Tensor networks demonstrate the robustness of localization and symmetry protected topological phases

TL;DR

This paper uses tensor network methods to prove that many-body localized symmetry protected topological systems with time reversal symmetry exhibit robust four-fold degeneracy in their entanglement spectra, indicating stable topological order.

Contribution

It introduces a tensor network framework demonstrating the robustness of topological phases in many-body localized systems with time reversal symmetry.

Findings

Eigenstates have four-fold degenerate entanglement spectra in the thermodynamic limit.

Local symmetries in matrix product operators define a robust $$ topological index.

Topological order remains stable under perturbations that preserve time reversal symmetry and localization.

Abstract

We prove that all eigenstates of many-body localized symmetry protected topological systems with time reversal symmetry have four-fold degenerate entanglement spectra in the thermodynamic limit. To that end, we employ unitary quantum circuits where the number of sites the gates act on grows linearly with the system size. We find that the corresponding matrix product operator representation has similar local symmetries as matrix product ground states of symmetry protected topological phases. Those local symmetries give rise to a $\mathbb{Z}_2$ topological index, which is robust against arbitrary perturbations so long as they do not break time reversal symmetry or drive the system out of the fully many-body localized phase.