Topological Order at Non-zero Temperature

TL;DR

This paper defines topological order at non-zero temperature, proving that 2D commuting Hamiltonians are trivial at T>0, but 4D toric code retains nontrivial topological properties at finite temperature.

Contribution

It introduces a new definition of topological order at non-zero temperature and demonstrates its implications for different models, including 2D and 4D systems.

Findings

2D commuting Hamiltonians are not topologically ordered at T>0

Trivial states cannot reliably store quantum information with certain operators

4D toric code exhibits nontrivial topological order at finite temperature

Abstract

We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at $T>0$. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.