor spins. If the spins on the loop can be flipped and still remain in the ground state

then it is easy to check that the sum of the spins on this loops is zero. If we have a configu-

ration with a lot of flippable loops, it has a small value of B and if we flip some of the loops

we get different configuration with the same B (if the loops lies in the region that we use for

coarse graining). Thus small Bs are more probable. Let us estimate the P( B). The most

probable value is B = 0. If we look at widely separated regions, but still within the coarse

graining area, it is reasonable to expect that they are becoming independent (except for the

constraint

B = 0) and can point equally in either direction. The B is then roughly a sum

of random vectors. The resulting distribution is that of a random walk, namely the binomial

distribution, or gaussian for large number of spins in the coarse graining area. Thus we

expect P( B) eK2

B2d2x where K is a constant. This is the lowest order action that does

not include for example the gradient terms that arise from correlations of different coarse

graining regions. To verify that this action is correct, we generated all the ground states in

a 10x10 planar pyrochlore lattice and plotted the logarithm of the integrated histogram of

the B2

, which should be linear if the action above is correct. This is indeed what we find,


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Figure. 2.7.

0 1000 2000 3000 4000 50000

5

10

15

20

25

B2

log

B2

P(x)dx

Figure 2.7: Integrated distribution P(B2). If P(x) eKx/2 then the graph should be astraight line.

Thus we have found two facts: B = 0 and P( B) eK2

B2dS. It is a standard

calculation to show that the correlations are dipolar (that of an electric dipole)

Bi(0)Bj(x) xixj 2ijx2

x4

Bi(q)Bj(q) qiqj ijq2

q2

Also note that one can write B = A where A has a gauge invariance A A + .Thus the zero temperature properties are described by a U(1) gauge theory.

Now let us ask what happens at T > 0. We expect that the system will disorder -

that the correlations will be dipolar up to some distance and then decay exponentially.

The simplest action representing this can be obtained as follows. At T > 0 but small, the

condition B = 0 is relaxed weakly. For a general B we can always write B = BT + BLwith BT = 0 and BL = 0. We need to suppress the BL. However we need tosuppress it at large q where we need to preserve the dipolar correlations but not at small q

where we need to destroy them. The lowest order action representing this

S =K

2

B2T +

BL (1 + ()2) BLd2x (2.13)

where K is a function of temperature that goes to a nonzero constant as T 0. The

action gives gives dipolar correlation up to approximately distance which then decay


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B2qx,qy = x,y

eiqxx+iqy(y+1/2)B2x,y+1/2

in other words we use the exact positions in space when Fourier transforming the spins.

Let R be a symmetry of a square transformation. For example for the rotation by 90

degrees R(rx, ry) = (ry, rx). Its action on the Fourier-transformed spin is

Bq =x

eiqxBx

x eiq

x

(signR,)BR

Rx

=x

eiRqRx(signR,)BRRx

= (signR,)BRRq

where signR, is the appropriate sign. For example for the rotation, B1qx,qy B2qy,qx ,

B2qx,qy B1qy,qx .The correlation function is clearly proportional to the identity in the spin indices and

will not display them. It doesnt change if we replace each B by its transformed value

C(q) = signR,signR,CRR(Rq)

Using all these symmetries we can derive relations from which all others relations follow

C11(qx, qy) = C11(qx, qy) = (2.14)

C22(qy, qx) = C22(qy, qx) (2.15)

and

C12(qx, qy) = C12(qx, qy) = (2.16)

C12(qy, qx) = C21(qy, qx) (2.17)

These imply that it is possible to express the correlation function in terms of two functions

only

C(qx, qy) =

C11(qx, qy) C

12(qx, qy)

C

12

(qx, qy) C

11

(qy, qx)

(2.18)


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Let us now discuss what the restriction to the ground state configurations implies. In

the ground state, the sum of the spins on each tetrahedron is zero. In momentum space

this gives

B1q sin(qx/2) + B2q sin(qy/2) = 0 (2.19)

Hence the vector (sin(qx/2), sin(qy/2)) is an eigenvector of the correlation function with

zero eigenvalue. This implies that the correlation function can be written in the form

C(qx, qy) = g(qx, qy) sin2 qy

2 sin qx2 sin qy2 sin qx2 sin qy2 sin2 qx2

(2.20)where g is some function. The correlation function would be dipolar if we can show that

g(q) 1/q2 at small q.At positive temperature the correlation function can also be further restricted from

(2.18). To derive that formula we have only used the fact that the Hamiltonian respects

all the symmetries of the lattice, but we havent used the form of the Hamiltonian. The

Hamiltonian has a special property that it depends only on the sum of the spins on each

tetrahedron. To use it we proceed as follows. We decompose the B into its lattice transverse

and longitudinal parts B = BT + BL. The lattice version of divergence is the sum of the

four spins on a tetrahedron, or the square with the crossings. The lattice version of curl

is the sum of the four spins on the square without crossing. It is easy to verify that in

the continuum limit these translate into divergence an curl of the continuous field. The

change of variables between the B and its transverse and longitudinal parts has a constant

determinant.

