Numerical Studies of Topological phases Thesis by Scott Geraedts In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2015 (Defended May 27, 2015) ii c 2015 Scott Geraedts All Rights Reserved iii Abstract The topological phases of matter have been a major part of condensed matter physics research since the discovery of the quantum Hall effect in the 1980s. Recently, much of this research has focused on the study of systems of free fermions, such as the integer quantumHall effect, quantum spin Hall effect, and topological insulator. Though these free fermion systems can play host to a variety of interesting phenomena, the physics of interacting topological phases is even richer. Unfortunately, there is a shortage of theoretical tools that can be used to approach interacting problems. In this thesis I will discuss progress in using two different numerical techniques to study topological phases. Recently much research in topological phases has focused on phases made up of bosons. Unlike fermions, free bosons form a condensate and so interactions are vital if the bosons are to realize a topological phase. Since these phases are difficult to study, much of our understandingcomes from exactly solvable models, such as Kitaev’s toric code, as well as Levin-Wen and Walker-Wang models. We may want to study systems for which such exactly solvable models are not available. In this thesis I present a series of models which are not solvable exactly, but which can be studied in sign-free Monte Carlo simulations. The models work by binding charges to point topological defects. They can be used to realize bosonic interacting versions of the quantum Hall effect in 2D and topological insulator in 3D. Effective field theories of “integer” (non-fractionalized) versions of these phases were available in the literature, but our models also allow for the construction of fractional phases. We can measure a number of properties of the bulk and surface of these phases. Few interacting topological phases have been realized experimentally, but there is one very important exception: the fractional quantum Hall effect (FQHE). Though the fractional quantum Hall effect we discov- ered over 30 years ago, it can still produce novel phenomena. Of much recent interest is the existence of non-Abelian anyons in FQHE systems. Though it is possible to construct wave functions that realize such particles, whether these wavefunctions are the ground state is a difficult quantitative question that must be answered numerically. In this thesis I describe progress using a density-matrix renormalization group algo- rithm to study a bilayer system thought to host non-Abelian anyons. We find phase diagrams in terms of experimentally relevant parameters, and also find evidence for a non-Abelian phase known as the ‘interlayer Pfaffian’. iv Acknowledgements The success of my PhD relied on support from a whole bunch of people. Especially I’d like to thank my office mate and friend Paraj Titum. I’d also like to thank my advisor, Lesik Motrunich, and also Roger Mong. Finally I’d like to thank my parents for their constant support. v Contents Abstract iii 1 Introduction 1 1.1 Introduction to Topological Phases ............................... 1 1.1.1 Short-ranged Entangled Topological Phases ...................... 3 1.1.2 Long-ranged entanglement ............................... 3 1.2 Tractable Models of Interacting Topological Phases ...................... 4 1.2.1 Integer and Fractional Quantum Hall Effect for Bosons ................ 5 1.2.2 Topological Insulator of Bosons ............................ 8 1.3 Topological Quantum Computing and Numerical Studies of the Fractional Quantum Hall Effect 11 1.4 Overview of All Projects and Their Relationships ....................... 14 2 Methods for studying two species of bosons with mutual statistical interactions 17 2.1 One species of bosons ...................................... 17 2.1.1 Trotter Decomposition ................................. 17 2.1.2 Reformulations in terms of φ variables ......................... 19 2.1.3 Reformulations in terms of vortices .......................... 20 2.2 Two species of Bosons: Modular Transformations ....................... 23 2.3 Monte Carlo Techniques .................................... 26 3 Numerical Study of the Boson Integer and Fractional Hall Effects 28 3.1 Introduction ........................................... 28 3.2 Explicit Models with Integer and Fractional Quantum Hall Effect ............... 29 3.3 Monte Carlo Study of a Model with a Boundary ........................ 32 3.4 Numerical Evidence for Gapless Edge ............................. 35 3.5 Phase diagrams ......................................... 41 12 3.5.1 Modelswith c/d = n, σxy =2n ............................ 