# Edge mode locality in perturbed symmetry protected topological order

## TL;DR

This paper investigates how local interactions and disorder affect edge zero modes in symmetry-protected topological spin chains, revealing that disorder does not universally stabilize edge modes and that control over localization is more crucial.

## Contribution

It introduces numerical tools for constructing locally conserved operators and challenges the idea that disorder always stabilizes topological edge modes.

## Key findings

- Disorder has no effect on edge modes in the Anderson localized regime.
- Disorder only localizes a subset of edge modes in the many-body interacting regime.
- One edge mode operator remains unaffected by disorder, acting as if non-interacting.

## Abstract

Spin chains with symmetry-protected edge zero modes can be seen as prototypical systems for exploring topological signatures in quantum systems. These are useful for robustly encoding quantum information. However in an experimental realization of such a system, spurious interactions may cause the edge zero modes to delocalize. To stabilize against this influence beyond simply increasing the bulk gap, it has been proposed to harness suitable notions of disorder. Equipped with numerical tools for constructing locally conserved operators that we introduce, we comprehensively explore the interplay of local interactions and disorder on localized edge modes in the XZX cluster Hamiltonian. This puts us in a position to challenge the narrative that disorder necessarily stabilizes topological order. Contrary to heuristic reasoning, we find that disorder has no effect on the edge modes in the Anderson localized regime. Moreover, disorder helps localize only a subset of edge modes in the many-body interacting regime. We identify one edge mode operator that behaves as if subjected to a non-interacting perturbation, i.e., shows no disorder dependence. This implies that in finite systems, edge mode operators effectively delocalize at distinct interaction strengths. In essence, our findings suggest that the ability to identify and control the best localized edge mode trumps any gains from introducing disorder.