An expectation value of some function of the B variables f(B) is proportional to the

integral off(B) multiplied by the Boltzmann factor over B at every site, with the restriction

that each B has a unit length. If this restriction was not there then, because the Hamiltonian

depends only on BL, the BL and BT would be uncorrelated BL(0)BT(x) = 0.Let us derive what would the last expression imply for the correlation function if it was

also true for the restricted spins. Fourier transforming the lattice version of BT = 0

gives us the condition (2.19). Fourier transforming the lattice version of BL = 0 gives


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us B1q sin(qy/2)

B2q sin(qx/2) = 0. Taking the correlation of these two equations implies

that if

v1 = c1q ( sin(qy/2), sin(qx/2))t (2.21)

v2 = c1q (sin(qx/2), sin(qy/2))

t (2.22)

then vt1 C v2 = 0. Define 2x2 matrix

Q = (v1v2) =1

cq

sin(qy/2) sin(qx/2)sin(qx/2) sin(qy/2)

(2.23)

Then CD = QtCQ is diagonal and hence v1 and v2 are the eigenvectors of C. This

works also in the opposite direction and thus we have the statement: QtCQ is diagonal if

and only if BL(0)BT(x) = 0.To get that this correlation is zero we assumed that the spins are unrestricted (dont

have fixed length) which is not true. However it is plausible that this correlation will still be

zero even if the spins are of fixed length as this constraint might average out to zero. We

were not able to give an analytic proof of this statement. However from the Monte-Carlo

simulations (described below), we found that CD is diagonal and hence this statement is

true.

The CD has a general form

CD =

g1(q) 0

0 g2(q)

(2.24)

where g1(qx, qy) and g2(qx, qy) are symmetric functions. The general form of C is then

C =1

c2q

sin2

qy2 sin

qx2 sin

qy2

sin qx2 sinqy2 sin2 qx2

g1(q)

1c2q

sin2 qx2 sinqx2 sin

qy2

sin qx2 sinqy2 sin

2 qy2

g2(q) (2.25)


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2.4.3 N =

correlations

We calculate the N = correlations on the planar pyrochlore lattice using the procedureoutlined above. These correlations were obtained before in [24]. However for some reason

their approach is very complicated. In addition we write them in a nice explicit form.

The J matrix has the form

Jq = 2

sin2 qx2 sin qx2 sin

qy2

sin qx2 sinqy2 sin

2 qy2

1 0

0 1

Its eigenvalues are

1q = 1

2q = 1 + 2c2q

where c2q = sin2 qx

2 +sin2 qy

2 . As expected from the fact that there is large number of ground

states, there is a flat band at the minimum.

The correlations can be calculated from the matrix C0 = T(J )1 and are given by

(2.25) with

g10(q) =T

g20(q) =T

2c2q(2.26)

where = (T) = (T) + 1 is given by the saddle point condition

1

2+

1

2

q

1

2c2q=

1

T(2.27)

In the limit T

0, we have that

T /2, the second term in the correlations formula

is negligible compared to the first one and the correlations are dipolar.

For T > 0 we rewrite the correlations conveniently as

C0 = T

sin2 qx2 sin qx2 sin

qy2

sin qx2 sinqy2 sin

2 qy2

1

2c2q 1

From this we see that the correlations decay exponentially with correlation length diverging

as 1/T as T 0 and for distances sufficiently shorter then this length they are

dipolar.


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2.4.4 1/N expansion

We are going to use the method of 1/N expansion outlined above to show that to any order

in perturbation theory the following statements are correct

1. At T = 0 the correlations at long distance are dipolar

2. For T > 0 there is a distance and temperature such that for distances larger then this

one and temperatures lower then this one, the correlations are dipolar up to approx

distance squared T /(1

0T /) and then decay exponentially. The 0 is a number

defined below.

The basic steps in the argument are the following. Let be a one particle irreducible

diagram in a two point correlation function, with the two external legs removed. We will

show that

1) has the same symmetries as C eq. (2.14)-(2.17)

2) is continuous

Then from the symmetry 12

(qx, qy) = 12

(qx, qy) it follows that the off-diagonal entriesgo to zero as q 0 and the diagram has the form

(q) = 0I + 1(q)

where 0 is a number and 1(q) 0 as q 0 and it is continuous. We will define 0 asthe sum of 0 for all the diagrams (assuming it converges) and similarly for 1.

