Anyon Theory in Gapped Many-Body Systems from Entanglement

Anyon theory in gapped many-body systems from entanglement Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Bowen Shi, B.S. Graduate Program in Department of Physics The Ohio State University 2020 Dissertation Committee: Professor Yuan-Ming Lu, Advisor Professor Daniel Gauthier Professor Stuart Raby Professor Mohit Randeria Professor David Penneys, Graduate Faculty Representative <ul style="display: flex;"><li style="flex:1">c</li><li style="flex:1">ꢀ Copyright by </li></ul>Bowen Shi 2020 Abstract In this thesis, we present a theoretical framework that can derive a general anyon theory for 2D gapped phases from an assumption on the entanglement entropy. We formulate 2D quantum states by assuming two entropic conditions on local regions, (a version of entanglement area law that we advocate). We introduce the information convex set, a set of locally indistinguishable density matrices naturally defined in our framework. We derive an isomorphism theorem and structure theorems of the information convex sets by studying the internal self-consistency. This line of derivation makes extensive usage of information-theoretic tools, e.g., strong subadditivity and the properties of quantum many-body states with conditional independence. The following properties of the anyon theory are rigorously derived from this framework. We define the superselection sectors (i.e., anyon types) and their fusion rules according to the structure of information convex sets. Antiparticles are shown to be well-defined and unique. The fusion rules are shown to satisfy a set of consistency conditions. The quantum dimension of each anyon type is defined, and we derive the well-known formula of topological entanglement entropy. We further identify unitary string operators that create anyon pairs and study the circuit depth. We define the topological S-matrix and show it satisfies the Verlinde formula. It follows that the mutual braiding statistics of the sectors are nontrivial (they are anyons); moreover, the underlying anyon theory is modular. Three additional things, closely related to this framework, are presented: (1) The framework on a discrete lattice; (2) A calculation of information convex set based on solvable Hamiltonians; (3) A conjecture concerning the generality of our assumptions. ii To my mother, who brought me images iii Acknowledgments It takes unexpected good luck and a fruitful journey for me to meet this wonderful research project and participate in it. None of this work would have been possible without the help of three people: Stuart Raby, Yuan-Ming Lu, and Isaac Kim. Stuart was my research advisor for my first three years at OSU. I gained valuable experience in research during that time. As a person who seeks the best experience for students, Stuart encouraged me to look into possible connections to different sub-fields of physics that I was interested in. I got to know some connection between high-energy theory and quantum many-body physics at that time. I am also very grateful to Stuart’s genuine support during my transition of research direction. Yuan-Ming is an enthusiastic junior faculty who introduced me to the fast-developing ideas in the research field of quantum many-body physics. I feel lucky that Yuan-Ming had moved to OSU and brought the related research directions to our department. Ever since I knocked on his door, we have done a few exciting works together. We had a reading club on anyons, in which I learned the valuable background of anyon theory. Isaac is my friend and a wonderful collaborator. After I read his compact and thought-provoking original works, my mind filled up with questions. The attempts to answer some of these questions during the years had gradually brought me to the exciting research project that this thesis describes. I thank all the physics education I got at OSU. The knowledge and idea matter so does the way to communicate these ideas. Here are some memorable courses. The quantum mechanics course by Mohit Randeria has a clear focus on the physical principle and logical reasoning. The statistical mechanics course by Ciriyam Jayaprakash is always filled with intriguing physical examples and intuitive explanations. In Stuart’s elementary particle physics courses, he always iv welcomes all kinds of questions from the students, and he shows an amazing ability to answer these questions; he is very generous in his time for the course and other (reading) courses. Chris Hirata has an admirable ability to break down a complex phenomenon into much simpler ones. Samir Mathur is both knowledgeable and humble; he pays great attention to what a student may say. I thank Joseph McEwen for organizing the “geometry club” during my 1st and 2nd year. It was an informal graduate student meeting on any topic we find interesting. Most of the topics are related to geometry and topology. I shared this good experience