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arxiv: 2604.28026 · v2 · pith:Q4G6LRMFnew · submitted 2026-04-30 · ❄️ cond-mat.stat-mech

Order by disorder up to arbitrarily high temperature

Ravish Mehta This is my paper

Pith reviewed 2026-05-19 16:48 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords order by disorderhigh temperature orderingclassical lattice modelscheckerboard orderPirogov-Sinai theoryPeierls boundentropic mechanism
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The pith

A class of classical lattice models exhibits long-range checkerboard order at high temperatures through a purely entropic mechanism.

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

This paper proves that certain lattice models with spins taking non-negative integer values and nearest-neighbor interactions develop long-range checkerboard order even at very high temperatures. The ordering is driven by entropy rather than energy. This matters because it shows that order can survive where thermal disorder is expected to dominate, in dimensions two and higher. The result applies to a class of models including one recently studied by other authors, and is established using contour arguments from Pirogov-Sinai theory.

Core claim

We prove that a class of classical lattice models on Z^d (d ≥ 2) with on-site space N_0 and nearest neighbour interaction exhibits long-range checkerboard order at sufficiently high temperature. The ordering mechanism is purely entropic. The class of models contains the recently introduced model of Han--Huang--Komargodski--Lucas--Popov. The proof uses Pirogov--Sinai theory and the key input is a Peierls bound.

What carries the argument

Pirogov-Sinai theory with a Peierls bound as input, which suppresses the formation of interfaces and stabilizes the checkerboard ordered phase at high temperature.

If this is right

  • Models satisfying the conditions for the Peierls bound will show this high-temperature ordered phase.
  • The specific model of Han et al. is included and thus orders at high temperature.
  • Order by disorder can be realized in classical systems without any energetic preference for the ordered state.

Where Pith is reading between the lines

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

  • If the Peierls bound can be established for other interaction types, the result extends to those cases.
  • Computer simulations could verify the persistence of order by measuring correlations between distant sites at increasing temperatures.
  • This mechanism might inspire searches for analogous entropic ordering in quantum lattice models.

Load-bearing premise

The models must admit a Peierls bound that makes the free energy cost of domain walls high enough to prevent them from proliferating.

What would settle it

In a Monte Carlo simulation of a model in this class, compute the checkerboard order parameter as temperature is increased; the claim is falsified if the order parameter approaches zero.

Figures

Figures reproduced from arXiv: 2604.28026 by Ravish Mehta.

Figure 1
Figure 1. Figure 1: A configuration with two contours γ1 and γ2 with the boundary Bpωq denoted by the dark grey region and the patterned regions representing boundaries of the contours. The outer boundary has the phase η1 and the interior boundary has the same phase as that of the interior components along which they are drawn. 8 view at source ↗
read the original abstract

We prove that a class of classical lattice models on $\mathbb{Z}^d$ ($d \geq 2$) with on-site space $\mathbb{N}_0$ and nearest neighbour interaction, exhibits long-range checkerboard order at sufficiently high temperature. The ordering mechanism is purely entropic. The class of models contains the recently introduced model of Han--Huang--Komargodski--Lucas--Popov (arXiv:2503.22789), by which our work is inspired. The proof uses Pirogov--Sinai theory and the key input is a Peierls bound.

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

1 major / 2 minor

Summary. The manuscript establishes that a class of classical lattice models on Z^d (d ≥ 2) with on-site space N_0 and nearest-neighbor interactions exhibits long-range checkerboard order at arbitrarily high temperatures. The ordering is driven by entropy rather than energy minimization. The proof proceeds by verifying the hypotheses of Pirogov-Sinai theory, with the key quantitative input being a Peierls bound that supplies a temperature-independent contour suppression factor.

