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
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-
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 =
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
), σz =
be the Pauli matrices. For each vertex v, define an
operator Av on LΓ =
as a tensor product of σz’s and identities: Av acts