with friends Alexander Davis, Noah Charles, Zaq Carson, Nicholas Mazzucca, Waylon Chen, and several others. These meetings sparkled with our curiosity for science, which makes it memorable. I attempted to work in several research areas. This experience gave me a valuable chance to meet and learn from people working on different research areas: cosmology, high energy physics, condensed matter physics, quantum information, and math. I thank Mingzhe Li and Xuejun Guo for some memorable instructions on research dating back to when I was at Nanjing University. I thank Eric Braaten, Chris Hirata, and Chris Hill to be my committee members during the time I was doing research on high energy physics. I thank Daniel Gauthier, Mohit Randeria, and Stuart Raby to be my committee members after I switched to condensed matter theory. I thank Junko Shigemitsu, Chris Hirata, Linda Carpenter, and Ulrich Heinz for everything I learned during the time I work with them as a grader. I thank Tin-Lun Ho, Nandini Trivedi, Ilya Gruzberg, and Brian Skinner for things I learned from them during various discussions. I am grateful to have learned many things about tensor categories from researchers in the math department at OSU, including Yilong Wang, David Penneys, Corey Jones, and Peter Huston. Among them, I first met Yilong during a conference at Indiana University Bloomington; I attended a course on quantum algebra taught by Dave; Corey is an energetic postdoc who can easily switch back and forth between math and physics terminology. In an occasional conversation with Peter, he encouraged me to talk about my work in the math department. v I have been fortunate to have the opportunity to travel to multiple places for short term summer (winter) schools and conferences. Furthermore, I thank Beni Yoshida and Timothy Hsieh for inviting me to the Perimeter Institute. I got to know many friends during these travels, including: Meng Hua, Shuoguang Liu, Rui-Xing Zhang, Isaac Kim, Xueda Wen, Liujun Zou, Zhehao Dai, Hassan Shapourian, Byungmin Kang, Shudan Zhong, Aaron Szasz, Lukasz Fidkowski, Junyi Zhang, Lauren McGough, Huan He, Jie Wang, Yunqin Zheng, Kohtaro Kato, Xie Chen, Tian Zhang, Kevin Slagle, Sharmistha Sahoo, Xiao Chen, Alex Thomson, Shubhayu Chatterjee, Zhen Bi, Yingfei Gu, Wenbo Fu, Akhil Sheoran, Zhu-Xi Luo, Hao-Yu Sun, ZiWen Liu, Sam Roberts, Dominic Williamson. Your presence enriched my travel experience. Especially, I was fortunate to have met Kohtaro Kato in a conference at KITP, during which he taught me how to merge quantum Markov states. Later on, this merging technique turned out to be crucial in the derivations in “fusion rules from entanglement”, a work we collaborated with Isaac. Finally, I wish to thank my fellow physics graduate students and postdocs for everything I learned from them as well as their accompanying in various weekly events/activities related to physics during different stages of my career as a graduate student, including Yunlong Zheng, Yan Yan, Yiming Pan, Qing Wang, Yifei He, Archana Anandakrishnan, B. Charles Bryant, Zijie Poh, Xiao Fang, Shaun Hampton, Bin Guo, Liping He, Lipei Du, Hong Zhang, Bei Zhou, James Rowland, Jiaxin Wu, Cheng Li, Fuyan Lu, Biao Huang, David Ronquillo, Tamaghna Hazra, Wenjuan Zhang, X. Y. Yin, Saad Khalid, Yanjun He, Xiaozhou Feng, Shuangyuan Lu, ChangYan Wang, Mohammed Karaki, Yonas Getachew, Joseph Szabo, Xin Dai, Alex Rasmussen, Wayne Zheng and many others too numerous to mention. Being able to discuss with friends makes physics even more fun. vi Vita 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Physics, Nanjing University Publications Research Publications 7. B. Shi, Verlinde formula from entanglement, [arXiv: 1911.01470] 6. B. Shi, K. Kato, and I. H. Kim, Fusion rules from entanglement, [arXiv: 1906.09376] 5. B. Shi, Seeing topological entanglement through the information convex, Phys. Rev. Re- search 1, 033048 (2019) [arXiv: 1810.01986] 4. B. Shi and Y.-M. Lu, Characterizing topological orders by the information convex, Phys. Rev. B 99, 035112 (2019) [arXiv: 1801.01519] 3. B. Shi and Y.-M. Lu, Deciphering the nonlocal entanglement entropy of fracton topological orders, Phys. Rev. B 97, 144106 (2018) [Editors’ suggestion] [arXiv: 1705.09300] 2. F. Lu, B. Shi, Y.-M. Lu, Classification and surface anomaly of glide symmetry protected topological phases in three dimensions, New J. Phys. 19, 073002 (2017) [arXiv: 1701.00784] 1. B. Shi and S. Raby, Basis invariant descriptions of chemical equilibrium with implications for a recent axionic leptogenesis model, Phys. Rev. D 92, 085008 (2015) [arXiv: 1507.08392] Fields of Study Major Field: Department of Physics vii Table of Contents Page <ul style="display: flex;"><li style="flex:1">ii </li><li style="flex:1">Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . </li></ul>Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii <ul style="display: flex;"><li style="flex:1">1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . </li><li style="flex:1">1</li></ul>1.1 Organization of the thesis: a road map and how to read . . . . . . . . . . . . 