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arXiv:2603.24671

Inequality for Strong-Weak Spontaneous Symmetry Breaking in Fermionic Open Quantum systems

Abhijat Sarma, Cenke Xu

quant-phcond-mat.str-elSubmitted Mar 25, 2026• Announced Mar 27, 2026• 0 citations
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Abstract

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.

AI Summary

Dephasing generally turns Gaussian fermionic states into non-Gaussian mixed states. This paper asks whether their correlations can nevertheless be constrained analytically, focusing on strong-to-weak spontaneous symmetry breaking (SW-SSB) of charge $U(1)$. For initially degenerate $N$-flavor free-fermion ground states, including disordered tight-binding models, it derives an inequality valid at arbitrary dephasing strength. In a doubled-space representation followed by a particle–hole transformation, correlations of onsite bilinears $O=c^\dagger\sigma^\mu\Omega c$ satisfy $|C(x,y)|\leq\mathcal C_*(x,y)$, where $\mathcal C_*$ is the correlation of $\mathcal O=c^\dagger\sigma^1\Omega c$, a Rényi-2 proxy for charge-$U(1)$ SW-SSB. The full-correlator result requires traceless Hermitian flavor matrices $\Omega$; even $N$ is sufficient for the positive measure used in the proof. The derivation combines Hubbard–Stratonovich decoupling, an algebraic relation between the doubled fermion operator and its adjoint, and Cauchy–Schwarz inequalities, paralleling Weingarten inequalities. Extensions cover certain flavor/color decoherence channels and continuum Dirac fermions. A related QED$_3$ argument bounds the scaling dimension of a flavor-nonsinglet mass bilinear by $\Delta\leq2$, assuming a conformal infrared theory and Maxwell gauge dynamics without a Chern–Simons term. Connections to Gutzwiller-projected states are discussed. The inequality establishes relative correlation dominance, not long-range order or a SW-SSB transition. Non-onsite extensions rely on coarse-graining arguments for asymptotic behavior. Rényi-2 correlations are explicitly only a proximate diagnostic; extending the result to fidelity or Rényi-1 correlations remains open.

No sign of AI-generated textAI use disclosedOverclaimsStrong assumptionsPossible error
We also acknowledge generative-AI for discussions and generating Fig.~\ref{diagram}.

Error found: The onsite theorem's sufficient even-flavor and traceless-flavor-matrix assumptions are omitted from the abstract, although they are stated in the body. The correlator inequality establishes relative dominance but does not establish enhancement with decoherence or the development of SW-SSB. The non-onsite result rigorously bounds correlations by shifted diagnostic correlators, while their identification with the same-position asymptotic behavior requires additional regularity assumptions. The appendix confuses a zero-frequency, time-integrated current with a current on the temporal defect and additionally infers a condensate from a conclusion that only excludes short-range correlations.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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6 pages, 2 figures

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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.
Dephasing generally turns Gaussian fermionic states into non-Gaussian mixed states. This paper asks whether their correlations can nevertheless be constrained analytically, focusing on strong-to-weak spontaneous symmetry breaking (SW-SSB) of charge U(1)U(1)U(1).
For initially degenerate NNN-flavor free-fermion ground states, including disordered tight-binding models, it derives an inequality valid at arbitrary dephasing strength. In a doubled-space representation followed by a particle–hole transformation, correlations of onsite bilinears O=c†σμΩcO=c^\dagger\sigma^\mu\Omega cO=c†σμΩc satisfy ∣C(x,y)∣≤C∗(x,y)|C(x,y)|\leq\mathcal C_*(x,y)∣C(x,y)∣≤C∗​(x,y), where C∗\mathcal C_*C∗​ is the correlation of O=c†σ1Ωc\mathcal O=c^\dagger\sigma^1\Omega cO=c†σ1Ωc, a Rényi-2 proxy for charge-U(1)U(1)U(1) SW-SSB. The full-correlator result requires traceless Hermitian flavor matrices Ω\OmegaΩ; even NNN is sufficient for the positive measure used in the proof.
The derivation combines Hubbard–Stratonovich decoupling, an algebraic relation between the doubled fermion operator and its adjoint, and Cauchy–Schwarz inequalities, paralleling Weingarten inequalities. Extensions cover certain flavor/color decoherence channels and continuum Dirac fermions. A related QED3_33​ argument bounds the scaling dimension of a flavor-nonsinglet mass bilinear by Δ≤2\Delta\leq2Δ≤2, assuming a conformal infrared theory and Maxwell gauge dynamics without a Chern–Simons term. Connections to Gutzwiller-projected states are discussed.
The inequality establishes relative correlation dominance, not long-range order or a SW-SSB transition. Non-onsite extensions rely on coarse-graining arguments for asymptotic behavior. Rényi-2 correlations are explicitly only a proximate diagnostic; extending the result to fidelity or Rényi-1 correlations remains open.
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.
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\mathbb{Z}_2Z2​ 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)U(1)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\tilde{R}_1\sim t^{-1}R~1​∼t−1 at early times and R~1∼t−1/z\tilde{R}_1\sim t^{-1/z}R~1​∼t−1/z at long times, with zzz set by the underlying hydrodynamic dynamics. As a numerical verification, we consider fermionic chains with z=2z=2z=2 and z=4z=4z=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 [1][2][3][4][5][6][7][8][9][10][11][12][13][14].It also provides a unifying language for phenomena as disparate as the decodability of topological quantum memories [1,15,16] and the emergence of classical hydrodynamics from decohered quantum dynamics [17,18].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.
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)U(1)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≥2d\geq 2d≥2, as confirmed numerically and experimentally in d=2d=2d=2, and quasi-long-range SWSSB in d=1d=1d=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≥3d\geq 3d≥3, or quasi-long-range SWSSB in d=2d=2d=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.
Mixed quantum states can exhibit unconventional forms of symmetry breaking without pure-state counterparts. A prominent example is strong-to-weak spontaneous symmetry breaking (SW-SSB), which cannot be diagnosed by observables linear in the density matrix. Here, we experimentally resolve finite-size critical signatures of SW-SSB transitions on a superconducting quantum processor. We develop a measurement protocol that combines quantum measurements with classical processing based on the known dephasing channel to reconstruct nonlinear mixed-state correlators. Using only two measurement settings, the protocol reconstructs fidelity, Rényi-2, and linear correlators, as well as string observables. Applying this approach to matrix-product states subjected to power-law-correlated dephasing, we observe finite-size crossings and scaling collapses in nonlinear correlators, accompanied by decaying linear correlations. We further map the estimated SW-SSB transition points across initial-state and channel parameters. Measurements of linear and Rényi-2 string observables further reveal signatures of mixed-state symmetry-protected topological (SPT) phases, with a nonlinear string-order crossover aligned with an independently estimated SW-SSB transition point. Our results establish a quantum-classical approach to probing SW-SSB critical behavior and mixed-state SPT order.
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\mathbb{Z}_2Z2​-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.