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arxiv: 2606.11561 · v1 · pith:FTQFUHOKnew · submitted 2026-06-10 · 🪐 quant-ph · cond-mat.stat-mech

Diffusive Relaxation of Participation Entropy in U(1)-symmetric Dynamics

Hanchen Liu , Tianci Zhou , Xiao Chen This is my paper

Pith reviewed 2026-06-27 09:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords participation entropyU(1) symmetryhydrodynamic modesdiffusive relaxationdensity correlationsquantum circuitsmany-body dynamicsconservation laws
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The pith

U(1) conservation slows participation entropy relaxation to t to the minus one half via diffusive density correlations.

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

The paper shows that U(1) charge conservation imprints slow hydrodynamic modes onto the relaxation of participation entropy, which measures how spread a many-body wavefunction is across configurations. After local density variations smooth out, the remaining entropy deficit is set by the square of connected density correlations. These correlations spread by diffusion, producing a power-law decay of the deficit as one over square root of time in the hydrodynamic regime. The decay then crosses over to an exponential form governed by system size squared. The relation is checked with exact diagonalization and tensor-network methods on U(1)-conserving circuits.

Core claim

After local density inhomogeneities decay, the leading participation entropy deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving ΔS(t)∼t^{-1/2} in the hydrodynamic regime and crossing over to ∼exp[-O(t/L^2)] when t≥L^2.

What carries the argument

Cluster expansion around equilibrium that isolates the participation entropy deficit as the square of connected density correlations.

If this is right

  • Participation entropy becomes a direct probe of hydrodynamic memory in conserved quantum systems.
  • Slow relaxation of this entropy follows generically from the presence of conservation laws.
  • The functional form crosses from power-law to exponential decay at diffusion time across the system size.
  • The same correlation-squared relation holds across different U(1)-conserving circuit models.

Where Pith is reading between the lines

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

  • Similar hydrodynamic slowing should appear in other spread measures such as entanglement or operator entanglement when conservation laws are present.
  • In higher dimensions the decay exponent would change with the diffusive scaling of density correlations.
  • Adding weak breaking of the U(1) symmetry should restore faster relaxation once the induced length scale is reached.

Load-bearing premise

The cluster expansion around equilibrium stays valid with higher-order terms negligible once local density inhomogeneities have decayed.

What would settle it

A measurement in a U(1)-conserving circuit showing that the participation entropy deficit fails to track the square of connected density correlations or deviates from t to the minus one half decay in the hydrodynamic regime would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2606.11561 by Hanchen Liu, Tianci Zhou, Xiao Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. Entropy density relaxation in the SSEP brickwork [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Brickwork update pattern for the one dimensional [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Local and conservation diagnostics for the infi [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diffusive scaling diagnostics for the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic of the infinite chain iMPS implementation. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. PE relaxation in the Haar random U(1) brickwork [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Local and conservation diagnostics for the infinite [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Schematic tensor contractions used for the two [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Entropy deficit relaxation for the random circuit [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Diffusive correlation diagnostics for the fully random [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Entropy deficit relaxation for dynamics described [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Diffusive correlation diagnostics for the structured [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Finite size exact evolution entropy-deficit collapse [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Infinite chain iMPS entropy-deficit relaxation for [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Diffusive collapse test for the connected charge corre [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
read the original abstract

Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that $\mathrm{U}(1)$ conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving $\Delta S(t)\sim t^{-1/2}$ in the hydrodynamic regime and crossing over to $\sim \exp[-O(t/L^2)]$ when $t\geq L^2$. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum $\mathrm{U}(1)$ conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.

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 claims that U(1) conservation laws slow the relaxation of participation entropy (PE) by coupling it to hydrodynamic modes. After local density inhomogeneities decay, a cluster expansion around equilibrium shows that the leading PE deficit ΔS is dominated by the square of connected two-point density correlations. This implies diffusive relaxation ΔS(t)∼t^{-1/2} in the hydrodynamic regime, crossing over to ∼exp[-O(t/L²)] for t≥L². The relation is confirmed via exact diagonalization and infinite-system tensor-network simulations on U(1)-conserving quantum circuits.