1.2 Physics of quantum many-body systems . . . . . . . . . . . . . . . . . . . . . 1.2.1 Quantum many-body systems . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Quantum Phases: its universal properties and classifications . . . . . . 1.2.3 Emergent laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Anyon theory, 2D gapped phases, and entanglement . . . . . . . . . . . . . . 366899 1.3.1 Algebraic theory of anyons and chiral central charge . . . . . . . . . . 11 1.3.2 Anyon theory and quantum entanglement . . . . . . . . . . . . . . . . 14 2. Entangled quantum states: background . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Quantum entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Quantum state and convex set . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Quantifying the distance between quantum states . . . . . . . . . . . . . . . . 22 2.4 Properties of the von Neumann entropy . . . . . . . . . . . . . . . . . . . . . 24 2.5 Quantum Markov states: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 viii 3. Formulating quantum states with an area law . . . . . . . . . . . . . . . . . . . . . 28 3.1 Setup and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Renormalization group fixed point . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Anyon data are the “order parameters” of topological orders . . . . . . . . . . 36 4. Information convex set and isomorphism theorem . . . . . . . . . . . . . . . . . . . 39 4.1 Information convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Elementary steps and Isomorphism Theorem . . . . . . . . . . . . . . . . . . . 44 4.3 Merging of elements in the information convex sets . . . . . . . . . . . . . . . 54 4.3.1 An alternative formulation of the information convex set . . . . . . . . 54 4.3.2 Merging in the alternative formulation . . . . . . . . . . . . . . . . . . 55 4.3.3 Equivalence of the definitions . . . . . . . . . . . . . . . . . . . . . . . 57 5. Fusion rules from entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Superselection sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2 Fusion rules and fusion spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Derivation of the axioms of the fusion rules . . . . . . . . . . . . . . . . . . . 68 5.4 Extreme points (details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4.1 Implication of the orthogonality . . . . . . . . . . . . . . . . . . . . . . 82 5.5 Fusion space (details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6. Topological entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.1 The derivation of TEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7. String operators and circuit depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.1 Heuristic discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 String operators from entanglement . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 Circuit depth of the string operators . . . . . . . . . . . . . . . . . . . . . . . 97 8. Verlinde formula from entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.1 Our definition of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.2 The proof of the Verlinde formula . . . . . . . . . . . . . . . . . . . . . . . . . 104 9. Discrete version of the framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 9.1 Setup and axioms on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.2 Reference state properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.3 The information convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 9.4 The isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 ix 9.5 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.6 Merging with a change of topology . . . . . . . . . . . . . . . . . . . . . . . . 117 9.7 Topological entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . 119 10. Information convex set and Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 121 10.1 Information convex set for frustration-free Hamiltonian . . . . . . . . . . . . . 122 10.2 Equivalence between Σ(Ω|σ) and Σ(Ω|H) under conditions . . . . . . . . . . . 125 10.3 Explicit calculation for quantum double models . . . . . . . . . . . . . . . . . 126 11. On the generality of area law: an RG point of view . . . . . . . . . . . . . . . . . . 132 11.1 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.2 Testing the conjecture with known examples . . . . . . . . . . . . . . . . . . . 133 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 xList of Tables <ul style="display: flex;"><li style="flex:1">Table </li><li style="flex:1">Page </li></ul>5.1 Physical data that can be extracted from disks with different number of holes. . 60 5.2 A partition of B used in the proof of Theorem 5.5. . . . . . . . . . . . . . . . . 