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

TL;DR

This work analyzes the stability of steady-state mixed-state SPT order in open quantum systems by constructing a Z2S×Z2W-symmetric parent Lindbladian whose steady state is a decohered cluster state with nontrivial mixed-state SPT order. Through a dual CZ mapping to a solvable reaction-diffusion model and DMRG/Clifford simulations, the authors show that generic symmetric perturbations induce strong-to-weak spontaneous symmetry breaking, destroying the mixed-state SPT order, while perturbations creating only weak-string defects leave the order intact. They identify an eightfold steady-state degeneracy under certain boundary conditions, characterize the phase diagram with explicit string-order and correlation diagnostics, and provide an exactly solvable perturbation class demonstrating SW-SSB. A Clifford-channel realization reproduces the essential physics and enables efficient classical simulation, highlighting a practical path to exploring steady-state phases via parent Lindbladians rather than density matrices. The results suggest a broad instability of 1d mixed-state SPT steady states to SW-SSB and motivate exploration of higher dimensions and bosonic systems where defect dynamics may stabilize the order.

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.

Figures (15)

• Figure 1: Summary of the main results of this work. a) In \ref{['sec_perturbed_Lindbladian']}, we study the steady-state phase diagram of a Lindbladian $\mathcal{L}_\lambda$ which interpolates between $\mathcal{L}_\mathcal{C}$, whose steady states exhibit mixed SPT order and $\tilde{\mathcal{L}}_\mathcal{C}$, whose steady states are trivially ordered. Rather than a finite phase of either order, we find that the entire intermediate region exhibits SW-SSB. b) We show in \ref{['sec:weak_defects']} that there exists a special class of perturbations to which the steady-state mixed SPT order is stable. c) In \ref{['sec_clifford_simulation']}, we study a parameterized quantum channel $\mathcal{E}_\lambda$ which approximates the dynamics of $\mathcal{L}_\lambda$ while being amenable to efficient classical simulation.
• Figure 2: An illustration of the $U_{\text{CZ}}$ circuit appearing in \ref{['eq:U_CZ']}.
• Figure 3: $(a)$ Rényi-1 strong-string correlator $C_{\text{I}, nm}^S$ as a function of the length of the string. $C_{\text{I}, nm}^S$ decays exponentially with $|m-n|$ for any $\lambda>0$. The curve for $\lambda = 1$ is not visible within the limits of the plots because $C_{\text{I}, nm}^S=0$. $(b)$ Log of the Rényi-2 strong-string correlator $C_{\text{II}, nm}^S$ as a function of the length of the string. $C_{\text{II}, nm}^S$ decays exponentially with $|m-n|$. The curve for $\lambda = 1$ is not visible within the limits of the plots because $C_{\text{II}, nm}^S=0$. $(c)$ Rényi-2 weak-string correlator $C_{\text{II}, nm}^W$ as a function of the length of the string. $C_{\text{II}, nm}^W$ remains nonzero for all $0 \leq \lambda < 1$.
• Figure 4: Connected correlator $B_{\text{II}, nm}$ for the Lindbladian that interpolates between the parent Lindbladian $\mathcal{L}_{\mathcal{C}}$ and its CZ dual, $\tilde{\mathcal{L}}_{\mathcal{C}}$ shown in Eq. \ref{['eq:interpolating_lind']}. We find that this correlator is nonzero for $0 < \lambda < 1$. This, taken together with $A_{\text{I}, nm} = A_{\text{II}, nm} = 0$, indicates that the strong symmetry on the even sites is broken to the weak symmetry. In the inset, we show $B_{\text{II}, nm}$ as a function of $|m-n|$ for various values of $\lambda$. We find that $B_{\text{II}, nm}$ is independent of $|m-n|$, but depends on $\lambda$, as shown in the main figure.
• Figure 5: Phase diagram of the steady state of the Lindbladian $\mathcal{L}(\lambda_0,\lambda_1,\lambda_2)$ given in \ref{['eqn: L012']}. The nontrivial SPT order is stable against weak-string defects, introduced by the perturbation $\tilde{L}_{0}$. However, introducing any strong-string defects such as $\tilde{L}_2$ leads to strong-to-weak SSB. Note the brown dashed line is the interpolated Lindbladian $\mathcal{L}_{\lambda}$ in \ref{['eq:interpolating_lind']}.

Theorems & Definitions (4)

• Definition 1: Fast Driving
• Definition 2: Lindbladian Phase of Matter
• Theorem 1
• proof