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 Rényi-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 Rényi-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 Z2-symmetric dynamics, the Rényi-2 correlation length grows exponentially in time—even when the spectrum is gapless—yielding an effective transition time tc~ln L and enabling rapid preparation of the Z2 SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, tc~Lα, with a filling-dependent dynamical exponent: ballistic growth (α ≈ 1) at finite filling crosses over to diffusive scaling (α ≈ 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.
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 Rényi-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 Rényi-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 Z2-symmetric dynamics, the Rényi-2 correlation length grows exponentially in time—even when the spectrum is gapless—yielding an effective transition time tcln L and enabling rapid preparation of the Z2 SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, tcLα, with a filling-dependent dynamical exponent: ballistic growth (α ≈ 1) at finite filling crosses over to diffusive scaling (α ≈ 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.
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 Rényi-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 Rényi-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 Z2-symmetric dynamics, the Rényi-2 correlation length grows exponentially in time--even when the spectrum is gapless--yielding an effective transition time tc∝lnL and enabling rapid preparation of the Z2 SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, tc∝Lα, with a filling-dependent dynamical exponent: ballistic growth (α≈1) at finite filling crosses over to diffusive scaling (α=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.
Strong-to-weak spontaneous symmetry breaking (SW-SSB) is a distinctive phenomenon of mixed quantum states, characterized by nonlinear long-range order associated with strong symmetry. In realistic open quantum systems, however, strong symmetry is macroscopically fragile and can be explicitly reduced to weak symmetry through local charge exchange with the environment. In this paper, we investigate how the SW-SSB order decays under such strong-symmetry-breaking Liouvillian dynamics. For discrete Z2 symmetry, a local symmetry-breaking perturbation drives an exponential decay of the Rényi-1 long-range correlator, revealing the fragility of discrete SW-SSB order. By contrast, for continuous U(1) symmetry, local charge exchange broadens the distribution over symmetry-charge sectors and leads to parametrically slower algebraic relaxation. We establish a rigorous algebraic lower bound on the connected Rényi-1 long-range correlation, demonstrating the robustness of this slow relaxation. In the thermodynamic limit, we further uncover a two-stage dynamical scaling: R~1∼t−1 at early times and R~1∼t−1/z at long times, with z set by the underlying hydrodynamic dynamics. As a numerical verification, we consider fermionic chains with z=2 and z=4, which are excellent agreement with the theoretical predictions.
Symmetry-based classification of quantum phases of matter is one of the most foundational organizing principles in physics; however, an analogous framework for mixed, decohered quantum states has only begun to emerge. A central new concept is strong-to-weak spontaneous symmetry breaking (SW-SSB), a sharp transition in mixed quantum states that is invisible to any observable linear in the density matrix and that has since been predicted across a broad class of open and monitored quantum systems. It also provides a unifying language for phenomena as disparate as the decodability of topological quantum memories and the emergence of classical hydrodynamics from decohered quantum dynamics. Here we report the first experimental observation of SW-SSB, in dephased single-component fermionic matter imaged by a quantum gas microscope. A quantum-classical estimator built on a machine-learned Gaussian reference state gives direct access to the nonlinear Rényi-1 and Rényi-2 correlators that diagnose SW-SSB, and reveals long-range Rényi order in the dephased Fermi liquid. Adding a commensurate superlattice drives the underlying fermions through a metal-to-insulator transition that, after full dephasing, manifests as a sharp SW-SSB phase transition. Our results uncover the symmetry principle behind information-theoretic transitions in open quantum systems, and extend Landau's symmetry paradigm into the regime of real, decohering quantum devices.
Symmetry in mixed quantum states can manifest in two distinct forms: , where each individual pure state in the quantum ensemble is symmetric with the same charge, and , which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the . We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices.
Strong-to-weak spontaneous symmetry breaking (SWSSB) is a novel phase transition in mixed states associated with the loss of global charge information. In this work we study fermions with U(1) symmetry under infinite density dephasing and derive the SWSSB properties of the resulting mixed states. Our central observation is that at infinite dephasing every Renyi correlator is exactly the spin-spin correlator of a compact XY model, whose Boltzmann weight is the squared full-counting-statistics generating function of a classical ensemble built from the parent state. For the Renyi-2 correlator it is the full counting statistics of the parent state itself. We complement the infinite dephasing mapping with a novel diagrammatic expansion for the Renyi-1 correlator, applicable to any diagonal mixed state, which computes the couplings of the Renyi-1 XY model. We also study a replica field theory for even Renyi index quantities, applicable at finite dephasing. Our analysis indicates that fully dephased metals display long-range SWSSB in d≥2, as confirmed numerically and experimentally in d=2, and quasi-long-range SWSSB in d=1. For insulating parent states both an SWSSB and a trivial phase are possible at infinite dephasing depending on UV details. Long-range SWSSB in d≥3, or quasi-long-range SWSSB in d=2, survives above a nonuniversal, Renyi-index-dependent threshold and is lost when the parent state is deep in the insulating phase, with the exception of quantum Hall insulators which generically have quasi-long-range SWSSB. For even Renyi indices we also determine the stability of the SWSSB phase away from infinite dephasing and derive the universality class of the SWSSB transition.
We explore the onset of spontaneous strong-to-weak symmetry breaking (SW-SSB) under U(1)-symmetric (i.e., charge-conserving) open-system dynamics. We define this phenomenon for quantum states and classical probability distributions, and explore it in three complementary models, one of which exhibits nontrivial quantum coherence at short times. Our main conclusions are as follows. In one dimension, the strong symmetry is not spontaneously broken at any finite time; however, correlators probing strong-to-weak symmetry breaking develop order on length scales that grow linearly in time, parametrically faster than charge diffusion. We provide numerical evidence for this scaling in multiple distinct probes of SW-SSB, and derive it from a field-theory analysis. Moreover, we relate this scaling to the problem of inferring the charge inside a subregion by measuring its surroundings, and construct explicit decoding protocols that illustrate its origin. In two dimensions, field theory and numerical simulations support a finite-time Berezinskii-Kosterlitz-Thouless-like SW-SSB transition. Within continuum hydrodynamics, by contrast, SW-SSB happens at infinitesimal time in two or more dimensions. The SW-SSB transition time can thus be interpreted as marking the emergence of a continuum hydrodynamic description, or (more precisely) the timescale beyond which non-hydrodynamic information such as discrete particle worldlines can no longer be inferred. We support this picture by analyzing a model in which we exploit SW-SSB to derive a classical stochastic hydrodynamic description from the underlying quantum dynamics.