84 xi List of Figures <ul style="display: flex;"><li style="flex:1">Figure </li><li style="flex:1">Page </li></ul>1.1 A road map to the thesis is illustrated. The boxes labeled by numbers are the <ul style="display: flex;"><li style="flex:1">chapters. The arrows indicate the relationships between these chapters. . . . . . </li><li style="flex:1">5</li></ul>2.1 Examples of convex sets: (a) A simplex; (b) A solid ball. . . . . . . . . . . . . . 21 3.1 The reference state σ of a 2D quantum many-body system. Some of the µ-disks are shown. The zoomed-in depiction of µ-disk b with partition BC (BCD) relevant to Axiom A0 (A1). Three relevant length scales ꢀ, r and ∆ are illustrated. All of them are larger than the correlation length and they can be much smaller than the system size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 An illustration of the growth procedure of a disk from AB to ABC. Here A can be large and BCD is contained in a µ-disk in a manner similar to Fig. 3.1. . . . 33 3.3 The extension of the axioms. A disk is divided into either BC or BCD. A µ-disk is on a smaller length scale, i.e., the small dashed circle surrounding the colored region. These figures represent three ways of enlarging C by a small step. (a) bc ⊆ B and d ⊆ C; (b) bc ⊆ B and d ⊆ C; (c) bc ⊆ B, d ⊆ C and d0 ⊆ D. . . . . 34 3.4 Subsystem E is a square of length scale ∆. F is an annulus around it. The thickness of F is at least ꢀ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1 This figure is a schematic depiction of regions involved in the definition of information convex set Σ(Ω|σ). Here Ω is the annulus between the black circles. Annulus Ω0 ⊇ Ω is a thickening of Ω. It is the region between the gray circles. 0˜ Any element in Σ(Ω |σ) is consistent with the reference state σ on every µ-disk b contained in Ω0. We choose Ω to be an annulus for illustration purposes. Other topologies are allowed as well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 A µ-disk can grow until it covers a larger disk ω. . . . . . . . . . . . . . . . . . . 43 xii 4.3 Smooth deformations of subsystems (ABC ↔ ABCD). The subsystem labels are chosen to make the (later) discussion of merging convenient. (a) A smooth deformation of a disk. (b) A smooth deformation of an annulus. The annulus topological is chosen for illustration purposes. Other topologies are allowed as well. 45 4.4 A schematic depiction of the merging process (Lemma 4.6). A pair Markov states ax ABCD <ul style="display: flex;"><li style="flex:1">ρaABC and λxBCD are merged into yet another quantum Markov state τ </li><li style="flex:1">. . . . 47 </li></ul>4.5 A schematic depiction of subsystem ABCD. The partition B0C0 = BC is chosen such that no µ-disk overlaps with both AB0 and CD. Note that, the subsystems A, B, C, D are allowed to take a variety of topologies. . . . . . . . . . . . . . . . 49 4.6 The detailed partition relevant to the elementary step of deformation Ω ↔ Ω0 is illustrated, where Ω = ABC and Ω0 = ABCD. BCD is a disk, and CD is contained in a µ-disk. Only a part of A is shown. The topology of A can be arbitrary. B0C0 = BC. A and D are separated by at least 2r + ꢀ, so that on µ-disk overlaps with both AB0 and CD. . . . . . . . . . . . . . . . . . . . . . . 51 ˆ 4.7 A partition of the subsystem Ω for defining Σ(Ω); see the second condition of Definition 4.3. Let Ω = ABC where BC is a subsystem contained in a µ-disk. The horizontal line is the boundary of Ω. Only part of A is shown for illustration purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ˆ 4.8 A partition of the subsystem Ω for defining Σ(Ω); see the third condition of Definition 4.3. Let Ω ⊇ ABC where BC is a subsystem contained in a µ-disk. The horizontal line is the boundary of Ω. Only part of A is shown for illustration purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1 (a) Division of an annulus X into three thinner annuli L, M, R. (b) A path (extensions-extensions-restrictions-restrictions) which generates an isomorphism <ul style="display: flex;"><li style="flex:1">∼</li><li style="flex:1">∼</li></ul><ul style="display: flex;"><li style="flex:1">Σ(L) </li><li style="flex:1">Σ(X) </li><li style="flex:1">Σ(R). (c) A schematic depiction of the simplex structure of </li><li style="flex:1">=</li><li style="flex:1">=</li></ul>Σ(X). The extreme points are the “corners” of the simplex. If the annulus X is contained in a disk, then one of the extreme points has the vacuum label “1”. . 61 5.2 A pair of annuli X0 and X1 on a torus. They cannot be connected by any smooth path because X0 is contractible and X1 is non-contractible. . . . . . . . . . . . 63 <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">1</li></ul><ul style="display: flex;"><li style="flex:1">˜</li><li style="flex:1">˜</li></ul>5.3 Both C and BC are disks. B is an annulus. X , X , X and X are annuli. Note

Anyon Theory in Gapped Many-Body Systems from Entanglement

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support