CHAPTER 8

Topological phases of matter

This chapter covers mathematical models of topological phases of matter: toric

code and Levin-Wen models for quantum doubles, and wave functions for FQH

liquids. In the end, we briefly discuss the inherent fault-tolerance of topological

quantum computers.

Since the discovery of the fractional quantum Hall effect(s), a new mathe-

matical framework to describe topological phases has become necessary. FQH

liquids have been modeled with wave functions, quantum Chern-Simons theory,

CFT/TQFT/MTC, and others. Each approach yields insights into these new

phases. While it is not the right moment to coin a definition of topological phases

of matter, several working definitions have been proposed: through code-subspace

properties, gapped Hamiltonians. We will use the lattice version of gapped Hamil-

tonian here.

A topological phase of matter is a state of matter whose low-energy effective

theory is a TQFT. There are two kinds of (2+1)-TQFTs which are well-studied:

quantum doubles, or Drinfeld centers, and Chern-Simons theories. Quantum dou-

bles are well-understood theoretically, as exemplified by Kitaev’s toric code model,

but their physical relevance is unclear at the moment. Chern-Simons theories are

the opposite: their physical relevance to FQH liquids is established, while their

Hamiltonian formulation on lattices is a challenge. Quite likely a content lattice

Hamiltonian formulation giving rise to Chern-Simons TQFTs from slave particles

does not exist. Our physical system lives on a compact oriented surface Y , possibly

with boundary. We will consider only closed Y , i.e., no anyons present. If there are

anyons, i.e., Y has punctures, then boundary conditions are necessary.

8.1. Doubled quantum liquids

8.1.1. Toric code. In this chapter, a lattice is an embedded graph Γ ⊂ Y

whose complementary regions are all topological disks. In physics, vertices of Γ are

called sites; edges, bonds or links; and faces, plaquettes.

Definition 8.1. Given an integer l 1, to each lattice Γ ⊂ Y we associate

the Hilbert space LΓ,l =

edges

Cl

with the standard inner product.

(1) A Hamiltonian schema (HS) is a set of rules to write down a Hermitian

operator HΓ on LΓ,l for each Γ ⊂ Y .

(2) A HS is k-local if there exists a constant k such that HΓ is a sum of

Hermitian operators Ok of the form id ⊗ A ⊗ id, where A acts on at most

k factors of LΓ,l. We will call a local HS a quantum theory.

Example 8.2 (The toric code schema). In the celebrated toric code, l = 2.

Let σx = ( 0 1

1 0

), σz =

(

1 0

0 −1

)

be the Pauli matrices. For each vertex v, define an

operator Av on LΓ =

edges

C2,

as a tensor product of σz’s and identities: Av acts

97

http://dx.doi.org/10.1090/cbms/112/08