Chapter 1: Introduction and Ancient History
Abstract: How Lord Kelvin and Peter Tait tried to describe atoms with topology and created the field of knot theory.
Keywords: Lord Kelvin, Peter Tait, Knot Invariant, Quantum Topology, Vortex, Smoke Ring, Periodic Table
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Chapter 2: Kauffman Bracket Invariant and Relation to Physics
Abstract: We introduce the idea of a knot invariant using the Kauffman bracket as an example and explore a few key ideas of knot theory. We then make the relationship to physical topological quantum systems whose dynamics are determined by knot invariants. We show the relationship to quantum mechanics and briefly argue that quantum computation can be done with such topological quantum systems. We mention some of the experimental systems that are topological, particularly including fractional quantum Hall effect.
Keywords: Kauffman Bracket, Knot Invariant, Knot Theory, Spin-Statistics, Twist, Quantum Computation, Fractional Quantum Hall Effect, Isotopy
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Chapter 3: Particle Quantum Statistics
Abstract: We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the ``quantum dimension" of an anyon which describes the size of the many anyon Hilbert space.
Keywords: Path Integral, Exchange, Identical Particles, Braid Group, Permutation Group, Fundamental Group, Anyons, Abelian Anyons, Nonabelian Anyons, Quantum Computation, Quantum Dimension, Parastatistics.
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Chapter 4: Aharonov-Bohm Effect and Charge-Flux Composites
Abstract: We give a basic introduction to the Aharonov-Bohm effect, and use this effect to construct a toy-model of anyons known as the "charge-flux" model. We explore some of the basic properties of this model, including fusion of anyons, antiparticles, and the degeneracy of the ground state on a torus, which is related to the number of species of anyons. This degeneracy can in principle serve as a quantum memory.
Keywords: Aharonov-Bohm Effect, Charge-Flux Composite, Fusion, Antiparticles, Anti-anyons, Vacuum, Number of Species of Anyons, Degenerate Ground States, Torus, Quantum Memory,
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Chapter 5: Chern-Simons Theory Basics
Abstract: We give a brief introduction to Chern-Simons theory starting with Abelian Chern-Simons theory and making the connection to the charge-flux model. We discuss the vacuum partition function and the idea of a manifold invariant. We then move on to discuss Nonabelian Chern-Simons theory and the idea of Wilson loop operations, and Witten's famous connection between knot invariants and Chern-Simons theory. The Chern-Simons theory is a paradigm for topological quantum field theories or TQFTs.
Keywords: Chern-Simons Theory, Topological Quantum Field Theory, TQFT, Gauge Theory, Manifold Invariant, Partition Function, Abelian Chern-Simons Theory, Nonabelian Chern-Simons Theory, Wilson Loop, Knot Invariant, Gauge Transformation, Pontryagin Index, K-matrix Formalism.
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Chapter 6: Short Digression on Quantum Gravity
Abstract: A brief discussion of how and why Chern-Simons theory was pursued in the context of quantum gravity.
Keywords: Quantum Gravity, Chern-Simons Theory, Einstein-Hilbert Action, 2+1 Dimensional Gravity
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Chapter 7: Defining Topological Quantum Field Theory
Abstract: Atiyah systematized the general principles of topological quantum field theories in terms of the mathematics of gluing together manifolds and combining vector spaces. A TQFT roughly is a partition function Z(M) with M a manifold, that returns a number of M has no boundary and a wavefunction on the boundary if M has a boundary. Manifolds can be glued together on their boundaries by taking inner products of wavefunctions. We roughly describe this approach and describe how it relates to the physics of anyons as we have discussed in prior chapters. We also introduce the important notion of the modular S-matrix as a change of basis.
Keywords: Topological Quantum Field Theory, Atiyah Axioms, Partition Function, Manifold, Cobordism, Gluing, Modular S-matrix.
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Chapter 8: Anyon Basics
Abstract: In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.
Keywords: Hilbert Space, Fusion, Vacuum, Antiparticles, Nonabelian Fusion, Locality, Fibonacci Anyons, Ising Anyons, Fusion Multiplicity Matrix, N-matrices, Associativity, Dimension of Hilbert Space on 2-Manifolds.
