arxiv: 2605.05288 · v1 · submitted 2026-05-06 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· quant-ph

Recognition: unknown

Charge Scrambling in Strong-to-Weak Spontaneous Symmetry Breaking

Jong Yeon Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnquant-ph

keywords strong-to-weak spontaneous symmetry breakingRényi-1 correlatorcharge variancecontinuous symmetriescharge scramblingtwist overlapWigner-Yanase skew information

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The pith

Long-range Rényi-1 order with rapid saturation forces extensive subsystem charge variance for continuous symmetries.

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

The paper establishes that, for continuous symmetries, a long-range Rényi-1 correlator that reaches its nonzero value sufficiently quickly implies an extensive lower bound on the variance of charge inside a spatial block. This indefiniteness is equivalent to extensive curvature in the truncated symmetry expectation value and supplies a static signature of charge scrambling. A sympathetic reader cares because the result connects a nonlinear diagnostic of strong-to-weak spontaneous symmetry breaking directly to measurable fluctuations of a conserved quantity, without invoking time evolution. The implication is conditional and non-reversible: the paper supplies explicit counterexamples where slow saturation permits subextensive variance despite the long-range order, and where extensive variance exists without local Rényi-1 order.

Core claim

For continuous symmetries, the combination of a long-range Rényi-1 correlator and a sufficiently rapid approach to its nonzero asymptotic value forces the block-charge variance to obey an extensive lower bound, or equivalently forces extensive curvature in the truncated symmetry expectation. This supplies a precise static footprint of charge scrambling. The implication is shown to be conditional by constructing dephased superfluids that retain Rényi-1 SWSSB yet keep subextensive charge variance when the tail decays too slowly, and sparse fixed-charge projectors that exhibit extensive variance but lack long-range conditional mutual information or local charge-transfer Rényi-1 order.

What carries the argument

The Rényi-1 correlator, which diagnoses strong-to-weak spontaneous symmetry breaking and, when long-range and rapidly saturating, enforces an extensive lower bound on block-charge variance.

If this is right

Block-charge variance acquires an extensive lower bound under the stated conditions on the Rényi-1 correlator.
The truncated symmetry expectation develops extensive curvature as an equivalent diagnostic.
The newly introduced twist overlap correlator decomposes local charge fluctuations into strong- and weak-symmetry channels.
The weak-symmetry channel isolates coherent charge fluctuations and equals the Wigner-Yanase skew information.

Where Pith is reading between the lines

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

The separation between nonlinear order, total charge variance, and coherent fluctuations may help design static probes that distinguish scrambling from other forms of symmetry breaking in lattice models.
Because the twist overlap works for both discrete and continuous symmetries, it could serve as a uniform diagnostic across a wider range of symmetry-protected phases.
The counterexamples indicate that specific forms of decoherence or charge projection can decouple these quantities, suggesting that phase diagrams may contain additional intermediate regimes not captured by either diagnostic alone.

Load-bearing premise

The Rényi-1 correlator must approach its nonzero asymptotic value sufficiently rapidly.

What would settle it

A concrete model with long-range Rényi-1 order but subextensive charge variance when the correlator tail decays slowly, or a model with extensive charge variance but no long-range Rényi-1 order, as realized by the dephased superfluid and sparse projector constructions.

Figures

Figures reproduced from arXiv: 2605.05288 by Jong Yeon Lee.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

read the original abstract

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 R\'enyi-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 R\'enyi-1 SWSSB with subextensive charge variance when the R\'enyi-1 tail is too slow, while sparse fixed-charge projectors have extensive charge variance but no local charge-transfer R\'enyi-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.

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.

