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arxiv: 2603.06363 · v2 · submitted 2026-03-06 · ❄️ cond-mat.mes-hall · cond-mat.str-el· quant-ph

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

Chang Shu , Kai Zhang , Zhu-Xi Luo , Yizhi You , Kai Sun This is my paper

Pith reviewed 2026-05-15 15:00 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elquant-ph
keywords strong-to-weak spontaneous symmetry breakingLindbladian evolutionRényi-2 correlatorsopen quantum systemsdynamical scalingsymmetry classesmixed-state phases of matter
0
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The pith

Symmetry class alone sets whether the SWSSB transition is exponential or algebraic in open quantum systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that the late-time dynamical scaling toward strong-to-weak spontaneous symmetry breaking in one-dimensional Lindblad-evolved systems is fixed entirely by the symmetry of the evolution. In Z2-symmetric cases the Rényi-2 correlation length grows exponentially, producing a transition time that grows only logarithmically with system size. U(1)-symmetric evolution instead produces algebraic growth whose exponent depends on particle filling. These scalings hold regardless of whether the underlying Liouvillian spectrum is gapped or gapless. A sympathetic reader cares because the result replaces the usual reliance on spectral properties with a simpler symmetry-based rule for controlling nonequilibrium phases in open systems.

Core claim

The emergence of strong-to-weak spontaneous symmetry breaking in open quantum systems is marked by the growth of the Rényi-2 correlation length at late times. This growth follows an exponential law for Z2-symmetric Lindbladians and an algebraic law for U(1)-symmetric ones, with the algebraic exponent varying from ballistic at finite filling to diffusive at zero filling. The scaling depends only on the symmetry class and remains the same whether the Liouvillian spectrum is gapped or gapless, so that the transition time scales as ln L for Z2 symmetry and as a power of L for U(1) symmetry.

What carries the argument

The Rényi-2 correlation length, whose late-time growth under symmetry-constrained Lindblad evolution determines the SWSSB transition scaling independent of spectral details.

If this is right

  • Z2 symmetry enables rapid preparation of SWSSB states with transition time logarithmic in system size.
  • U(1) symmetry leads to slower power-law scaling of the transition time.
  • The dynamical exponent for U(1) changes with filling fraction from approximately 1 to 2.
  • SWSSB is reached only in the infinite-size limit even though the scaling is symmetry-controlled.
  • Conventional spectral-gap arguments do not apply to this late-time regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Different symmetry classes might allow distinct strategies for stabilizing mixed-state order in experimental platforms.
  • The result could extend to higher dimensions where the algebraic exponents might change.
  • Measuring nonlinear correlators in open-system simulations could test the predicted symmetry-driven distinction directly.
  • Gapless systems with appropriate symmetry could still achieve fast symmetry breaking without closing gaps.

Load-bearing premise

That the late-time growth of the Rényi-2 correlation length is controlled purely by the symmetry class of the Lindbladian.

What would settle it

A calculation or simulation showing that two different Lindbladians with the same symmetry but different spectral gaps produce different growth laws for the Rényi-2 length would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.06363 by Chang Shu, Kai Sun, Kai Zhang, Yizhi You, Zhu-Xi Luo.

Figure 1
Figure 1. Figure 1: FIG. 1. Gapless [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Modified gapless [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Tripartition of the system into [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The full Liouvillian spectrum of strongly symmetric U(1) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Implementation of the Choi-Jamiołkowski isomorphism [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Universality of the ballistic dynamical scaling. (a) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the late-time dynamics of strong-to-weak spontaneous symmetry breaking (SWSSB) in one-dimensional Lindblad open quantum systems. By tracking the Rényi-2 correlation length, the authors claim that the functional form of its growth (exponential for Z2-symmetric Lindbladians, algebraic for U(1)) is dictated solely by the symmetry class on the doubled Hilbert space, independent of whether the Liouvillian spectrum is gapped or gapless. This yields t_c ~ ln L for Z2 and t_c ~ L^alpha (alpha filling-dependent, ~1 or 2) for U(1), with explicit Lindblad constructions, exact-diagonalization, and tensor-network data supporting the scaling.

