Classification of symmetry-protected topological many-body localized phases in one dimension

TL;DR

This paper classifies symmetry-protected topological phases in one-dimensional many-body localized systems, showing that all eigenstates share the same topological index as ground states and introducing a tensor network approach for fermionic systems.

Contribution

It provides a comprehensive classification scheme for MBL SPT phases in 1D spin and fermionic systems, including a tensor network formulation and stability analysis.

Findings

Eigenstates have the same topological index as ground states.

MBL phases are labeled by elements of the second cohomology group.

Fermionic MBL phases classified by generalized cohomology, with some limitations.

Abstract

We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a $\mathbb{Z}_2$ classification (cf. [Phys. Rev. B 98, 054204 (2018)]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a $\mathbb{Z}_2$ topological index expected from the classification of fermionic SPT ground states. Furthermore, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.