Desk Editor's Note private letter to a colleague

Rényi-1 SWSSB forces extensive charge variance only with fast approach to the limit, plus a new twist overlap that splits strong and weak channels.

read the letter

The paper shows that for continuous symmetries, a nonzero long-range Rényi-1 correlator implies an extensive lower bound on block-charge variance only when it reaches its asymptotic value fast enough. This is the direct static link they draw to charge scrambling. They also give an equivalent statement in terms of curvature of the truncated symmetry expectation. The claim is explicitly conditional, and they demonstrate necessity with two sets of examples: dephased superfluids that keep Rényi-1 order but have slow tails and only subextensive variance, and sparse fixed-charge projectors that produce extensive variance without local Rényi-1 order or long-range conditional mutual information. They introduce the twist overlap correlator as a practical analogue that works for both discrete and continuous symmetries and decomposes local fluctuations into strong- and weak-symmetry channels, with the weak channel isolating coherent fluctuations and matching the Wigner-Yanase skew information. The derivations follow from the definitions without circularity or hidden fitting parameters. The counterexamples are chosen to isolate the rate condition cleanly. One minor limitation is that the paper does not give quantitative bounds on how fast the approach must be for typical lattice models, so checking the condition in new systems may require extra work. The overall structure is tight and the math checks out on its own terms. This is for readers in condensed-matter statistical mechanics or quantum information who already work with nonlinear symmetry diagnostics and want a clearer separation between order, charge indefiniteness, and coherent fluctuations. It deserves a serious referee.

Referee Report

2 major / 3 minor

Summary. The manuscript shows that for continuous symmetries, a long-range Rényi-1 correlator approaching its nonzero asymptotic value sufficiently rapidly implies an extensive lower bound on block-charge variance (equivalently, extensive curvature of the truncated symmetry expectation). This supplies a static fluctuation signature of charge scrambling in strong-to-weak spontaneous symmetry breaking. Counterexamples establish that the implication is conditional and non-reversible: dephased superfluids retain Rényi-1 SWSSB with subextensive variance when the tail decays too slowly, while sparse fixed-charge projectors exhibit extensive variance without local Rényi-1 order. The authors introduce a twist-overlap correlator that decomposes local charge fluctuations into strong- and weak-symmetry channels and relates the weak channel to the Wigner-Yanase skew information.

Significance. If the central implication holds, the work supplies a precise, falsifiable link between nonlinear SWSSB diagnostics and conserved-charge fluctuations, together with a symmetry-channel decomposition that applies to both continuous and discrete cases. The explicit counterexamples (dephased superfluids and sparse projectors) and the connection to skew information are concrete strengths that clarify distinctions among order, indefiniteness, and coherence. These results should be useful for classifying symmetry-protected phases and for diagnosing scrambling in many-body systems.

major comments (2)

[Section deriving the variance lower bound from the Rényi-1 correlator] The positive implication (rapid Rényi-1 approach forces extensive variance) is stated as conditional on a quantitative rate; the manuscript should give an explicit bound (e.g., exponential decay faster than 1/r^α with α>1) in the theorem statement, because the counterexamples only demonstrate necessity of some rate and do not quantify the threshold.
[Paragraph stating the equivalence for continuous symmetries] The equivalence between the extensive block-charge variance bound and extensive curvature of the truncated symmetry expectation is asserted to follow directly from the definitions for continuous symmetries; an explicit one-line derivation (or reference to the relevant identity) should be inserted to confirm there is no hidden assumption on the support of the charge distribution.

minor comments (3)

[Abstract] The abstract packs three distinct results into its final sentence; splitting the sentence or adding a clause separator would improve readability.
[Counterexample section] In the dephased-superfluid counterexample, state the precise form of the dephasing channel (e.g., local phase damping rate) so that the slow tail can be reproduced independently.
[Definition of the twist overlap correlator] The twist-overlap correlator is defined via an auxiliary twist operator; add a short remark comparing its scaling to the standard Rényi-1 correlator when the symmetry is continuous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive suggestions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section deriving the variance lower bound from the Rényi-1 correlator] The positive implication (rapid Rényi-1 approach forces extensive variance) is stated as conditional on a quantitative rate; the manuscript should give an explicit bound (e.g., exponential decay faster than 1/r^α with α>1) in the theorem statement, because the counterexamples only demonstrate necessity of some rate and do not quantify the threshold.