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Chapter 9: Change of Basis and F-Matrices
Abstract: Continuing our exploration of the detailed structure of anyon theories, we explain how one can describe the same Hilbert space with different basis sets. Changing between these bases is done with the so-called F-matrix which is a gauge dependent quantity. The F-matrices satisfy a set of consistency equations known as the pentagon equations which are "rigid" up to Gauge transformations (i.e., solutions are discrete and cannot be deformed small amounts to produce other solutions). We again use Fibonacci and Ising anyons as simple examples.
Keywords: Change of Basis, F-matrix, Fibonacci Anyons, Ising Anyons, Pentagon Equations, Rigidity, Gauge Transformation
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Chapter 10: Exchanging Identical Particles
Abstract: Having established the structure of the Hilbert space of multiple anyons, we now discuss the effects of braiding the anyons around each other. We introduce the idea of an R-matrix and emphasize the importance of locality. Using R and F together we can fully describe any braiding of identical particles. We again use Fibonacci and Ising anyons as simple examples.
Keywords: R-matrix, F-matrix, Exchange, Braid, Locality, Ising Anyons, Fibonacci Anyons
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Chapter 11: Computing with Anyons
Abstract: We introduce some basic ideas of quantum computing including the idea of a quantum circuit model and the idea of a universal quantum computer --- implementing arbitrary gates arbitrarily accurately. We discuss how braiding of certain anyons can be used as a universal quantum computer, and we illustrate this using Fibonacci anyons as an example. We then turn to the task of designing braids to perform particular computations, which is known as Topological Quantum compiling.
Keywords: Quantum Computing, Topological Quantum Computing, Universal Quantum Computer, Quantum Circuit Model, Fibonacci Anyons, Kitaev-Solovay Algorithm.
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Chapter 12: Planar Diagrams
Abstract: We return to study planar diagram algebras as in chapters 8 and 9, but now more formally. We define diagrams as being operators that can be "stacked" on top of each other. We pay particular attention to properties of diagrams such as completeness of states and orthonormality of basis vectors, and we again review the use of F-matrices for change of basis. We discuss the implications of causality for these diagrams, and to what extent diagrams can be deformed in the plane. We give explicit detailed rules for evaluating planar diagrams.
Keywords: Fusion category, Planar Diagram Algebra, Operators, F-matrices, Isotopy, Causality, Completeness, Orthonormality.
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Chapter 13: Braiding Diagrams
Abstract: We continue the development of diagrammatics now considering diagrams living in 3 dimensions. We generalize the discussion of the R-matrix for braiding (non-identical) particles around each other, and discuss the Hexagon equations which provide a consistency condition for the R-matrices. As with the pentagon, Hexagon solutions are also rigid.
Keywords: Braiding, R-matrix, Hexagon Equations, Braided Fusion Category, Rigidity.
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Chapter 14: Seeking Isotopy
Abstract: The diagram algebras that we have developed are closely related to knot invariants. To make this connection we want the algebras to be "isotopy invariant", menaing that the diagrams can be freely deformed as long as strands are not cut. To achieve this we change the diagram normalization to use so-called ``Isotopy Normalization", where a particle loop is weighted by the quantum dimension. This almost achieves isotopy invariance of certain diagrams, except for a possible sign known as the Frobenius-Schur indicator. We carefully discuss this sign, its origin, physical meaning, and how one nonetheless achieves isotopy invariant diagrams. Having handled this sign, we achieve what essentially amounts to a knot invariant --- an isotopy invariant mapping from a labeled knot or link diagram to a complex number output. Even given this success it may still be the case that diagrams with branching may incur phases from deforming an up-branching to a down-branching. We briefly discuss other properties of unitary fusion categories including the spherical and pivotal properties.
Keywords: Isotopy, Isotopy Invariance, Isotopy Normalization, Loop Weight, Quantum Dimension, Frobenius-Schur Indicator, Fusion Category
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Chapter 15: Twists
Abstract: An important process (discussed already in chapter 2) is the process by which a particle twists around its own axis. For a quantum mechanical particle this accumulates a phase associated with the particle's spin. The particle spin is related in several ways to the R-matrix, and satisfies a number of important identities, including the ribbon identity.
Keywords: Twist, Twist Factor, Conformal Scaling Dimension, Spin, Topological Spin, R-matrix, Ribbon, Ribbon Identity, Spin-Statistics.