43 3.5.2 Modelswith d =1 ................................... 45 6 3.6 Hamiltonian formulation .................................... 47 vi 3.7 Discussion ............................................ 52 4 Numerical Study of Bosonic Topological Insulator and Fractional Topological Insulator 54 4.1 Introduction ........................................... 54 4.2 Realizing the topological insulator by binding bosons to hedgehogs of SO(3) spins ..... 56 4.2.1 Model and its Bulk Phase Diagram .......................... 57 4.2.1.1 Importance of Discrete Symmetry ...................... 60 4.2.1.2 Binding of Multiple Bosons to a Hedgehog ................. 62 4.2.2 Phase Diagram on the Boundary Between the Binding Phase and a Trivial Insulator . 63 4.2.3 Surface with Zeeman Field ............................... 66 4.3 Realizing the topological insulator by binding bosons to hedgehogs of an easy-plane CP 1 model .............................................. 68 4.3.1 Bulk Phase Diagram .................................. 69 4.3.1.1 Symmetries When The Spins Are Represented By An Easy-plane CP 1 Model .................................... 71 4.3.2 Observation of a Witten effect ............................. 73 4.3.3 Surface Phase Diagram ................................. 79 4.3.4 Symmetric Surface Phase with Topological Order ................... 83 4.3.5 Time-Reversal Breaking and Hall Effect on the Surface ................ 85 4.4 Realizing symmetry-enriched topological phases by binding multiple hedgehogs to a boson . 87 4.5 Discussion and Conclusions .................................. 91 5 DMRG study of a quantum Hall bilayer with filling ν =1/3+1/3 94 5.1 Introduction ........................................... 94 5.2 Experimental Background ................................... 97 5.3 Density Matrix Renormalization Group Applied to Quantum Hall systems .......... 98 5.3.1 Reviewof DMRG ................................... 99 5.3.2 DMRG applied to quantum Hall systems ....................... 101 5.3.3 The bilayer model ................................... 103 5.3.4 Entanglement invariants for the identification of FQH phases ............. 104 5.4 Abelian Phase Diagram ..................................... 105 5.4.1 Determination of the phases .............................. 106 5.4.2 Order of the transitions ................................. 108 5.4.3 Spin polarization .................................... 113 5.5 Non-Abelian phase ....................................... 113 5.5.1 The interlayer-Pfaffian state .............................. 116 5.5.2 Exact-diagonalization overlaps ............................. 117 vii 5.5.3 Spin-charge separation ................................. 119 5.5.4 Non-Abelian signatures ................................ 122 5.6 Conclusion ........................................... 125 6 Conclusions and Future Directions 127 Bibliography 129 1 Chapter 1 Introduction During the course of my graduate research, I have worked on two sets of projects. In the first, I developed a set of analytical and numerical tools for studying a certain class of statistical mechanics models. These models potentially have a number of applications, but the application I focused on was topological phases of bosons. In the end I was able to produce numerically tractable models of topological phases in one, two and three dimensions. This series of projects occupied the majority of my time during my degree. More recently, I have been working on projects related to the numerical study of the fractional quantum Hall effect. What these two sets of projects have in common is that they are both studies of topological phases of matter. Therefore this thesis begins with a discussion and motivation for the study of topological phases. 1.1 Introduction to Topological Phases A central concept is condensed matter physics is that of phase transitions, which are locations in parameter space where the free energy is not analytic. From this we can arrive at the definition of a ‘phase of matter’: two states of a material are in the same phase if there is a path in parameter space that connects them and that does not cross a phase transition or break a symmetry. It has long been known that different phases of matter can have qualitatively different properties. This understandingwas put on a firmer footing by Landau[1], who pointed out that these qualitatively different properties arise because different phases have different symme- tries. One example is a paramagnetic phase with full spin-rotation symmetry that can have a ferromagnetic phase with less symmetry, for example the ferromagnetic phase my have spin rotation symmetry about only one axis. Liquids have continuous translational