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More information about this series at http://www.springer.com/series/8790 Adam Iaizzi

Magnetic Field Effects in Low-Dimensional Quantum Magnets

Doctoral Thesis accepted by Boston University, MA, USA

123 Adam Iaizzi Department of Physics National Taiwan University Taipei, Taiwan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-01802-3 ISBN 978-3-030-01803-0 (eBook) https://doi.org/10.1007/978-3-030-01803-0

Library of Congress Control Number: 2018960268

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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland . . . science is more than a body of knowledge. It’s a way of thinking, a way of skeptically interrogating the universe with a fine understanding of human fallibility. If we are not able to ask skeptical questions, to interrogate those who tell us that something is true, to be skeptical of those in authority, then we’re up for grabs for the next charlatan, political or religious, who comes ambling along. Carl Sagan

Timestamp 3:52 from this May 1996 interview with Charlie Rose https://charlierose.com/videos/9094. This thesis is dedicated to: Dr. Arianna Wright Rosenbluth The author of the first-ever implementation of Metropolis Algorithm Monte Carlo Supervisor’s Foreword

A goal of computational materials science is to reproduce or predict physical properties as precisely as possible. However, in many cases it is impossible to include all interactions and quantum mechanical effects and still be able to carry out reliable calculations. From the standpoint of fundamental condensed matter physics, many details pertaining to specific materials are mere distractions, and one would instead like to construct the simplest possible models to describe physical phenomena of interest. A successful approach to studying quantum phases of matter and the phase transitions taking place between them is to construct field theories that are only constrained by relevant symmetries. If a given field theory can be solved, or treated with sufficiently good approximations, a wealth of information can be extracted on possible ground states and their excitations. However, solving quantum field theories is also no easy task, and it is not always possible to draw definite conclusions. Another approach is to formulate simplified lattice models inspired by real materials but not using the exact lattice structure and interactions of any specific material. If successful, these models (Hamiltonians) host the low-energy quantum states of interest, and they can sometimes be characterized in an approximation- free manner by numerical simulations. Since the low-energy physical properties of the lattice models should correspond exactly to some quantum field theory, direct connections can be made with theory as well as with experiments probing generic features. Even simplified lattice models are not automatically amenable to efficient numerical simulations. In particular, Monte Carlo simulations of quantum systems are often hampered by mixed signs of the weight function characterizing the configuration space—the infamous sign problem. Models tailored both from the perspective of interesting physics and computational tractability have been termed “designer Hamiltonians”1 and are playing an increasingly important role in quantum many-body physics. Adam Iaizzi’s thesis is devoted to constructing and studying a class of designer Hamiltonians in the field of quantum magnetism.

1R.K. Kaul, R.G. Melko, and A.W. Sandvik, Annu. Rev. Condens. Matter Phys. 4, 179 (2013).

ix x Supervisor’s Foreword

The work starts from the J -Q family of S = 1/2 quantum spin models on the 2D square lattice.2 These models contain the common Heisenberg exchange interaction of strength J and a multi-spin interaction of strength Q. When Q = 0 the system has an antiferromagnetic ground state. The Q interaction favors the formation of spatially correlated singlets, and, above a critical value of Q/J ,a spontaneously dimerized ground state. The transition separating the antiferromag- netic and dimerized phases is an intriguing “deconfined” quantum critical point, at which the excitations become fractionalized and exhibit unusual behaviors normally not found in 2D antiferromagnets. The J -Q model has been extensively studied in this context. Iaizzi expands on the J -Q model in both one and two dimensions by adding an external magnetic field, thereby accessing a broader range of physical phenomena. The quantum Monte Carlo algorithms, which can be adapted to many other models as well, are described in detail in this thesis and many new results are presented. Even in the conventional (Q = 0) Heisenberg model the field effects have not previously been fully characterized; the Q interactions introduce further rich behaviors that mimic some of the properties expected in traditional frustrated quantum magnets (which are difficult to study because of the aforementioned sign problems). Iaizzi studies the experimentally important transition into a fully polar- ized ferromagnetic state and discovers a case of metamagnetism—a discontinuous jump in the magnetization versus field above a threshold value of Q/J .This threshold value itself corresponds to a kind of quantum criticality, which is here characterized quantitatively, including an exact analytical solution in one dimension. A previous theory for the continuous transition into full polarization for lower Q/J is tested in detail in both one and two dimensions. At weaker fields and Q/J close to the deconfined critical point, it is interesting to study the consequences of the spinons (the putative deconfined excitations) induced by the field in two dimensions. Iaizzi’s work here aims to resolve the thermodynamic behaviors distinguishing spinons from the conventional magnon (spin waves) excitations. In addition to solving some key problems, the thesis points to intriguing open questions that could be addressed in future work.