Significance. If the central entropy-correlation relation holds, the work establishes PE as a sensitive probe of hydrodynamic memory and shows that slow relaxation is a generic consequence of conservation laws. Credit is due for deriving the relation via an independent cluster expansion (rather than fitting) and for providing numerical confirmation across multiple circuit families and system sizes.

major comments (2)
  1. [cluster expansion around equilibrium] The central claim rests on truncating the cluster expansion to the two-point level after local density relaxation. Because the connected two-point density correlator itself decays only as t^{-d/2} in the diffusive regime, it is not obvious that three-point and higher connected cumulants remain parametrically smaller; an explicit scaling bound or order-of-magnitude estimate justifying the neglect of these terms is required (see the derivation of the PE deficit in terms of density correlations).
  2. [numerical confirmation] The hydrodynamic scaling ΔS(t)∼t^{-1/2} and the subsequent finite-size crossover are load-bearing predictions. The numerical evidence must demonstrate that the observed power law is not contaminated by residual local-density relaxation or by higher-order cumulants; quantitative comparison of the prefactor extracted from the two-point function versus the measured ΔS(t) would strengthen the result.
minor comments (2)
  1. Notation for the participation entropy deficit ΔS and the connected density correlator should be introduced with a single consistent definition early in the text to avoid ambiguity when the cluster expansion is applied.
  2. Figure captions for the tensor-network data should explicitly state the bond dimension, truncation error, and how the infinite-system limit is taken.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [cluster expansion around equilibrium] The central claim rests on truncating the cluster expansion to the two-point level after local density relaxation. Because the connected two-point density correlator itself decays only as t^{-d/2} in the diffusive regime, it is not obvious that three-point and higher connected cumulants remain parametrically smaller; an explicit scaling bound or order-of-magnitude estimate justifying the neglect of these terms is required (see the derivation of the PE deficit in terms of density correlations).

    Authors: We thank the referee for highlighting this important point regarding the validity of the truncation. In the cluster expansion of the participation entropy deficit, after local density relaxation the two-point term is leading because higher-order connected cumulants enter as products involving three or more correlation functions. Each additional factor decays as t^{-d/2} (with spatial support growing as t^{d/2}), so that the three-point contribution scales as t^{-3d/2} or faster while the two-point squared term yields the observed t^{-d/2} after spatial summation. We will add an explicit order-of-magnitude estimate and scaling bound in a revised section on the cluster expansion to make this rigorous. revision: yes

  2. Referee: [numerical confirmation] The hydrodynamic scaling ΔS(t)∼t^{-1/2} and the subsequent finite-size crossover are load-bearing predictions. The numerical evidence must demonstrate that the observed power law is not contaminated by residual local-density relaxation or by higher-order cumulants; quantitative comparison of the prefactor extracted from the two-point function versus the measured ΔS(t) would strengthen the result.

    Authors: We agree that a quantitative prefactor comparison would strengthen the evidence. The existing numerics (exact diagonalization and tensor-network simulations) already confirm the scaling and crossover across multiple circuit families, with local density relaxation occurring at much earlier times than the hydrodynamic regime. In the revised manuscript we will add a direct comparison of the prefactor obtained from the independently computed two-point density correlator against the prefactor fitted to ΔS(t), together with explicit checks that residual density inhomogeneities and higher-cumulant contributions remain negligible in the fitting window. revision: yes

Circularity Check

0 steps flagged

No circularity: central relation derived via independent cluster expansion around equilibrium

full rationale

The paper derives the leading PE deficit relation to squared connected density correlations explicitly from a cluster expansion around equilibrium after local density relaxation. This is a theoretical expansion step, not a fit, redefinition, or self-citation chain. The result is then checked against independent exact diagonalization and tensor-network simulations on U(1)-conserving circuits. No load-bearing self-citations, ansatze smuggled via prior work, or fitted inputs renamed as predictions appear in the provided text. The truncation of the cluster expansion is a modeling assumption whose validity can be tested externally, but it does not reduce the claimed relation to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the cluster expansion around equilibrium and the assumption that the squared connected density correlation term dominates the PE deficit after local density decay; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Cluster expansion around equilibrium accurately isolates the leading PE deficit as the square of connected density correlations after local density decay
    Invoked to derive the hydrodynamic scaling from the correlation function.

pith-pipeline@v0.9.1-grok · 5692 in / 1269 out tokens · 20669 ms · 2026-06-27T09:55:51.889573+00:00 · methodology

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Forward citations

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Reference graph

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