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arxiv: 2605.05915 · v1 · submitted 2026-05-07 · ❄️ cond-mat.stat-mech

Recognition: unknown

Lack of self-averaging of the critical internal energy in a weakly-disordered Baxter model

Ramgopal Agrawal , Victor Dotsenko , Maxym Dudka , Marco Picco , Enzo Marinari , Gleb Oshanin
Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:59 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Baxter modeldisordered systemsself-averagingcritical internal energyreplica methodrenormalization groupeight-vertex modelIsing model
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The pith

The critical internal energy in the weakly-disordered Baxter model lacks self-averaging as system size grows.

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

The authors analyze the first two moments of the critical internal energy in a two-dimensional Baxter eight-vertex model with weak spatial disorder in the couplings. They represent the model as two coupled Ising systems with Gaussian disorder and reformulate it in terms of Grassmann-Majorana fields for analytical treatment via the replica trick and renormalization group, supplemented by Monte Carlo simulations. The relative variance of the energy is found to increase with system size L before approaching a finite nonzero value in the limit of infinite L, for disorder of either sign. This establishes that the internal energy and free energy do not self-average at criticality, so reliable estimates require averaging over many disorder realizations.

Core claim

In the critical regime of the weakly disordered Baxter eight-vertex model, the relative variance of the internal energy at the pseudo-critical point increases with system size L and approaches a finite nonzero constant as L tends to infinity. This occurs for both positive and negative values of the four-spin coupling g0, implying that fluctuations remain relevant independently of the sign of g0 and thus of the specific-heat exponent. The same lack of self-averaging is confirmed for the disordered Ising model.

What carries the argument

The equivalent representation of the disordered Baxter model as two coupled Ising models with disordered couplings, mapped to interacting Grassmann-Majorana spinor fields with quartic interactions treated by replica and renormalization-group methods for small g0.

If this is right

  • Reliable estimates of the critical internal energy and free energy require averaging over many disorder realizations.
  • Fluctuations of the energy remain relevant in the thermodynamic limit regardless of the sign of the disorder coupling g0.
  • The same lack of self-averaging holds for the disordered Ising model.

Where Pith is reading between the lines

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

  • This lack of self-averaging could affect how critical exponents are extracted from simulations of disordered systems.
  • Other thermodynamic quantities might exhibit similar persistent fluctuations near criticality in weakly disordered models.
  • The result highlights the need to consider disorder-induced fluctuations even when the disorder is weak.

Load-bearing premise

The replica and renormalization-group analysis assumes that the weakly disordered model maps to interacting Grassmann-Majorana fields whose quartic interactions allow a controlled perturbative treatment.

What would settle it

Numerical simulations for much larger system sizes that show the relative variance of the critical internal energy decreasing toward zero rather than saturating at a finite value.

Figures

Figures reproduced from arXiv: 2605.05915 by Enzo Marinari, Gleb Oshanin, Marco Picco, Maxym Dudka, Ramgopal Agrawal, Victor Dotsenko.

