Topological Phases, Entanglement and Boson Condensation Huan He 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 Adviser: Bogdan Andrei Bernevig June 2019 c Copyright by Huan He, 2019. All rights reserved. Abstract This dissertation investigates the boson condensation of topological phases and the entanglement entropies of exactly solvable models. First, the bosons in a \parent" (2+1)D topological phase can be condensed to obtain a \child" topological phase. We prove that the boson condensation formalism necessarily has a pair of modular matrix conditions: the modular matrices of the parent and the child topological phases are connected by an integer matrix. These two modular matrix conditions serve as a numerical tool to search for all possible boson condensation transitions from the parent topological phase, and predict the child topological phases. As applications of the modular matrix conditions, (1) we recover the Kitaev's 16-fold way, which classifies 16 different chiral superconductors in (2+1)D; (2) we prove that in any layers of topological theories SO(3)k with odd k, there do not exist condensable bosons. Second, an Abelian boson is always condensable. The condensation formalism in this scenario can be easily implemented by introducing higher form gauge symmetry. As an application in (2+1)D, the higher form gauge symmetry formalism recovers the same results of previous studies: bosons and only bosons can be condensed in an Abelian topological phase, and the deconfined particles braid trivially with the condensed bosons while the confined ones braid nontrivially. We emphasize again that the there exist non-Abelian bosons that cannot be condensed. Third, the ground states of stabilizer codes can be written as tensor network states. The entanglement entropy of such tensor network states can be calculated exactly. The 3D fracton models, as exotic stabilizer codes, are known to have several features which exceed the 3D topological phases: (i) the ground state degeneracy generally increase with the system size; (ii) the gapped excitations are immobile or only mobile in certain sub-manifolds. In our work, we calculate, for the first time, the entanglement entropy for the fracton models, and show that the entanglement iii entropy has a topological term linear to the subregions' sizes, whereas the topological phases only have constant topological entanglement entropies. iv Acknowledgements First and foremost, I would like to thank my advisor Prof. B. Andrei Bernevig for guidance and support. I have benefited enormously from his curiosity, dedication to physics and perseverance. He provides me with a complete freedom to work on the problems and topics that I am interested in, and more importantly, he never allows me to give up easily until we all find a satisfying answer. It is fair to say that rigorous and solid research is the trademark of our group, which I now completely agree is truly valuable. I will bear this spirit in mind for the rest of my career. During the journey, I am indebted to the help from many professors and friends who make me a better physicist. Among all of them, I owe the most to Prof Nicolas Regnault, Prof German Sierra and Yunqin Zheng. Many papers were not possible without their help and efforts. In fact, Nicolas and German are the two aspects of modern physicists. Nicolas's numerical technologies are unbelievable. I still remember how I was amazed when he finished a new program and computation before I finished speaking my speculations. It turned out that I was wrong and he was correct. I am lucky to meet and learn from German. Every time I discussed with German, I would enjoy the aesthetic of his physics and formulas, which I have almost forgotten over the years of research. His formulas reminded me of my initial motivation of becoming a physicist: the simple and beautiful laws of nature. Yunqin has been my major collaborator since the end of my first year in Princeton. In fact, we have known each other since undergraduate (10 years ago from now). I would never forget all the time we spent together with chalks on blackboards, pens on papers, and Mathematica on screens. The PCTS program and our department certainly bring in brilliant and optimistic young physicists globally. I am grateful for the collaborations with the postdoctoral researchers in Princeton: Prof Curt von Keyserlingk, Prof Titus Neupert, Prof Barry Bradlyn, Prof Jennifer Cano, Prof Yi Li, Dr Abhinav Prem, Dr Lian Biao, Dr Yizhi v You and Dr Kiryl Pakrouski. I hope that they also enjoy discussing and working with me. It was joyful and also surprisingly efficient to work with Prof Jia-Wei Mei and Dr. Ji-Yao Chen. A powerful tensor network MATLAB program was delivered in only a few weeks over the summer. I need to thank Xue-Da Wen, Apoorv Tiwari, Peng Ye for analytically calculating entanglement entropies, and being reliable and helpful collaborators for this project. Particularly, Xue-Da is the person who I would like to discuss all the time because of his enthusiastic passion. It would be my fault if I