Strong-to-weak spontaneous symmetry breaking (SWSSB) is diagnosed by nonlinear correlators, but its direct static implication for conserved charge fluctuations is not automatic. We show that, for continuous symmetries, long-range Rényi-1 correlator, together with a sufficiently rapid approach to its nonzero asymptotic value, forces subsystem charge indefiniteness: the block-charge variance has an extensive lower bound; equivalently, the truncated symmetry expectation has extensive curvature. This gives a precise static fluctuation footprint of charge scrambling. We construct examples to show that the implication is conditional and non-reversible: dephased superfluids retain Rényi-1 SWSSB with subextensive charge variance when the Rényi-1 tail is too slow, while sparse fixed-charge projectors have extensive charge variance but no local charge-transfer Rényi-1 order or long-range conditional mutual information. Finally, we introduce a \emph{twist overlap} correlator, which serves as an analogue of charge variance applicable to both discrete and continuous symmetries. This naturally decomposes local block-charge fluctuations into strong- and weak-symmetry channels. We found that the weak-symmetry channel isolates coherent charge fluctuations and is directly related to the Wigner--Yanase skew information. Taken together, these results give a unified understanding for distinguishing nonlinear SWSSB order, local charge indefiniteness, and coherent charge fluctuations.
Strong-to-weak spontaneous symmetry breaking (SWSSB) is diagnosed by nonlinear correlators, but its direct static implication for conserved charge fluctuations is not automatic. We show that, for continuous symmetries, long-range Rényi-1 correlator, together with a sufficiently rapid approach to its nonzero asymptotic value, forces subsystem charge indefiniteness: the block-charge variance has an extensive lower bound; equivalently, the truncated symmetry expectation has extensive curvature. This gives a precise static fluctuation footprint of charge scrambling. We construct examples to show that the implication is conditional and non-reversible: dephased superfluids retain Rényi-1 SWSSB with subextensive charge variance when the Rényi-1 tail is too slow, while sparse fixed-charge projectors have extensive charge variance but no local charge-transfer Rényi-1 order or long-range conditional mutual information. Finally, we introduce a \emph{twist overlap} correlator, which serves as an analogue of charge variance applicable to both discrete and continuous symmetries. This naturally decomposes local block-charge fluctuations into strong- and weak-symmetry channels. We found that the weak-symmetry channel isolates coherent charge fluctuations and is directly related to the Wigner--Yanase skew information. Taken together, these results give a unified understanding for distinguishing nonlinear SWSSB order, local charge indefiniteness, and coherent charge fluctuations.
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.
Strong-to-weak spontaneous symmetry breaking (SWSSB) can occur in a maximally mixed charge sector even when ordinary charged correlations vanish. We study this order when a charge sector splits into dynamically disconnected fragments. In a one-dimensional t-Jz chain with species-resolved dephasing, each fragment is labeled by a conserved spin word. A charged transfer between separated finite intervals preserves that word only if the intervening spins satisfy a periodicity constraint. In the full-state two-point sense, we use this constraint to prove the absence of long-range fidelity and Rényi-1 order in typical fixed-composition fragments, uniformly over all norm-bounded charged operators in any prescribed finite interval. A fragment with primitive spin-word period k supports a k-particle channel with exact long-distance value νk(1−ν)k/k2, and no charged operator on fewer than k sites can carry this order. A transverse exchange connects the fragments while preserving the global charges and restores the full-sector steady state at finite size. The conserved spin word therefore selects the local channels that can support mixed-state order.
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.
Recently, strong-to-weak spontaneous symmetry breaking (SW-SSB) has been established as a framework for describing quantum phases in open systems. However SW-SSB cannot be characterized by conventional charged two-point correlators. In this work, we show that, for strongly Z2-symmetric states with local charged involutions, SW-SSB manifests as a persistent deficit from the maximal quantum Fisher information (QFI) under symmetry-neutral pair rotations. A model-independent inequality hierarchy establishes the equivalence of this deficit with fidelity and Rényi-1 correlators as asymptotic long-range-order diagnostics. It thereby connects finite-angle state overlaps to a constraint on the infinitesimal Bures geometry along the same pair-generated orbit. For symmetry-projected Gibbs states, we further derive a lower bound on the deficit governed by the mean energy injected by the pair operation. Crucially, locality keeps this energy cost uniformly bounded in system size and endpoint separation, maintaining a finite gap below the maximal QFI at any fixed finite temperature. When ordinary charged long range order is absent, this persistent geometric constraint provides a quantitative information-geometric characterization of thermal SW-SSB.
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) 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.