Topological Order at Nonzero Temperature

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.

Figure 1

Illustration of holes in the lattice. The solid lines indicate the division of the lattice into squares of size $l_{\beta }$, and only some of the squares are illustrated. Solid circles indicate holes in the lattice, with one hole per square. Dashed lines connect holes (dashed line extending outside the solid lines connecting to only one hole are intended to indicate connections to holes in other square which are not illustrated). Symbols $0,1,2,\dots$ indicate different subsets of the lattice with holes in it, with the dashed lines indicating divisions between those subsets.