Protection of topological order by symmetry and many-body localization

TL;DR

This paper investigates which symmetry protected topological (SPT) orders can be realized in highly excited, many-body localized states, revealing that all bosonic SPTs are localizable while certain fermionic phases are not, especially under weak interactions.

Contribution

It introduces a criterion for many-body localizability of SPT orders and demonstrates that all bosonic SPTs are localizable, whereas some fermionic SPTs face fundamental obstructions.

Findings

All bosonic SPTs in 1, 2, 3 dimensions are MB localizable.

Chiral phases like quantum Hall fluids are not MB localizable.

Some 2D topological superconductors are MB localizable when related to bosonic SPTs.

Abstract

In closed quantum systems, strong randomness can localize many-body excitations, preventing ergodicity. An interesting consequence is that high energy excited states can exhibit quantum coherent properties, such as symmetry protected topological (SPT) order, that otherwise only occur in equilibrium ground states. Here, we ask: which types of SPT orders can be realized in highly excited states of a many-body (MB) localized system? We argue that this question is equivalent to whether an SPT order can be realized in an exactly solvable lattice model of commuting projectors. This perspective enables a sharp definition of MB localizability. Using this criterion, it is straightforward to establish that whereas all bosonic SPTs in spatial dimensions $d=1,\,2,\,3$ are MB localizable, chiral phases (e.g. quantum Hall fluids) are not. We also show that free fermion SPTs in $d >1$ (including topological insulators and superconductors) cannot be localized if interactions are weak. A key question is whether strong interactions can render them MB localizable, which we study in the context of a class of $d=2$ topological superconductors. Using a decorated domain wall (DDW) approach we show that some phases in this class are MB localizable, when they correspond to bosonic SPT orders. However, a similar DDW approach faces a fatal obstruction to realizing certain intrinsically fermionic SPT orders, an issue we argue may persist beyond this specific construction.