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 $\mathbb{Z}_2$-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.
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.
We propose a local notion of strong-to-weak spontaneous symmetry breaking (SW-SSB), through a local one-point fidelity correlator. Compared with the previous definition in terms of a two-point fidelity correlator, our local formulation offers two key advantages: (1) it is easier to detect in large systems: for a system of size N and with poly(N) amount of resources, one can detect the local fidelity order up to volume scale O(log(N)); and (2) the local SW-SSB order remains well defined in the thermodynamic limit, where the density matrix itself is not well defined. We show that key features of SW-SSB, including stability under finite-depth symmetric channels and long-range conditional mutual information, persist within this local framework. Our definition is conceptually analogous to local thermalization, as exemplified by pure states obeying the eigenstate thermalization hypothesis (ETH). For critical states, the local one-point fidelity correlator defines an interesting class of defect problems. We demonstrate the applicability of the local formulation through several concrete examples, and derive the universal scaling behavior of the local fidelity correlator in a range of critical systems, including ground states of conformal field theories as well as ballistic and diffusive free-fermion metals.
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.
Under decoherence, an initial Gaussian (free-fermion) state evolves into a non-Gaussian mixed state, so the resulting decohered fermionic state is not exactly solvable in general. We show through an inequality that a class of Rényi-2 correlators of the decohered fermion state are upper-bounded by the Rényi-2 correlator serving as a proximate diagnostic of strong-weak spontaneous symmetry breaking (SW-SSB) of the charge-U(1) symmetry. This inequality holds for arbitrary decoherence strength and suggests that decoherence drives fermionic quantum matter toward U(1) SW-SSB. We also make connections between our inequality and other subjects such as projected quantum spin Hall insulator and Dirac spin liquid states.
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.
We develop an operational approach to strong-to-weak spontaneous symmetry breaking (sw-SSB) of a non-Abelian symmetry group G. We formulate it as the decoding task of distinguishing two strongly symmetric states that differ by a pair of inserted irreps, after both are corrupted by strongly symmetric decoherence. For finite G, this task is dual to distinguishing logical states of the quantum double D(G) under pure-charge noise. We derive a hierarchy of statistical mechanics models describing optimal decoding from the fusion outcomes measured on successively larger nested regions. As for maximum-likelihood decoders of Abelian codes the measured irreps act as quenched disorder obeying a Nishimori-like condition. Strikingly, because local irreps do not determine how they fuse globally, the strong G×G symmetry is realized only nonlocally, and irreps trivially charged under Z(G) (the center of G) can hide inside noise-generated ones. As a result, the strong symmetry spontaneously breaks down to Z(G), times the weak symmetry. The only transition at nonzero error rate is then an ordering of the center, whose threshold cannot decrease as the hierarchy is ascended. This conclusion holds at any finite level of the hierarchy for non-nilpotent groups. We confirm these predictions with tensor-network calculations for S3 and SU(2). At the first level, the S3 model, whose center is trivial, shows no transition, and the fidelity correlations of all irreps provably decay to non-zero values. The SU(2) model exhibits a Z2 transition consistent with the Nishimori-Ising universality class, where the fidelity correlations of spin-1/2 irreps become long-range. The threshold rises from the first to the second level of the hierarchy by 16%. Our results provide a systematic framework for studying non-Abelian sw-SSB and for constructing decoders for non Abelian topological codes.
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.