We will briefly show how the conclusions of this subsection follow from these properties.

First the basic idea. To the lowest order at T = 0, the full correlation is C = C0 + C0C0.

Noting that C20 C0, if the leading term in is identity, the C will be proportional to C0to the lowest order in q and hence at large distances the correlations are dipolar.

Now more precisely and at T > 0 but small, write the correlation function (2.26) in the

form C0 = C1T + C2

T

11+2c2q/

. Then notice that C21 = C1, C21 = C2 and C1C2 = 0. Then

write the Dyson series for the correlations, pull out (C0 0I)1 term, use the propertiesof C1 and C2 and finally arrive at

C = (1 (C1

0 0I)1

1)1


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T

(1 0 T)

C1 + C21

1 2c2q(10T /)

The first term goes to identity as q 0 and that happens on the scale independent of T tothe lowest order. The last term is the N = correlation with temperature renormalizedand the second term is the overall renormalization of the correlations. This implies the

conclusions of this subsection.

Thus it remains to prove the properties of . We start with the symmetry. As seen

from the explicit expression the C0 has the same symmetries as C eq. (2.14)-(2.17) (this issimply true because it is also a correlation). From the definition of P (2.11) it is easy to

show that (P1)RR(Rp) = (P1)(p) where R is any of the symmetries of a square. It

is also easy to show that the inverse of the matrix also has these symmetries

PRR(Rp) = P(p)

These are the symmetries (2.14)-(2.17) but with no minus signs in the front.

It is easiest to understand the proof with a picture in mind, e.g. the diagram on Figure

2.8 which equals

(p) =

1,2

q1,q2

C10 (q1)C120 (q2)

C20 (p q1 + q2)P2(p q1)P1(q1 q2)

Figure 2.8: A diagram at order 1/N2

We would like to evaluate RR(Rp). Change all the internal momenta qi Rqi andall the internal indices i Ri. Use the symmetries of P and C0 to remove all the Rs.

Clearly we are going to get

(p) and it is the sign that we need to determine. Consider


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a specific R namely the transformation R(qx, qy) = (

qx, qy). R doesnt change the indices

. The C120 , C210 change sign, C

110 , C

220 dont and none of the Ps do. So we need to count

the number of C120 and C210 to determine the overall sign. There is always one line of C0s

going from to and some number closed loops. If we start and end with the same index,

the number of C120 , C210 or the minus signs is even and if we start with one index and end

with the other one, the number of minus signs is odd. For the closed loop we thus always

get plus and the diagram will get the minus sign iff and are different. Thus we showed

(

px, py) = (2

1)(px, py). Similar argument one can do for all the symmetries

and conclude that the has the same symmetries as C0, (2.14)-(2.17).

Now let us show that is continuous. First we show it for P1 which is a convolution

of two C0s. The C0 is not continuous, but it is bounded and discontinuous only at zero

and points equivalent to it in momentum space. Given > 0. Write

P1(p0 +p) P1(p0) =

dqC0(q)(C0(p0 +p q) C0(p0 q))

Clearly, if p0 is not a point of discontinuity the P will be continuous at that point, so lets

assume p0 is point of discontinuity. Pick a region small enough around all the discontinuity

points so that the integral over these is smaller then /2 no matter what p is (this is possible

because C0 is bounded). The rest of the region can be thought of as compact, because the

C0 is periodic. A continuous function on compact set is uniformly continuous. Thus we can

pick a so that for all points |p| < the above integral is smaller then /2. Thus the leftside of the formula is smaller then and so the P1 is continuous.

If P is invertible then by the inverse mapping theorem, it is continuous. We were not

able to show it analytically, but we did numerically and assume that this is true.

The full is made of a lot of convolutions of Ps and C0s and by similar argument to

the one above it is continuous.

2.4.5 Numerical Simulations

We have used heat bath Monte Carlo method to simulate Heisenberg spins on planar py-

rochlore lattice.


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We Fourier transform the result from the numerical simulations and then transform

the resulting matrix using Q in (2.23) to obtain QD = QtCQ. As promised this matrix

is diagonal. The off-diagonal terms are not zero for every configuration but are zero on

average. In out typical simulation they were decreasing with time, in our typical simulation

time they came down to about 0.5% of the diagonal terms and their shape looked like a

random noise. The functions g1 and g2 are plotted on the Figure 2.9.

0

2

4

0

1

2

karol gregor- aspects of frustrated magnetism and topological order

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