Professor of Physics Anders W. Sandvik Boston University, Boston, MA, USA

2A.W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007). Acknowledgments

My work was supported by the NSF under grants No. DMR-1710170 and DMR- 1410126. The collaboration with Kedar Damle at the Tata Institute of Fundamental Research in Mumbai was supported by a travel grant from the APS-IUSSTF Physics PhD Student Visitation Program. My travel for presentations and other collaboration was supported by the NSF, the Simons Foundation, the Boston University Graduate Student Organization, the American Physical Society, and the Computational Science Research Center—Beijing (Natural Science Foundation of China Grant No. U1530401). The computational work reported in this paper was performed in part on the Shared Computing Cluster administered by Boston University’s Research Computing Services. Obviously I must thank my advisor Anders Sandvik for his patience and guidance and for providing me with so many opportunities to grow as a scientist over the past 5 years. I also must thank David Campbell for introducing me to quantum many-body physics, for his mentorship as I navigated graduate school and postdoc applications, and for nominating my thesis for a Springer Thesis Award. I am deeply grateful to my collaborators Kedar Damle and Dries Sels for their assistance and insights. I owe a great thanks to my collaborators Harley D. Scammell and Oleg P. Sushkov for their help with Chap. 4. In addition, I would like to thank the following people for useful conversations: Cenke Xu, Hai-Qing Lin, Sylvain Capponi, Ling Wang, Oleg Sushkov, and Harley Scammell. A special thanks to Hong Gang Luo, Bin-Bin Mao, and Chen Cheng for alerting me to an error in one of my preprints. I would also like to thank all members of the Sandvik group at Boston University, especially Phillip Weinberg, Pranay Patil, Na Xu, Thomas Lang, Maro Suwa, Nvsen Ma, Tim Khouw, and Ying Tang. Thanks also to my readers Anders Sandvik and David Campbell, along with the rest of my committee: Claudio Chamon, Rob Carey, and Shyam Erramilli. I owe a great debt to my family, especially my parents, Tracey Iaizzi and Garry Iaizzi, and my grandmother, Marion Heider. And also to my friends, whose kindness, compassion, and homework help have kept me sane throughout graduate

xi xii Acknowledgments school. Last and certainly not least, I would like to thank my wife, Vanessa Calaban, for her love and support throughout this seven-year process, all the way from applying to graduate school to finding a postdoc and writing this very document; I cannot imagine having a better partner in my life.

Taipei, Taiwan Adam Iaizzi [email protected] www.iaizzi.me https://orcid.org/0000-0001-5570-8282 Contents

1 Introduction ...... 1 1.1 HowtoReadThisDissertation ...... 2 1.2 What Is Computational Physics? ...... 3 1.2.1 A Brief History of Computational Physics...... 4 1.2.2 DevelopmentoftheMetropolisAlgorithm ...... 5 1.2.3 Toward a More Detailed Balance...... 7 1.3 Condensed Matter Physics ...... 11 1.4 Classical Phase Transitions ...... 13 1.4.1 2D Ising Model...... 14 1.5 Quantum Phase Transitions ...... 18 1.5.1 Deconfined Quantum Criticality...... 22 1.5.2 WhatAreQuasiparticles?...... 22 1.6 Motivation ...... 24 References ...... 26 2 Saturation Transition in the 1D J -Q Model ...... 29 2.1 Introduction ...... 29 2.2 Methods ...... 31 2.3 Phase Diagram ...... 32 2.4 Metamagnetism in the J -Q Chain ...... 34 2.4.1 Origin of the Magnetization Jump...... 37 2.4.2 An Exact Solution at qmin ...... 40 2.4.3 Excluded Mechanisms for Metamagnetism...... 41 2.5 Metamagnetism in the J1-J2 Chain ...... 42 2.6 Zero-Scale-Factor Universality ...... 45 2.7 Conclusions and Discussion ...... 50 References ...... 52 3 Saturation Transition in the 2D J -Q Model ...... 55 3.1 Background ...... 56 3.2 Methods ...... 57 3.3 Phase Diagram ...... 57