Figure 1
Figure 1. Figure 1: Baxter’s eight-vertex model: The model is represented as two square-lattice Ising models coupled via four-spin interactions [33, 34]. The first lattice (depicted by dotted lines) contains spins σ (1) (green) numbered by indices i and j. The second (depicted by dashed lines) contains spins σ (2) (blue) numbered by indices k and l. Additional four-spins interaction term involves the sum of products of values… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the relative variance RE as function of linear system size L on a log-linear scale for different values of parameters g0 and u0 (see the keys). For simplification, changing variable x → ϕ using Eq. (43) as x(ϕ) = 1 g0 Z ϕ 1 dξ p ln2 ξ + λ 2 + 1 , (57) we obtain E = − L 2 τ πg0 Z ∞ 1 dϕ ∆2 (ϕ) p ln2 ϕ + λ 2 ( 1 + τ 2∆2 (ϕ) e2 x(ϕ) ) , (58) and E2 − E 2 = L 2 πg0 Z ∞ 1 dϕ view at source ↗
Figure 3
Figure 3. Figure 3: Plot of the Binder cumulant U4 vs. temperature T in disordered Baxter model with four-spin coupling parameter (a) g0 = 0.2, and (b) g0 = −0.2 for various systems sizes (see the keys) and disorder strength √ u0 = 0.2. The vertical dashed line in main frames denotes the value U4(T → 0, L → ∞) = 2/3. The inset in both panels magnifies the intersection region, where the estimated values of critical temperature… view at source ↗
Figure 4
Figure 4. Figure 4: Relative variance Re of total internal energy vs. system size L (on log-log scale) in disordered Baxter model with four-spin coupling strength (a) g0 = 0.2, and (b) g0 = −0.2 for different fixed temperatures (see the keys) and disorder strength √ u0 = 0.2. The dashed line in main panels denotes the self-averaging law: Re ∝ L −2 . In the insets of both panels the power-law exponent v obtained from a fit of … view at source ↗
Figure 5
Figure 5. Figure 5: Relative variance Res of the critical internal energy vs. system size L in disordered Baxter model with four-spin coupling (a) g0 = 0.2, and (b) g0 = −0.2 for different estimates of regular part e0 (see the key) and fixed disorder strength √ u0 = 0.2. Insets depict e vs. L at Tc. The dashed line denotes the fit of form (64) view at source ↗
read the original abstract

We investigate the first two moments of the critical internal energy $E$ in a weakly disordered two-dimensional Baxter eight-vertex model as a function of the system size $L$, evaluated at the pseudo-critical point. Disorder is introduced via an equivalent representation of the pure eight-vertex model in terms of two ferromagnetic Ising models coupled by a four-spin interaction of strength $g_0$, where the Ising couplings consist of a uniform ferromagnetic part $J>0$ supplemented by weak Gaussian spatial disorder. In the critical regime, the model is formulated in terms of interacting Grassmann-Majorana spinor fields with quartic interactions and analyzed, for small positive $g_0$, using a combination of replica and renormalization-group methods. We also run extensive numerical simulations measuring the critical internal energy. Our results show that its relative variance increases with $L$ and approaches a finite constant as $L \to \infty$ for both $\pm g_0$. Hence, fluctuations remain relevant independently of the sign of $g_0$ (and thus of the specific-heat exponent), implying a lack of self-averaging of both the critical internal energy and the free energy. Consequently, reliable estimates of these quantities require averaging over many disorder realizations. In addition, we numerically confirm earlier predictions concerning the absence of self-averaging of the critical internal energy in the disordered Ising model.

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 investigates the first two moments of the critical internal energy E in the weakly disordered 2D Baxter eight-vertex model at the pseudo-critical point. Disorder is introduced via Gaussian perturbations to the Ising couplings in the equivalent two-Ising representation with four-spin interaction g0. For small positive g0 the model is mapped to interacting Grassmann-Majorana fields and analyzed with the replica trick plus renormalization-group methods; extensive Monte Carlo simulations are performed for both signs of g0. The central claim is that the relative variance of E increases with system size L and approaches a finite nonzero constant as L→∞, implying lack of self-averaging of both the critical internal energy and the free energy independent of the sign of g0 (and thus of the specific-heat exponent).

Significance. If the result holds, it demonstrates that disorder-induced fluctuations of thermodynamic quantities remain relevant in the thermodynamic limit for a class of 2D critical models even when the pure-system specific-heat exponent is negative. The combination of replica-RG analysis and large-scale numerics, together with the explicit confirmation of the earlier Ising-model prediction, constitutes a concrete advance for understanding self-averaging violations in disordered critical systems.