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Chapter 16: Nice Theories with Planar or Three-Dimensional Isotopy
Abstract: Many of the theories that one runs into most often are particularly simple in that they have a higher degree of symmetry than a generic TQFT or anyon theory needs to have. In this chapter we construct a simplified set of rules for evaluating diagrams for such simpler theories.
Keywords: Planar Diagrams, Isotopy, F-matrix, Tetrahedral Diagram, Braiding, Twist.
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Chapter 17: Further Structure
Abstract: TQFTs, or anyon theories, have a large amount of addition mathematical structure which we explore in this chapter. We show that the quantum dimension which defines the weight of a loop in the diagram algebra is the same as the quantum dimension that describes the size of the many-anyon Hilbert space. We define the S-matrix in terms of diagrams, and the T-matrix in terms of twists. We show how the S- and T-matrices are related to the modular group of diffeomorphisms of the torus, and related further to the notion of central charge. We discuss how the many constraints on TQFTs enables one to develop a table of all possible sufficiently ``small" TQFTs given certain very general conditions. We give tables for both modular and super-modular theories. We introduce the Kirby Color or Omega Strand as an important tool.
Keywords: Quantum Dimension, S-matrix, T-matrix, Modular Group, Modular, Super-Modular, Central Charge, Table of TQFTs, Kirby Color, Omega Strand.
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Chapter 18: Some Simple Examples
Abstract: We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called ``prime" theories.
Keywords: Fusion Ring, F-matrices, R-matrices, Unitary Fusion Category, Unitary Braided Fusion Category, Z2 fusion, Boson, Fermion, Semion, Z3 fusion, ZN fusion, Ising Fusion, Ising Anyons, Fibonacci Fusion, Fibonacci Anyons, Toric Code, Fermionic Toric Code, ZN Anyons, Prime Anyon Theory.
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Chapter 19: Anyons From Discrete Group Elements
Abstract: Given a discrete group, we consider planar diagrams where each edge is labeled with an element of the group, and vertices must obey the rule that multiplication of the edges incident on the vertex (in order) must yield the identity. The possible F-matrices for such a diagram are discussed and are known as 3-cocycles from the formalism of group cohomology. We discuss a simple example of building anyons from the ZN group. Noncommutative groups can be considered although they cannot describe a braided theory.
Keywords: Discrete Group, Group Cohomology, 3-cocyle, F-matrix, ZN anyons.
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Chapter 20: Bosons and Fermions from Group Representations
Abstract: We consider building planar diagram algebras by labeling edges with group representations and vertices follow the fusion rules of the group representations. We review using character tables to deduce fusion rules and quantum dimensions. The groups S3 and Q8 (quaternions) are used as simple examples. We show how F-matrices can be constructed using generalized Clebsch-Gordan coefficients. We deduce the possible braidings for such theories and conclude that only bosons or fermions are possible. We reconsider the possibility of parastatistics and argue why, given the structure of diagram algebras, only bosons and fermions are possible in 3+1 dimensions.
Keywords: Group Theory, Group Representation, Group Character, Fusion of Representations, Clebsch-Gordan, Parastatistics.
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Chapter 21: Quantum Groups (In Brief)
Abstract: The mathematical structure known as a Quantum Group can be thought of as a deformation of a classical Lie group. The (deformed) representation theory of quantum groups gives a natural way to generate data (F-matrices, and R-matrices) for anyon theories. In many cases, this approach gives data equivalent to Chern-Simons theory based on the corresponding Lie group.
Keywords: Quantum Group, Lie Group, Deformation, Representation Theory, q-Deformation, Deformed Representation Theory, Chern-Simons.
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Chapter 22: Temperly-Lieb Algebra and Jones-Kauffman Anyons
Abstract: Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.
Keywords: Loop Weight, Quantum Dimension, Jones-Wenzl Projector, Temperly-Lieb Algebra, Z2 Loop Gas, Ising Anyons, F-matrix, R-matrix.
