Magnetic Field Effects in Low-Dimensional Quantum Magnets

ISBN-13: 9783030018023 / Angielski / Twarda / 2018 / 156 str.

Adam Iaizzi

Spis treści

1 Introduction 1

1.1 How to Read this Dissertation . . . . . . . . . . . . . . . . . . . . . . 2

1.2 What is Computational Physics? . . . . . . . . . . . . . . . . . . . . 3

1.2.1 A Brief History of Computational Physics . . . . . . . . . . . 5

1.2.2 Development of the Metropolis Algorithm . . . . . . . . . . . 7

1.2.3 Toward a More Detailed Balance . . . . . . . . . . . . . . . . 9

1.3 Condensed Matter Physics . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Classical Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 2D Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.1 Deconned Quantum Criticality . . . . . . . . . . . . . . . . . 31

1.5.2 What are Quasiparticles? . . . . . . . . . . . . . . . . . . . . . 32

1.6 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Saturation Transition in the 1D J-Q Model 38

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Metamagnetism in the J-Q Chain . . . . . . . . . . . . . . . . . . . 46

2.4.1 Origin of the Magnetization Jump . . . . . . . . . . . . . . . . 49

2.4.2 An Exact Solution at qmin . . . . . . . . . . . . . . . . . . . . 54

2.4.3 Excluded Mechanisms for Metamagnetism . . . . . . . . . . . 55

2.5 Metamagnetism in the J1-J2 Chain . . . . . . . . . . . . . . . . . . . 57

2.6 Zero-Scale-Factor Universality . . . . . . . . . . . . . . . . . . . . . . 61

2.7 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . 68

3 Saturation Transition in the 2D J-Q Model 71

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4 Metamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.1 Exact Solution for qmin . . . . . . . . . . . . . . . . . . . . . 77

3.4.2 Quantum Monte Carlo Results . . . . . . . . . . . . . . . . . . 80

3.5 Zero-Scale-Factor Universality in 2D . . . . . . . . . . . . . . . . . . 82

3.5.1 Form of the Low-Temperature Divergence . . . . . . . . . . . 85

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Signatures of Deconned Quantum Criticality in the 2D J-Q-h Model 93

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1.1 The Zero-eld J-Q Model . . . . . . . . . . . . . . . . . . . . 94

4.1.2 Anomalous Specic Heat . . . . . . . . . . . . . . . . . . . . . 96

4.1.3 BKT Transition . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4 Field-induced BKT Transition . . . . . . . . . . . . . . . . . . . . . 102

4.4.1 Spin Stiness . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.2 Non-monotonic m(T) Dependence . . . . . . . . . . . . . . . . 107

4.4.3 Estimation of TBKT . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5 Anomalous Specic Heat . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5.1 Contributions from the Gapless Modes . . . . . . . . . . . . . 115

4.5.2 QMC Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5 Methods 127

5.1 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.2.1 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . 132

5.2.2 What is a Markov Process? . . . . . . . . . . . . . . . . . . . 133

5.2.3 The Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . 135

5.2.4 Practical Considerations: Autocorrelations, Binning, Error Bars,

and Equilibration . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3 Quantum Monte Carlo: The Stochastic Series Expansion . . . . . . . 140

5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.3.2 Sampling Procedure . . . . . . . . . . . . . . . . . . . . . . . 146

5.4 The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4.1 Diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4.2 O-diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . 155

5.4.3 Observables in SSE . . . . . . . . . . . . . . . . . . . . . . . 160

5.5 The J-Q2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.5.1 Diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.2 O-diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . 167

5.6 The Heisenberg Model in an External Field . . . . . . . . . . . . . . 168

5.6.1 Diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.6.2 O-diagonal updates . . . . . . . . . . . . . . . . . . . . . . . 173

5.7 The J-Q-h Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.7.1 Diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.7.2 Directed Loop Updates . . . . . . . . . . . . . . . . . . . . . . 185

5.8 Supplementary Procedures . . . . . . . . . . . . . . . . . . . . . . . 185

5.8.1 Quantum Replica Exchange . . . . . . . . . . . . . . . . . . . 187

5.8.2 Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.9 Pseudorandom Number Generation . . . . . . . . . . . . . . . . . . . 191

6 Conclusions 192

A Supplementary Material for the 1D Few-magnon Expansion 194

A.1 Few Magnons in the J-Q-h Chain . . . . . . . . . . . . . . . . . . . 194

A.2 Derivation of the Magnetization Jump in the J1-J2 Chain . . . . . . 198

Bibliography 201

Curriculum Vitae 208

O autorze

Adam Iaizzi received his PhD from Boston University in 2018. He now holds a postdoctoral position at National Taiwan University.

Blurb

This thesis is a tour-de-force combination of analytic and computational results clarifying and resolving important questions about the nature of quantum phase transitions in one- and two-dimensional magnetic systems. The author presents a comprehensive study of a low-dimensional spin-half quantum antiferromagnet (the J-Q model) in the presence of a magnetic field in both one and two dimensions, demonstrating the causes of metamagnetism in such systems and providing direct evidence of fractionalized excitations near the deconfined quantum critical point. In addition to describing significant new research results, this thesis also provides the non-expert with a clear understanding of the nature and importance of computational physics and its role in condensed matter physics as well as the nature of phase transitions, both classical and quantum. It also contains an elegant and detailed but accessible summary of the methods used in the thesis—exact diagonalization, Monte Carlo, quantum Monte Carlo and the stochastic series expansion—that will serve as a valuable pedagogical introduction to students beginning in this field.