Significance. If the central symmetry-based derivation holds, the result reorients the understanding of SWSSB preparation away from spectral-gap assumptions toward representation-theoretic properties of the Lindbladian, providing a universal dynamical classification for mixed-state phases without pure-state analogs. The explicit models, numerical verification across gapped/gapless cases, and parameter-free scaling predictions constitute a concrete advance for nonequilibrium open-system physics.

major comments (2)
  1. [§4] §4, the U(1) algebraic scaling derivation: the reported crossover from ballistic (alpha≈1) to diffusive (alpha=2) at zero filling is tied to the representation theory, but the manuscript does not explicitly show how the filling dependence emerges from the symmetry action on the doubled space without additional assumptions on the jump operators; a step-by-step expansion of the relevant matrix elements would confirm the exponent is symmetry-protected rather than model-specific.
  2. [§3.1] §3.1, the Z2 exponential growth claim: while the data show exponential Rényi-2 length growth even for gapless Liouvillians, the effective time constant tau is extracted numerically; the text should derive its independence from microscopic rates directly from the symmetry representation to make the 'solely by symmetry' statement fully analytic rather than numerically supported.
minor comments (2)
  1. [Figure 2] Figure 2 caption: the legend for the gapless Z2 case should explicitly state the system sizes used for the finite-size scaling collapse to allow direct reproduction.
  2. [Eq. (3)] Notation: the definition of the Rényi-2 correlator in Eq. (3) uses a doubled-space trace; clarify whether the partial trace is over one copy or both to avoid ambiguity when comparing to standard purity measures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation for minor revision. The comments highlight opportunities to strengthen the analytic content of the symmetry-based arguments. We address each major comment below and will incorporate the requested expansions in the revised manuscript.

read point-by-point responses

  1. Referee: §4, the U(1) algebraic scaling derivation: the reported crossover from ballistic (alpha≈1) to diffusive (alpha=2) at zero filling is tied to the representation theory, but the manuscript does not explicitly show how the filling dependence emerges from the symmetry action on the doubled space without additional assumptions on the jump operators; a step-by-step expansion of the relevant matrix elements would confirm the exponent is symmetry-protected rather than model-specific.

    Authors: We agree that an explicit derivation would make the symmetry protection clearer. In the revised §4 we will insert a step-by-step expansion of the matrix elements of the effective superoperator that governs the Rényi-2 correlator. Starting from the U(1) action on the doubled Hilbert space, we show that the commutation relations alone enforce the filling-dependent dynamical exponent: at finite filling the leading term yields ballistic propagation (α≈1), while the zero-filling sector projects onto a diffusive channel (α=2) without invoking any model-specific properties of the jump operators beyond the symmetry constraint. This addition will be parameter-free and directly tied to representation theory. revision: yes

  2. Referee: §3.1, the Z2 exponential growth claim: while the data show exponential Rényi-2 length growth even for gapless Liouvillians, the effective time constant tau is extracted numerically; the text should derive its independence from microscopic rates directly from the symmetry representation to make the 'solely by symmetry' statement fully analytic rather than numerically supported.

    Authors: We thank the referee for this observation. The exponential functional form itself follows analytically from the Z2 representation on the doubled space, which generates a constant effective decay rate for the relevant off-diagonal blocks independent of the Liouvillian gap. However, the precise numerical prefactor tau does retain a dependence on microscopic rates in general. In the revision we will add a short analytic paragraph in §3.1 deriving the exponential scaling from the symmetry algebra and explicitly stating that only the functional form (exponential versus algebraic) is symmetry-protected, while tau may vary with rates. This clarifies the scope of the 'solely by symmetry' claim without overstatement. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs explicit Lindblad operators for Z2 and U(1) symmetry classes on the doubled Hilbert space, derives the functional form of Rényi-2 correlation length growth (exponential vs algebraic) from the representation theory of the symmetry, and verifies the scaling—including cases with closed Liouvillian gap—via exact diagonalization and tensor-network numerics. No derivation step reduces a claimed prediction to a fitted parameter by construction, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified. The central claim that symmetry alone dictates the dynamical scaling is supported by direct model constructions and data that remain independent of the target scaling laws.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions about Lindbladian dynamics and the diagnostic for SWSSB with no free parameters, invented entities, or ad hoc axioms introduced beyond domain conventions.

axioms (2)
  • domain assumption Dynamics are governed by a Lindbladian superoperator that preserves the specified symmetry class
    Standard framework for Markovian open quantum systems as used throughout the abstract.
  • domain assumption SWSSB is diagnosed by long-range order in the Rényi-2 correlator while linear correlations remain short-ranged
    Definition of the mixed-state phase as stated in the abstract.

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Reference graph

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