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Chapter 23: State Sum TQFTs
Abstract: Manifolds can be decomposed into simplicies (triangles in 2D or tetrahedra in 3D). TQFTs can be constructed as state sums over discrete quantum numbers on the on these discretized manifolds. Such sums appear like partition functions, statistical mechanics sums of Boltzmann weights. In order to for these sums to yield manifold invariants the sum must be independent on the particular simplicial decomposition (or triangulation) of the manifold. The so-called "Pachner Moves" describe all possible changes of the decomposition, so a sum which is unchanged under Pachner moves gives a manifold invariant. One such state sum is the the Turaev-Viro state sum, which takes as an input a (spherical) fusion category (F matrices satisfying the pentagon equation) and then allows one to assign a (scalar) manifold invariant to a 3D manifold. The corresponding TQFT is known as the quantum double, or Drinfeld double of the fusion category. Much of the study of the Turaev-Viro model has been in the context of so-called spin-network models of quantum gravity. A very similar state sum TQFT is the Dijkgraaf-Witten model, which (in 3D) takes as an input a group and a 3-cocyle (acting as F-matrices). The Dijkgraaf-Witten model is generalizable to any dimension.
Keywords: Simplicial Decomposition, Partition Function, Pachner Move, Manifold Invariant, Turaev-Viro, Dijkgraaf-Witten, Spin Network, Quantum Gravity, Quantum Double, Drinfeld Double, F-matrix, 3-Cocycle.
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Chapter 24: Formal Construction of ``Chern-Simons'' TQFT: Surgery and More Complicated Manifolds
Abstract: We have constructed diagrammatic rules for evaluating labeled knots and links (and graphs) embedded in S3. However, a 3D TQFT should be able to give us information about any 3-manifold. In order to describe arbitrary 3-manifolds we use Dehn surgery --- a manifold is represented as a link embedded in S3. To obtain the manifold, the strands of the link are thickened to a solid torus which is excised from the manifold and replaced with longitude and meridian exchanged. The Lickorish-Wallace theorem assures us that any closed orientable 3D manifold can be obtained in this way. The link representation a 3-manifold is not unique --- links related by a series of Kirby moves represent the same manifold. The Witten-Reshitikhin-Turaev invariant is a link invariant which remains unchanged by Kirby moves and therefore corresponds to a manifold invariant of the represented 3-manifold. This invariant is the formal construction of the Chern-Simons manifold invariant which does not require reference to ill-defined concepts like functional integrals. The surgery approach also gives a nice proof (using a construction known as Chain-Mail) of the Turaev-Walker-Roberts theorem that the Turaev-Viro invariant for a Chern-Simons theory is the absolute square of the Chern-Simons (or Witten-Reshitikhin-Turaev) invariant
Keywords: Surgery, Dehn-Surgery, Lickorish-Wallace Theorem, Kirby Calculus, Kirby Moves, Witten-Reshitikhin-Turaev Invariant, Chern-Simons, Turaev-Walker-Roberts Theorem, Chain-Mail.
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Chapter 25: Anyon Condensation
Abstract: A mechanism for constructing one anyon theory from another is anyon condensation, akin to Bose condensation. It is believed that any continuous phase transition between TQFTs must occur by such a condensation transition. We describe the main steps in such a condensation: Identification/Splitting, and Confinement. We give simple examples when the condensing boson is a simple current then describe the more general structure. We explain how the idea of a Chern-Simons coset can be understood under the framework of condensation. Finally we discuss the relationship between condensation and gappable boundaries.
Keywords: Condensation, Bose Condensation, Anyon Condensation, Identification, Splitting, Confinement, Coset, Gappable Boundary.
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Chapter 26: Introducing Quantum Error Correction
Abstract: We briefly introduce some basic ideas of information and quantum information. We start by comparing a classical memory with a quantum memory. Simple repetition codes can protect classical memories from error and we introduce the idea of a code space, physical bits, and logical bits. However, the quantum no-cloning theorem prevents a straightforward generalization to the quantum case. Nonetheless quantum error correction is indeed possible. We show the 9 qubit Shor code as an example.
Keywords: Classical Memory, Quantum Memory, Classical Error Correction, Code Space, Physical Bits, Logical Bit, Quantum No-Cloning Theorem, Quantum Error Correction, Shor Code.
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Chapter 27: Introducing the Toric Code
Abstract: We introduce the toric code as an example of a quantum error correcting code. The system is a lattice of spins on a torus. We use a basis of spin up/down spins which we translate into diagrams of colored (or uncolored) edges of the lattice. We introduce commuting vertex and plaquette operators, and explain that particular eigenstates of these operators define our code-space which in the diagram language corresponds to sums over loop configuration. The code space decomposes into four orthogonal wavefunctions which are distinguished by the parity of the number of colored loops around each of the two cycles of the torus. Quantum information is stored in the coefficients of these four wavefunctions in superposition. We then turn to consider error processes. We discover that errors occur in pairs locally and are moved apart from each other by "strings" of additional error operators. We discover that it is easy to identify when physical errors have occurred and correct them before a logical error occurs. We extend the discussion to consider irregular "lattices" of spins and surfaces with arbitrary genus (and make the connection to the Shor Code). Finally we turn to consider the ZN generalization of the toric code.
