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Interplay Between Symmetry and Topological Order in Quantum Spin Systems Interplay between Symmetry and Topological Order in Quantum Spin Systems by Hao Song B.S., Nanjing University, 2009 M.S., University of Colorado Boulder, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 2015 This thesis entitled: Interplay between Symmetry and Topological Order in Quantum Spin Systems written by Hao Song has been approved for the Department of Physics Prof. Michael Hermele Prof. Victor Gurarie Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Song, Hao (Ph.D., Physics) Interplay between Symmetry and Topological Order in Quantum Spin Systems Thesis directed by Prof. Michael Hermele In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general. Dedication To my parents. v Acknowledgements First and foremost, I am sincerely grateful to my advisor, Prof. Michael Hermele, for his gen- erous financial support, insightful guidance and warm encouragement during my graduate studies. He led me to the exciting field studying topological phases of quantum matter and has been making a lot of stimulating suggestions to my research projects. Without him, this thesis would not have been possible. Besides, Prof. Hermele has also provided a lot of professional help and advices on writing paper, preparing presentations and finding post-doctoral positions. Next, I want to thank my friends in the department of physics, Dr. Gang Chen, Dr. Andrew Essin, Sheng-Jie Huang, Yi-Ping Huang, Han Ma, Abhinav Prem, Zhaochuan Shen and Xiao Yin, for useful discussions and comments. Particular thanks are due to Sheng-Jie Huang for our ongoing inspiring collaborations. In addition, it is a pleasure to acknowledge the help of Dr. Hongcheng Ni on fixing a technical LaTex issue during my thesis writing. Moreover, I would like to thank Prof. Alexander Gorokhovsky, Prof. Victor Gurarie Prof. Leo Radzihovsky, Prof. Ana Maria Rey and Prof. Jonathan Wise for being in my committee of compre- hensive exam and defense. I would especially appreciate Prof. Gurarie who proofread this thesis as well. Finally, I want to express my deep gratitude to my parents who always support me no matter how far away I am. Also, I am grateful to all my friends that I have made in Boulder. Thank you all, sincerely. vi Contents Chapter 1 Introduction 1 1.1 Chiral Spin Liquids in Cold Atoms . .1 1.2 Topological Phases and Symmetries . .5 1.3 Anomalous Symmetry Fractionalizations and SPT Phases . .8 2 Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms in Three Dimensions 10 2.1 Theoretical Model . 11 2.2 Large-N Ground States . 14 2.2.1 Summary of the large-N mean-field results . 14 2.2.2 Detailed descriptions of the mean-field ground states . 16 2.2.3 Obtaining the mean-field results . 18 2.2.4 Relation between bilayer states and square lattice saddle points . 21 2.3 Discussion . 26 3 Crystal Symmetry Fractionalization 28 3.1 Outline and Main Results . 29 3.2 Review of Z2 Topological Order . 31 3.3 Review: toric code model on the square lattice . 34 3.4 Toric Codes on General Two-dimensional Lattices with Space Group Symmetry . 37 3.5 Fractionalization and Symmetry Classes . 44 vii 3.5.1 Review of fractionalization and symmetry classes . 44 3.5.2 Fractionalization and symmetry classes in the solvable models . 46 3.6 Symmetry Classes Realized by Toric Code Models . 49 3.6.1 Model realizing all e particle fractionalization classes . 50 3.6.2 Toric code models without spin-orbit coupling . 53 3.6.3 General toric code models . 58 3.7 Summary and Beyond Toric Code Models . 61 4 Topological Phases Protected by Reflections and Anomalous Crystalline Symmetry Frac- tionalization 65 4.1 Reflection Symmetry Protected Topological Phyases . 66 4.1.1 With a Single Reflection Symmetry . 66 4.1.2 With Two Orthogonal Reflection Symmetries . 68 4.2 Boundary Degrees of Freedom of SPT Phases . 69 4.2.1 Dimer model for 1d SPT . 69 4.2.2 CZX model for 2d SPT with onsite Z2 symmetry . 70 4.3 Model of Fractionalization Anomaly: a Single Reflection (ePmP) . 72 4.3.1 Spectrum of the model . 74 4.3.2 Symmetry behaviour of anyons . 77 4.4 Model of Fractionalization Anomaly: Two Orthogonal Reflections . 77 4.5 All Fractionalization Anomalies Associated with the Square Lattice Symmetries . 81 5 Summary and Perspective 83 5.1 Summary . 83 5.2 Perspetive . 84 viii Bibliography 86 Appendix A Complete set of commuting observables 92 B Symmetry-invariant ground states 94 C General construction of e and m localizations in toric code models 99 C.1 General constraints on symmetry classes in toric code models . 101 C.1.1 Toric codes without spin-orbital coupling . 101 C.1.2 Toric codes with spin-orbit coupling . 106 D Models in TC (G) 119 E A model of eTmT 125 ix Tables Table 2.1 Ground state saddle-point patterns of χrr0 , and the corresponding energies in units of NJ Ns for k = 2; 3;:::; 10. The different types of large-N ground states are described in the text, and depicted in figures as indicated. 16 2.2 This table contains information about our SCM numerical study on the cubic lattice (1st column), as well as the related problems of a single bilayer (2nd column), and single layer square lattice with k0 = k=2 (3rd column). On the left-hand side of each entry of the table, the range of unit cell dimensions is shown as an inequality. For every choice of lx;y;z within the given range, the number of times we ran the SCM algorithm with distinct random initial configurations of χrr0 is shown on the right-hand side of the entry (top). Also on the right-hand side is the minimum linear system size L (bottom, italics). 24 x 2.3 Comparison of energies of a variety of simple saddle points (top four rows), with the energy of the ground state found by SCM numerics (bottom row). All energies are in units of NJ Ns. Each row represents a class of saddle points, described below. For classes including multiple different saddle points, the energy shown is the lowest in the class. We considered the following classes of saddle points: Bilayer (Φ = 2πn=k). We considered a generalization of the CSL bilayer saddle point described in the main text, where the flux through each plaquette is Φ = 2πn=k, where n = 0; : : : ; k − 1. k-site cluster. The energy of a cluster with k sites is proportional to the number of bonds in the cluster,[1, 2] so the lowest-energy such state can be found by finding a k-site cluster containing the greatest number of bonds. Uniform real χ. This is the state where χrr0 is real and spatially constant. (2πnx;y;z=k)-flux. These states have 2πnx=k flux through every plaquette normal to the x-direction, and similarly for y and z, where 0 ≤ nx;y;z ≤ k − 1. Since most of these states break lattice rotation symmetry, the magnitude jχrr0 j is allowed to vary depending on bond orientation, but is fixed to be translation invariant.[3] . 25 3.1 Notation used in the chapter. 32 xi Figures Figure 2.1 Ground-state clusters for k = 2; 3; 4. Shaded bonds are those with χrr0 6= 0. Bonds with different shading (or color in online version) may have different magnitudes jχrr0 j. (a) The k = 2 ground state clusters are dimers and square plaquettes. The square plaquette is pierced by π-flux, and the ratio of jχrr0 j on light (pink online) and dark (blue online) bonds can be chosen arbitrarily. Setting jχrr0 j = 0 on the two light (pink) bonds breaks the plaquette into two dimers. (b) The k = 3 ground state cluster is a 6-site chain pierced by π-flux. On the cubic lattice, such chains can exist either as a flat rectangular loop (left), or as the same loop bent by 90◦ in the middle (right).
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