xiii xiv Contents

3.4 Metamagnetism ...... 58 3.4.1 Exact Solution for qmin ...... 59 3.4.2 Quantum Monte Carlo Results ...... 62 3.5 Zero-Scale-Factor Universality in 2D ...... 63 3.5.1 Form of the Low-Temperature Divergence ...... 65 3.6 Conclusions ...... 69 References ...... 70 4 Signatures of Deconfined Quantum Criticality in the 2D J -Q-h Model...... 73 4.1 Background ...... 74 4.1.1 The Zero-Field J -Q Model ...... 74 4.1.2 Direct Evidence of Spinons...... 75 4.1.3 BKT Transition ...... 76 4.1.4 Outline ...... 77 4.2 Methods ...... 77 4.3 Phase Diagram ...... 78 4.4 Field-Induced BKT Transition ...... 79 4.4.1 SpinStiffness ...... 80 4.4.2 Non-monotonic m(T ) Dependence ...... 83 4.4.3 Estimation of TBKT ...... 85 4.5 Anomalous Specific Heat ...... 87 4.5.1 Full Contributions from the Gapless Modes ...... 89 4.5.2 QMCResults...... 93 4.6 Conclusions ...... 95 References ...... 96 5 Methods ...... 99 5.1 Exact Diagonalization ...... 99 5.2 MonteCarlo ...... 100 5.2.1 Importance Sampling ...... 102 5.2.2 What Is a Markov Process?...... 103 5.2.3 TheMetropolis–HastingsAlgorithm...... 104 5.2.4 Practical Considerations: Autocorrelations, Binning, Error Bars, and Equilibration ...... 106 5.3 Quantum Monte Carlo: The Stochastic Series Expansion ...... 108 5.3.1 Formalism ...... 109 5.3.2 Sampling Procedure...... 113 5.4 The Heisenberg Model ...... 114 5.4.1 Diagonal Updates ...... 117 5.4.2 Off-Diagonal Updates...... 119 5.4.3 ObservablesinSSE ...... 122 5.5 The J -Q2 Model ...... 125 5.5.1 Diagonal Updates ...... 127 5.5.2 Off-Diagonal Updates...... 129 Contents xv

5.6 The Heisenberg Model in an External Field ...... 129 5.6.1 Diagonal Updates ...... 131 5.6.2 Off-Diagonal Updates...... 133 5.7 The J -Q-h Model ...... 140 5.7.1 Diagonal Updates ...... 141 5.7.2 Directed Loop Updates ...... 142 5.8 Supplementary Procedures ...... 143 5.8.1 Quantum Replica Exchange ...... 144 5.8.2 β Doubling ...... 146 5.9 Pseudorandom Number Generation ...... 147 References ...... 147 6 Conclusions ...... 149 References ...... 150

A Supplementary Material for the 1D Few-Magnon Expansion ...... 151 A.1 Few Magnons in the J -Q-h Chain ...... 151 A.2 Derivation of the Magnetization Jump in the J1-J2 Chain ...... 154 References ...... 156 xvii

Parts of this thesis have been published in the following journal articles:

1. Adam Iaizzi, Harley D. Scammell, Oleg P. Sushkov, and Anders W. Sand- vik. Direct numerical observation of Bose-Einstein condensation of spinons (in preparation) 2. Adam Iaizzi, Kedar Damle, and Anders W. Sandvik. Metamagnetism and zero- scale-factor universality in the two-dimensional J-Q model. Physical Review B 98, 064405 (2018). https://doi.org/10.1103/PhysRevB.98.064405. 3. Adam Iaizzi, Kedar Damle, and Anders W. Sandvik. Field-driven quantum phase S = 1 transitions in 2 spin chains. Physical Review B 95, 174436 (2017). https:// doi.org/10.1103/PhysRevB.95.174436 4. Adam Iaizzi and Anders W. Sandvik. 1D valence bond solids in a magnetic field. Journal of Physics: Conference Series 640(1), 012043 (2015). https://doi.org/10. 1088/1742-6596/640/1/012043 List of Abbreviations

AFM Antiferromagnet(ic) BEC Bose–Einstein condensate/condensation BKT (or KT) Berezinskii–Kosterlitz–Thouless (transition) DQC Deconfined quantum critical(ity) FM Ferromagnet(ic) LRO Long-range order QLRO Quasi-long-range order QMC Quantum Monte Carlo SSE Stochastic series expansion VBS Valence-bond solid

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