# Classification of symmetry-protected topological phases in two-dimensional many body-localized systems

## TL;DR

This paper introduces a method using low-depth quantum circuits to classify two-dimensional symmetry-protected topological phases in many-body localized systems, highlighting the role of cohomology classes as invariants.

## Contribution

The authors develop a classification scheme for 2D SPT MBL phases using tensor network techniques, extending the understanding of topological invariants in disordered quantum systems.

## Key findings

- Third cohomology class is a topological invariant for certain symmetries.
- Classification applies to quasi-periodic and strongly disordered systems.
- Potential extension to ground state classifications with rigorous proofs.

## Abstract

We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we show that the (generalized) third cohomology class of the symmetry group is a topological invariant; however our approach leaves room for the existence of additional topological indices. We argue that our classification applies to quasi-periodic systems in two dimensions and systems with true random disorder within times which scale superexponentially with the inverse interaction strength. Our technique might be adapted to supply arguments suggesting the same classification for two-dimensional symmetry-protected topological ground states with a rigorous proof.