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
****
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
****
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.
****
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,
****
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.
****
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
****
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.
****
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.
****
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
****
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
****
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.
****
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.
****
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.
****
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
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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.
****
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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