Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking

TL;DR

This paper studies the stability of mixed-state symmetry-protected topological order in open quantum systems, revealing that strong symmetric perturbations destabilize it, while weak symmetry defects do not.

Contribution

It constructs an exactly solvable Lindbladian model for mixed-state topological order and analyzes its stability under various symmetric perturbations.

Findings

Strong symmetric perturbations cause spontaneous symmetry breaking.

Weak symmetry defects do not destabilize the topological order.

A Clifford-based quantum channel replicates Lindbladian physics efficiently.

Abstract

Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the decohered cluster state -- a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry -- we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.