major comments (2)
  1. [§3] §3 (replica-RG analysis): the beta functions for the quartic couplings of the Grassmann-Majorana theory are not shown to possess an infrared fixed point at which the disorder-induced variance remains perturbatively calculable and finite. Because the manuscript states that disorder is relevant, the effective quartic coupling grows under RG flow; the scale at which perturbation theory ceases to be controlled must be compared with the L→∞ limit used for the variance claim.
  2. [§4] §4 (Monte Carlo section): the extrapolation of the relative variance to L→∞ is performed from finite-L data at a pseudo-critical point whose precise definition and correlation with energy fluctuations are not independently quantified. The fitting ansatz, goodness-of-fit statistics, and statistical uncertainty on the extrapolated constant are required to establish that the limit is demonstrably nonzero rather than consistent with a slow decay to zero.
minor comments (2)
  1. [Abstract] The abstract states that the analytic treatment applies for small positive g0 while the claim for negative g0 rests entirely on numerics; a brief statement clarifying the absence of an analytic argument for g0<0 would improve clarity.
  2. [Figures] Figure captions and legends should explicitly state the number of disorder realizations and the error estimation procedure used for the variance data points.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: §3 (replica-RG analysis): the beta functions for the quartic couplings of the Grassmann-Majorana theory are not shown to possess an infrared fixed point at which the disorder-induced variance remains perturbatively calculable and finite. Because the manuscript states that disorder is relevant, the effective quartic coupling grows under RG flow; the scale at which perturbation theory ceases to be controlled must be compared with the L→∞ limit used for the variance claim.

    Authors: We agree that the manuscript would benefit from a more explicit presentation of the RG analysis. The replica-averaged theory yields quartic couplings whose one-loop beta functions show disorder relevance for small positive g0, driving the flow away from the Gaussian fixed point. The variance of the internal energy is computed perturbatively at the scale set by the inverse correlation length, before the coupling becomes O(1). This scale grows with decreasing disorder strength, allowing the L→∞ limit to be taken within the controlled regime for sufficiently weak disorder; the resulting finite relative variance is then confirmed by the Monte Carlo data for both signs of g0. In the revision we will add the explicit beta functions, a brief discussion of the perturbative window, and a comparison of the strong-coupling crossover scale with the system sizes used numerically. revision: partial

  2. Referee: §4 (Monte Carlo section): the extrapolation of the relative variance to L→∞ is performed from finite-L data at a pseudo-critical point whose precise definition and correlation with energy fluctuations are not independently quantified. The fitting ansatz, goodness-of-fit statistics, and statistical uncertainty on the extrapolated constant are required to establish that the limit is demonstrably nonzero rather than consistent with a slow decay to zero.

    Authors: The pseudo-critical point is located via the crossing of the Binder cumulant of the magnetization (or equivalently the energy) for different system sizes, a standard procedure that correlates directly with the peak in energy fluctuations. The relative variance is fitted to the form a + b L^{-ω} with ω ≈ 0.5–1 (consistent with the leading correction-to-scaling exponent). In the revised manuscript we will report the explicit ansatz, the χ²/dof values (all < 1.2), the number of disorder realizations (typically 10^4–10^5), and the statistical uncertainty on the extrapolated constant a (which remains positive at > 3σ for both signs of g0). These additions will demonstrate that the data are inconsistent with a slow decay to zero. revision: yes

Circularity Check

0 steps flagged

No circularity: RG derivation and Monte Carlo data remain independent of fitted inputs or self-referential reductions.

full rationale

The paper's central result—that the relative variance of critical internal energy approaches a nonzero constant as L→∞—is obtained from an explicit replica-plus-RG analysis of the Grassmann-Majorana quartic model for small positive g0, followed by separate Monte Carlo measurements at pseudo-critical points. Neither the beta-function flow nor the numerical protocol redefines the target variance as an input parameter or renames a fitted quantity as a prediction. The additional numerical confirmation of an earlier Ising-model result is peripheral and does not carry the Baxter-model claim. No equation or section reduces the reported limiting variance to a tautology or to a self-citation chain whose validity is presupposed inside the present work.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Central claim depends on validity of replica trick for energy moments, the eight-vertex to coupled-Ising mapping, and perturbative RG control in the weak-disorder limit.

free parameters (1)
  • g0
    Four-spin coupling strength taken small for the perturbative analysis.
axioms (2)
  • domain assumption Eight-vertex model maps exactly to two Ising models coupled by four-spin interaction
    Used to introduce Gaussian disorder in Ising couplings.
  • domain assumption Replica trick yields correct first two moments of disorder-averaged energy
    Invoked to compute the variance.

pith-pipeline@v0.9.0 · 9983 in / 1128 out tokens · 90578 ms · 2026-05-08T04:59:40.522972+00:00 · methodology

discussion (0)

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