Authors: We agree that an explicit quantitative rate strengthens the theorem. In the revised manuscript we will state a sufficient condition in the theorem, for example that the deviation of the Rényi-1 correlator from its asymptotic value decays at least as fast as O(r^{-1-ε}) for ε>0 (which is weaker than exponential but stronger than the counterexample tails). This makes the implication precise while preserving the counterexamples as demonstrations that the rate condition is necessary. revision: yes
Referee: [Paragraph stating the equivalence for continuous symmetries] The equivalence between the extensive block-charge variance bound and extensive curvature of the truncated symmetry expectation is asserted to follow directly from the definitions for continuous symmetries; an explicit one-line derivation (or reference to the relevant identity) should be inserted to confirm there is no hidden assumption on the support of the charge distribution.

Authors: We will insert the requested one-line derivation: for a continuous U(1) symmetry generated by the total charge Q, the curvature of the truncated expectation value is ∂²/∂θ² ⟨e^{iθ Q_A}⟩|_{θ=0} = -Var(Q_A), which holds by direct differentiation under the trace (no assumption on the support of the charge distribution is required beyond the existence of the generator). A reference to this standard identity will be added. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a conditional implication: for continuous symmetries, a long-range Rényi-1 correlator that approaches its nonzero value sufficiently rapidly implies an extensive lower bound on block-charge variance (equivalently, extensive curvature in the truncated symmetry expectation). This follows directly from the definitions of the quantities involved, with the rapidity condition stated explicitly as necessary. Counterexamples (dephased superfluids with slow tails and sparse projectors) are constructed to demonstrate that the implication is non-reversible and does not hold without the condition. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the twist overlap correlator is introduced as a new diagnostic that decomposes fluctuations without circular renaming or ansatz smuggling. The central claim remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim relies on the definition of Rényi-1 correlator and the continuous symmetry assumption as background; no free parameters are mentioned.

axioms (1)

domain assumption Systems possess continuous symmetries such as U(1).
The main implication is stated to hold specifically for continuous symmetries.

invented entities (1)

twist overlap correlator no independent evidence
purpose: Analogue of charge variance for discrete and continuous symmetries that decomposes fluctuations into strong- and weak-symmetry channels.
Newly introduced in the paper to relate to Wigner-Yanase skew information.

pith-pipeline@v0.9.0 · 5554 in / 1316 out tokens · 74344 ms · 2026-05-08T15:44:22.357652+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Vanishing R´ enyi-1 Order yet Extensive Subsystem Charge Variance 11
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diagonal

Conditional mutual information: sparse projectors versus the uniform ensemble. 15 D. Structure Factor 18 Appendix A: Doubled Hilbert space For simplicity, here we assumeGis a finite Abelian symmetry group with unitary representationU g on a Hilbert spaceH. SinceGis Abelian,Hdecomposes into charge sectors H= M q∈ bG Hq,(A1) where bGis the character group. ...
[51]

X n Inyn 2 # = 1 r2

Vanishing R´ enyi-1 Order yet Extensive Subsystem Charge Variance In this appendix we show that the converse of Theorem 2 is false: extensive subsystem charge variance does not, by itself, imply SWSSB. The counterexample is a typical sparse classical projector inside a fixed-charge sector. Setup.Fix a filling fractionν, and consider a sequence of systems ...
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fLaj8UvQG/94ZZtesO+I7rJ7k20=

Conditional mutual information: sparse projectors versus the uniform ensemble. Another important diagnostic of SWSSB is the von Neumann conditional mutual information (CMI)I(C:E|B) for a tripartition of the system into a center regionC, a shielding regionB, and an environment regionE: <latexit sha1_base64="fLaj8UvQG/94ZZtesO+I7rJ7k20=">AAAVhnicjZjbbtw2EIa...