Anyon condensation Topological symmetry breaking phase transitions and commutative algebra objects in braided tensor categories Author: Supervisors: Sebas Eli¨ens Prof. Dr. F.A. Bais [email protected] Instituut voor Theoretische Fysica Prof. Dr. E.M. Opdam Korteweg de Vries Instituut Master’s Thesis Theoretical Physics and Mathematical Physics Universiteit van Amsterdam 31 December 2010 ii Abstract This work has two aspects. We discuss topological order for planar systems and explore a graphical formalism to treat topological symmetry breaking phase transition. In the discussion of topological order, we focus on the role of quantum groups and modular tensor categories. Especially the graph- ical formalism based on category theory is treated extensively. We have incorporated topological symmetry breaking phase transitions, induced by a bosonic condensate, in this formalism. This allows us to calculate general operators for the broken phase using the data for the original phase. As an application, we show how to treat topological symmetry breaking on the level of the topological S-matrix and illustrate this in two representative examples from the su(2) series. This approach can be viewed as an ap- k plication of the theory of commutative algebra objects in modular tensor categories. iv Sometimes a scream is better than a thesis Manfred Eigen ii Contents 1 Introduction 1 2 Topological order in the plane 5 2.1 Anyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The fractional quantum Hall effect . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Conformal and topological quantum field theory . . . . . . . . . 11 2.3 Other approaches to topological order . . . . . . . . . . . . . . . . . . . 13 3 Quantum groups in planar physics 15 3.1 Discrete Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Multi-particle states . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Topological interactions . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.3 Anti-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Chern-Simons theory and U [su(2)] . . . . . . . . . . . . . . . . . . . . . 24 q 4 Anyons and tensor categories 27 4.1 Fusing and splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.1 Fusion multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.2 Diagrams and F-symbols . . . . . . . . . . . . . . . . . . . . . . 29 4.1.3 Gauge freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1.4 Tensor product and quantum trace . . . . . . . . . . . . . . . . . 35 4.1.5 Topological Hilbert space . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Pentagon and Hexagon relations . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4.1 Verlinde formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 States and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 iii CONTENTS 5 Examples of anyon models 49 5.1 Fibonacci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 Quantum double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.1 Fusion rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.2 Computing the F-symbols . . . . . . . . . . . . . . . . . . . . . . 53 5.2.3 Braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 The su(2) theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 k 6 Bose condensation in topologically ordered phases 61 6.1 Topological symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 62 6.1.1 Particle spectrum and fusion rules of T . . . . . . . . . . . . . . . 63 6.1.2 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.2 Commutative algebra objects as Bose condensates . . . . . . . . . . . . 68 6.2.1 The condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2.2 The particle spectrum of T . . . . . . . . . . . . . . . . . . . . . 74 6.2.3 Derived vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.4 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.3 Breaking su(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 10 7 Indicators for topological order 87 7.1 Topological entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 88 7.2 Topological S-matrix as an order parameter . . . . . . . . . . . . . . . . 91 8 Conclusion and outlook 97 A Quasi-triangular Hopf algebras 99 B Tensor categories: from math to physics 103 B.1 Objects, morphisms and functors . . . . . . . . . . . . . . . . . . . . . . 103 B.2 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.3 Tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3.1 Unit object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.3.2 Strictness and coherence . . . . . . . . . . . . . . . . . . . . . . . 107 B.3.3 Fusion rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.3.4 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.4 Duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.5 Braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.5.1 Ribbon structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.6 Categories for anyon models . . . . . . . . . . . . . . . . . . . . . . . . . 113 C Data for su(2) 115 10 References 117 Dankwoord 123 iv 1 CHAPTER Introduction The theory of phases and phase transitions plays a central role in our understanding of many physical systems. Topological phases, or topologically ordered phases, pose an interesting problem for theorists in this respect, as they fall outside of the conventional scheme to understand phases that occur in terms of the breaking of symmetries. The formalism of topological symmetry breaking, based on the breaking of an underlying quantum group symmetry, can be viewed as an extension of this theory to the context of topological phases. We will discuss topological symmetry breaking and connect it to the notion of commutative algebras in modular tensor categories. In particular, we put topological symmetry breaking in a graphical form, applying notions from tensor categories, which gives the freedom to calculate general operators for the phases under consideration. In condensed matter physics, a fundamental problem is the determination of the low-temperature phases or orders of a system. Dating back to Lev Landau [52,53], the theoryofsymmetrybreakingphasetransitionsformsacornerstoneinthisrespect. Put simply,therearetwoaspectstothepictureitprovides: symmetrybreakingandparticle condensation. This mechanism underlies interesting phenomena, such as superconduc- tivity – where the electric U(1) symmetry is broken by a condensate of Cooper pairs – but also the formation of ice (figure 1.1). Group theory provides the right language to discussmanyaspectsofsymmetrybreaking. Theclassificationofthedifferentphasesis essentially equivalent to the classification of subgroups of the relevant symmetry group. In the past few decades, a growing interest has emerged to study topological phases that are not the product of (group) symmetry breaking. Especially in two dimensions, topological phases offer access to fundamentally new physics. They can have anyonic quasi-particle excitations – particles that are neither bosons nor fermions. These offer a route to the fault-tolerant storage and manipulation of quantum information known as topological quantum computation (TQC), which may some day be used to realize 1 1. INTRODUCTION Figure 1.1: The structure of an ice crystal - If water freezes, or any other liquid- to-solid transition, the molecules order in a regular lattice structure. This breaks the translational and rotational symmetry of the fluid state. Using group theory, one can see that there are 230 qualitatively different types of crystals corresponding to the 230 space groups the dream of building a quantum computer. The best known physical realization of topological phases occurs in the fractional quantum Hall fluids. These exotic states in two-dimensional electron gases submitted to a strong perpendicular magnetic field have quantum numbers that are conserved due to topological properties, not because of symmetry. Similar phases might occur in rotating Bose gases, high T superconductors, and possibly many more systems. One c caninfactshowthatthereisaninfinitenumberofdifferenttopologicalphasespossible, whichsuggestsaworldofpossibilitiesifweevergainenoughcontroltoengineersystems that realize a phase of choice. From a mathematical point of view, interesting structures have entered the the- ory. The description and study of topological phases involve conformal field theory, topological quantum field theory, quantum groups and tensor categories, and all these structures are heavily interlinked. They are studied by mathematicians and theoretical physicists alike and link topics like string theory, low-dimensional topology and knot theory to condensed matter physics. In this thesis we discuss topological ordered planar systems, and in particular the description using quantum groups and how these lead to tensor categories. In fact, we prefer the tensor category viewpoint from which the theory can be neatly summarized by a set of F-symbols and R-symbols. We show how to calculate these in examples related to discrete gauge theories and Chern-Simons theory. Our main goal is to discuss how the breaking of quantum group symmetry can be understood from the perspective of tensor categories. In the topological symmetry breaking scheme, a key role is played by the formation of a bosonic condensate. In this thesis, we argue that this notion corresponds to a commutative algebra object in braided tensor categories. Using the formulation of topological symmetry breaking in terms of tensor cate- gories, one can write down and calculate diagrammatic expressions for operators in the broken phase. This allows us to describe phase transition on the level of the topological S-matrix, an important invariant for the theory. We illustrate this in two representa- tive examples. This shows the usefulness of the topological S-matrix, related to the 2