Significance. If the central argument holds, the result provides a rigorous example of entropic stabilization of long-range order in the high-temperature regime, extending the model of Han-Huang-Komargodski-Lucas-Popov (arXiv:2503.22789) to a broader class. The use of standard Pirogov-Sinai machinery together with an explicit Peierls bound that remains effective as β → 0 constitutes a technically nontrivial contribution to the literature on order-by-disorder phenomena.

major comments (1)
  1. [§4] §4, Theorem 4.1 and the statement of the Peierls bound (Eq. (4.3)): the claimed lower bound on the contour weight τ must be shown to satisfy τ > log(2d) + o(1) uniformly in β. The current sketch derives the bound from the on-site degeneracy and nearest-neighbor structure, but it is not immediately clear whether the resulting τ remains bounded away from the expansion threshold when the inverse-temperature parameter is taken to zero; an explicit estimate independent of β (or growing with T) is required for the cluster expansion in the Pirogov-Sinai contour model to converge in the high-T regime.
minor comments (2)
  1. [§2.1] The definition of the model class in §2.1 could be stated more explicitly by listing the precise conditions on the interaction function that guarantee the Peierls bound; this would help readers verify membership for new examples.
  2. [§2] Notation for the on-site space N_0 and the checkerboard ground states should be introduced once in §2 and used consistently thereafter to avoid minor confusion in the contour construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for highlighting the need for greater clarity in the derivation of the Peierls bound. We agree that an explicit, β-independent lower bound on τ is essential to confirm convergence of the cluster expansion at arbitrarily high temperatures. We will revise the manuscript to address this point.

read point-by-point responses

  1. Referee: [§4] §4, Theorem 4.1 and the statement of the Peierls bound (Eq. (4.3)): the claimed lower bound on the contour weight τ must be shown to satisfy τ > log(2d) + o(1) uniformly in β. The current sketch derives the bound from the on-site degeneracy and nearest-neighbor structure, but it is not immediately clear whether the resulting τ remains bounded away from the expansion threshold when the inverse-temperature parameter is taken to zero; an explicit estimate independent of β (or growing with T) is required for the cluster expansion in the Pirogov-Sinai contour model to converge in the high-T regime.

    Authors: We agree that the current presentation leaves the uniformity of the bound implicit. The Peierls constant τ arises from an entropic penalty: for a contour separating regions of the two checkerboard phases, the on-site degeneracy (N_0 states) combined with the nearest-neighbor constraint produces a multiplicative suppression factor whose logarithm is bounded below by a positive constant determined solely by the ratio of admissible configurations on the contour versus the bulk. Because the underlying Hamiltonian is bounded and the interaction is nearest-neighbor, the energy contribution to the contour weight is O(β) and vanishes as β → 0. Consequently, liminf_{β→0} τ(β) equals the purely entropic lower bound, which is strictly larger than log(2d) for any fixed d ≥ 2 and any N_0 ≥ 2. In the revised manuscript we will insert a self-contained lemma in §4 that computes this limit explicitly, verifies τ(β) ≥ τ_0 > log(2d) for all β ∈ [0, β_0], and confirms that the o(1) error is uniform in β and vanishes with contour size. This supplies the required input for the Pirogov–Sinai cluster expansion at arbitrarily high temperature. revision: yes

Circularity Check

0 steps flagged

No circularity: mathematical proof via external Pirogov-Sinai theory with independent Peierls bound

full rationale

The derivation is a standard mathematical proof that invokes Pirogov-Sinai theory (an established external framework) and supplies a Peierls bound as the quantitative input. No step reduces the claimed high-temperature order to a fitted parameter, a self-definition, or a load-bearing self-citation; the cited model (Han et al.) is from different authors and serves only as inspiration. The argument remains self-contained against external benchmarks because the contour expansion convergence is controlled by the derived bound rather than by re-labeling the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Pirogov-Sinai theory to the defined class and the existence of a suitable Peierls bound; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Pirogov-Sinai theory can be applied to establish the phase transition in these models
    Explicitly stated as the proof method in the abstract.
  • domain assumption A Peierls bound holds for the models in the class
    Described as the key input to the proof.

pith-pipeline@v0.9.0 · 5611 in / 1534 out tokens · 64461 ms · 2026-05-19T16:48:19.950111+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The ordering mechanism is purely entropic: the checkerboard configurations are not energy minimisers, but are selected by the partial trace over occupation numbers in the β→0 limit. The proof uses Pirogov–Sinai theory and the key input is a Peierls bound.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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