Keywords: Toric Code, Vertex Operator, Plaquette Operator, Code Space, Quantum Error Correction, Error String, ZN Toric Code, Euler Characteristic.
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Chapter 28: The Toric Code as a Phase of Matter and a TQFT
Abstract: The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.
Keywords: TQFT, Quasiparticles, Hamiltonian, Charge-Flux Model, Toric Code, ZN Toric Code.
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Chapter 29: Robustness of Topologically-Ordered Matter
Abstract: Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.
Keywords: Gap, Toric Code, TQFT, Topological Order, Topologically-Ordered Matter, Topological Phase of Matter, Perturbation Theory, Rigidity.
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Chapter 30: Abstracting the Toric Code: Introduction to Tube Algebra
Abstract: The arguments about the toric code relied on diagrammatic reasoning about loops around handles of the torus, but the underlying lattice was not crucial to any of the arguments. We now decsribe the same model more abstractly, with diagrams in the continuum, with no reference to the underlying lattice at all. We find that we are using a planar diagram algebra (in this case a d=1, Z2 loop gas) and the toric code ground states arise as a sum over all possible such diagrams. We introduce the notion of the tube algebra. We build so-called "idempotent" states on an annulus (or "tube") and from these we can extract quasiparticle types, the S-matrix, the T-matrix, braiding, and fusion properties. We then generalize the construction to the ZN toric code.
Keywords: Toric Code, Loop Gas, Planar Diagram Algebra, Tube Algebra, Quasiparticle, Idempotent, T-matrix, S-matrix, Braiding, Fusion, ZN Toric Code.
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Chapter 31: Kitaev Quantum Double Model
Abstract: One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the "untwisted" case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle "ribbon operators" which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.
Keywords: Kitaev Quantum Double Model, Drinfeld Double, Twisted Kitaev Quantum Double Model, Vertex Operator, Plaquette Operator, Tube Algebra, Ribbon Operator, Gauge Theory, Lattice Gauge Theory.
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Chapter 32: Doubled-Semion Model
Abstract: The Doubled-Semion model is another simple example of generalizing the toric code. Here we start with a planar diagram algebra that is the d=-1, Z2 loop gas. I.e, here a loop is given the value of -1. This is also known as the nontrivial cocyle of Z2. This model can be thought of as the simplest case of a twisted Kitaev Quantum Double, and it is also the simplest example of the Drinfeld double of a modular anyon theory (the double of the semion theory). In this case we construct a Hamiltonian such again we have a loop gas, but now flipping over a plaqutte incurs a sign in the wavefunction if we change the parity of the number of loops. We again turn to the tube algebra to identify the excitations of the model, which are both right- and left-handed semions. We run into a problem related to the nontrivial Frobenius-Schur indicator of the input diagram algebra. However, here we can resolve the issue by choosing a convenient gauge.
Keywords: Twisted Kitaev Quantum Double, Drinfeld Double, Nontrivial Cocycle of Z2, Z2 Loop Gas, Tube Algebra, Gauge Choice.
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Chapter 33: Levin-Wen String-Net
Abstract: The Levin-Wen model is a general construction that uses an input planar diagram algebra to construct a Hamiltonian with a topologically ordered ground state (the Drinfeld double of the input category). The ground state (code) space consists of a sum over all diagrams weighted by the evaluation of the diagram. We work through the example of the doubled Fibonacci model. Here the input category is the Fibonacci fusion rules and F-matrices. We work with continuum diagrams and construct the full tube algebra of the model. We identify the quasiparticle types to include both left- and right-handed Fibonacci anyons. In the case that the input category admits a modular braiding, much of the analyis simplifies and the output is two copies of the input category of opposite chiralities (both left- and right-handed). We finally turn to the construction of the explicit Hamiltonian on the lattice, and construction of quasiparticle string (or ribbon) operators. We return to the reconsider the toric code and doubled semion models as simple examples.
Keywords: Levin-Wen Model, Quantum Double, Drinfeld Double, String-Net, Tube Algebra, Doubled Fibonacci Model, Doubled Semion Model, Toric Code.
