Universal Dynamical Scaling of Strong-to-Weak Spontaneous Symmetry Breaking in Open Quantum Systems

TL;DR

This paper reveals that the late-time dynamical scaling of strong-to-weak spontaneous symmetry breaking in open quantum systems is governed solely by the symmetry class of the Lindbladian, independent of spectral gap details.

Contribution

It uncovers a universal dynamical regime controlled by symmetry class, showing exponential or algebraic growth of correlations depending on the symmetry, regardless of Liouvillian spectrum gap.

Findings

Z2 symmetry leads to exponential growth of correlation length and rapid state preparation.

U(1) symmetry results in algebraic growth with filling-dependent exponents.

Symmetry class, not spectral gap, controls late-time dynamical scaling of SWSSB.

Abstract

Strong-to-weak spontaneous symmetry breaking (SWSSB) defines a mixed-state phase of matter--without a pure-state counterpart--in which nonlinear observables such as the R\'enyi-2 correlator develop long-range order while conventional linear correlations remain short-ranged. Here we study the emergence of SWSSB in one-dimensional open quantum systems governed by Lindbladian evolution, where the transition time diverges with system size and SWSSB appears only asymptotically in the steady state. By tracking the late-time growth of the R\'enyi-2 correlation length, we uncover a universal dynamical regime controlled purely by the symmetry class of the Lindbladian. Contrary to the conventional expectation that late-time dynamics are governed by the low-lying Liouvillian spectrum, we find that the time dependence of the SWSSB transition--exponential versus algebraic--is dictated solely by symmetry, independent of details of the Lindbladian, including whether the Liouvillian spectrum is gapped or gapless. For $\mathbb{Z}_2$-symmetric dynamics, the R\'enyi-2 correlation length grows exponentially in time--even when the spectrum is gapless--yielding an effective transition time $t_c \propto \operatorname{ln} L$ and enabling rapid preparation of the $\mathbb{Z}_2$ SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, $t_c \propto L^{\alpha}$, with a filling-dependent dynamical exponent: ballistic growth ($\alpha \approx 1$) at finite filling crosses over to diffusive scaling ($\alpha = 2$) in the zero-filling limit. These results establish symmetry--rather than spectral gap structure--as the controlling principle for SWSSB late-time dynamical scaling, and open a new route to nonequilibrium symmetry breaking in open quantum systems.