did not mention my group members, who I have in- teractions with, for many enlightening and refreshing conversations and discussions: Yang-Le Wu, Aris Alexandradinata, Sanjay Moudgalya, Zhi-Jun Wang, Jian Li, Zhi- Da Song, Fang Xie. I also would like to thank the physicists that I met during confer- ences, visit, emails and Skype: Rui-Xing Zhang, Chen Fang, Hong Yao, Yuan-Ming Lu, Meng Cheng, Yang Qi, Yuan Wan, Xie Chen, Giuseppe Carleo, Alan Morningstar, Yi-Chen Huang, Juven Wang, Yidun Wan, Heidar Moradi, Wen Wei Ho, Tian Lan and many others. My life would be miserable if without my friends in Princeton: Jing-Yu Luo, Jie Wang, Zheng Ma, Jing-Jing Lin, Yu Shen, Wu-Di Wang, Zhen-Bin Yang, Jun Xiong, Liang-Sheng Zhang, Bin Xu, Zhao-Qi Leng, Tong Gao, Suerfu, Yin-Yu Liu, Jia-Qi Jiang, Si-Hang Liang, Xin-Ran Li, Xiao-Wen Chen, Bo Zhao, Jun-Yi Zhang, Xue Song, Xin-Wei Yu, Hao Zheng, Zi-Ming Ji, Ho Tat Lam, Wen-Li Zhao, Po-Shen Hsin, Xiao-Jie Shi, Chaney Lin and many many others. I appreciate Prof Duncan Haldane, Prof Waseem Bakr and Prof Silviu Pufu for be- ing my committee members, and Prof Robert Austin for leading me the experimental project. vi Last but not the least, I appreciate the support from my parents during all these years. And most importantly, it is beyond my language to describe my gratitude for many years' accompany with my love, Yina. This dissertation is dedicated to you. vii To my love Yina viii Contents Abstract..................................... iii Acknowledgements...............................v List of Tables.................................. xiii List of Figures.................................. xiv 1 Preliminaries1 1.1 Stabilizer Codes..............................2 1.2 Entanglement Entropy and Tensor Network States...........4 1.3 Abelian Topological Field Theory....................7 1.4 Modular Tensor Category........................9 2 An Example: Toric Code Model 14 2.1 Stabilizer Code.............................. 14 2.2 Tensor Network State........................... 16 2.3 Modular Tensor Category........................ 18 2.4 BF Field Theory............................. 20 3 Boson Condensation 23 3.1 Definitions and Assumptions....................... 26 3.2 The Condensation Matrix Mab ...................... 29 3.2.1 Proof that M commutes with T matrix of the A theory.... 31 3.2.2 Proof that M commutes with S matrix of the A theory.... 31 ix 3.3 The Modular Tensor Category after condensation........... 33 3.4 Simple currents.............................. 36 3.4.1 Introduction to simple currents................. 36 3.4.2 Simple current condensation................... 38 3.5 One confined particle theories...................... 43 3.6 Formalism and implementation..................... 44 3.6.1 Solutions for M .......................... 45 3.6.2 Solutions for n .......................... 48 3.6.3 The modular matrices of the new theory............ 51 3.7 Layer constructions and uncondensable bosons............. 54 3.7.1 Theories with Zm-graded condensations............. 55 3.7.2 Theories with Z-fold way: Fibonacci TQFT.......... 60 3.8 Conclusions................................ 63 4 No-Go Theorem for Boson Condensation in Topologically Ordered Quantum Liquids 65 4.1 First No-Go Theorem........................... 66 4.2 Example (i): Multiple layers of Fibonacci................ 69 4.3 Example (ii): Single layer of SO(3)k ................... 71 4.4 Example (iii): Multiple layers of SO(3)k ................ 74 4.5 Summary................................. 74 5 Abelian Boson Condensation in Field Theory 76 5.1 Abelian Boson Condensation Formalism................ 77 5.2 Condensations in K-Matrix Chern-Simons Theories.......... 81 5.2.1 Condensable Condition...................... 82 5.2.2 Confinement/Deconfinement................... 84 x 6 Fracton Models, Tensor Network States and Their Entanglement Entropies 87 6.1 Stabilizer Code Tensor Network States................. 91 6.1.1 Notations............................. 91 6.1.2 Stabilizer Code and TNS Construction............. 94 6.1.3 TNS Norm............................ 98 6.1.4 Transfer Matrix.......................... 100 6.2 Entanglement properties of the stabilizer code TNS.......... 102 6.2.1 TNS as an exact SVD...................... 102 6.2.2 Summary of the results...................... 107 6.3 3D Toric Code............................... 109 6.3.1 Hamiltonian of 3D Toric Code Model.............. 109 6.3.2 TNS for 3D Toric Code...................... 111 6.3.3 Concatenation Lemma...................... 115 6.3.4 Entanglement........................... 117 6.3.5 Transfer Matrix as a Projector.................. 121 6.3.6 GSD and Transfer Matrix.................... 124 6.4 X-cube Model............................... 128 6.4.1 Hamiltonian of X-cube Model.................. 129 6.4.2 TNS for X-cube Model...................... 131 6.4.3 Concatenation Lemma...................... 134 6.4.4 Entanglement........................... 136 6.4.5 Transfer Matrix as a Projector.................. 141 6.4.6 GSD and Transfer Matrix...................