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Chapter 34: Topological Entanglement
Abstract: If we partition a Hilbert space we can define the entanglement entropy (both von Neumann and Renyi) between the two pieces. We consider breaking up a system into two or more distinct spatial regions and ask about the entanglement between these two regions. Studying the toric code ground state as an initial example, we show that there is an entanglement that scales as the length of the cut between the two regions, but there is also a sub-leading term, known as the topological entanglement entropy, which depends only on the topological properties of the system. While the leading term is nonuniversal, the topological term is robust and can be generalized to any other TQFT. We give three different derivations of the topological entanglement entropy.
Keywords: Entanglement, Entanglement Entropy, Topological Entanglement Entropy, Toric Code, Surgery, Entanglement Hamiltonian.
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Chapter 35: SPT Phases of Matter
Abstract: We refine our notion of a topoogical phase of matter to account for symmetry protection. We describe SPT (symmetry protected topological or trivial) phases and give some history of this idea. We focus on the case of on-site symmetries. The case of one-dimensional SPTs with on-site symmetry is discussed. For understanding two-dimensional SPTs we focus on the case of Z2 symmetry and argue that there are two possible Z2 paramagnetic phases, i.e., gapped phases where the Z2 symmetry is unbroken. These two phases are in one-to-one correspondence with the two possible Z2 loop gases. We argue that with any discrete group symmetry, we generally classify SPT phases with the possible 3-cocyles. We briefly discuss the idea of symmetry defects and explain that such particles are the excitations of the corresponding twisted Kitaev quantum double model of the same cocyle. We briefly discuss the edge modes of SPT phases. Finally we argue that the SPT phases can be considered as the "ungauging" of the corresponding Dijkgraaf-Witten model.
Keywords: Topological Phase of Matter, Symmetry Protected Topological Phase of Matter, Symmetry Protected Trivial Phase of Matter, Paramagnet, Z2 Symmetry, Loop Model, Twisted Kitaev Quantum Double Model, Symmetry Defect, Dijkgraaf-Witten Model, Ungauging.
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Chapter 36: Anyon Permuting Symmetry
Abstract: A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.
Keywords: Anyon Permuting Symmetry, Anyon Condensation, Symmetry Defect, Toric Code, Majorana Zero mode, G-crossed, Symmetry Enriched Topological Phase.
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Chapter 37: Experiments (In Brief!)
Abstract: We give a brief account of some of the experimental systems believed to harbor anyons (or that might someday be coaxed to harbor anyons). We discuss Fractional Quantum Hall Effects --- abelian and nonabelian --- in some detail. We mention Fractional Chern Insulators, and Bosonic Fractional Quantum Hall Effects. We discuss gapped spin liquids --- particular the Kitaev Honeycomb model and frustrated antiferromagents. We briefly discuss conventional superconductors before turning to so-called Majorana materials. Finally we discuss the recent advances in quantum simulation of matter.
Keywords: Experiments, Fractional Quantum Hall Effect, Abelian Fractional Quantum Hall Effect, Non-Abelian Fractional Quantum Hall Effect, Fractional Chern Insulators, Bosonic Fractional Quantum Hall Effects, Gapped Quantum Spin Liquids, Kitaev Honeycomb Model, Frustrated Antiferromagnets, Superconductors, Majoranas, Quantum Simulation.
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Chapter 38: Final Comments
Abstract: We give a brief discussion of some things that have been left out of the book. In particular this includes detailed theory of fractional quantum Hall effect, connection to conformal field theory, tensor networks, and topological insulators.
Keywords: Fractional Quantum Hall Effect, Conformal Field Theory, Tensor Networks, Topological Insulators.
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Chapter 39: Appendix: Kac and Other Resources for TQFTs
Abstract: A number of tools for determining properties of TQFTs are described. In particular we outline the use of the computer program "Kac". We also give pointers to other useful resources.
Keywords: TQFTs, Kac.
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Chapter 40: Appendix: Some Mathematical Basics
Abstract: This chapter is included to introduce some basic mathematical concepts. This includes basic notions about manifolds, basic ideas about groups, group theory, Lie groups, Lie algebras, representations, and the topological idea of a fundamental group.
Keywords: Manifold, Group, Representation, Lie Group, Lie Algegbra, Fundamental Group.
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