On the stability of topological order in tensor network states

TL;DR

This paper demonstrates the stability of a tensor network representation of the 3d toric code ground state against certain local perturbations by relating it to the phase diagram of the 3d Ising gauge theory, and explores how virtual symmetries influence topological stability.

Contribution

It constructs a stable tensor network representation of the 3d toric code and analyzes the conditions for stability based on virtual symmetries and excitations.

Findings

Stability linked to virtual symmetries and topological excitations.

Representation of 3d toric code is stable under specific perturbations.

Dual and alternative representations like X-cube are unstable.

Abstract

We construct a tensor network representation of the 3d toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3d Ising gauge theory, which has a non-zero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual operators that break those symmetries. In particular, a dual representation of the 3d toric code ground state, as well as representations of the X-cube and cubic code ground states, for which point-like excitations are created by such operators, are found to be unstable.