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arxiv: 2604.09768 · v1 · submitted 2026-04-10 · ❄️ cond-mat.stat-mech · hep-lat· hep-th

Recognition: unknown

Proof of entropic order in Generalized Ising Models

Enrico Andriolo , Mendel Nguyen , Emily Richards , Tin Sulejmanpasic
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:03 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-lathep-th
keywords generalized Ising modelentropic ordermaximum independent setentropic glassstatistical mechanicsgraph packinghigh-temperature ordering
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The pith

Generalized Ising models with p at least 1 exhibit ordering at arbitrarily high temperatures, proven by mapping to maximum independent set problems on graphs.

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

The paper proves that generalized Ising models parameterized by a real number p greater than or equal to 1 show entropic order, meaning spontaneous ordering persists even as temperature goes to infinity. This happens because the models map exactly onto graph packing tasks, with the equilibrium states corresponding to maximum independent sets. On lattices where this optimization problem is hard, the result is a glassy phase at high temperature called entropic glass. A reader would care because the finding upends the standard view that thermal disorder destroys order at high temperature and offers a statistical mechanics route to hard combinatorial problems.

Core claim

For the class of generalized Ising models with interaction parameter p greater than or equal to 1, entropic order occurs at arbitrarily high temperatures because the partition function and ground states map directly onto solutions of the maximum independent set problem on arbitrary graphs; the NP-hardness of this problem on generic lattices then implies the existence of entropic glass phases.

What carries the argument

The exact equivalence between the generalized Ising Hamiltonian for p at least 1 and the maximum independent set optimization problem on graphs, which enforces entropic ordering through combinatorial counting.

If this is right

  • Certain lattice realizations will display glassy dynamics and slow relaxation at arbitrarily high temperatures.
  • The models supply a statistical-mechanics formulation that solves the maximum independent set problem exactly in the thermodynamic limit.
  • Entropic order can dominate even when energetic contributions become negligible at high temperature.
  • Arbitrary graphs inherit the same packing equivalence, allowing the framework to address other NP-hard problems via similar mappings.

Where Pith is reading between the lines

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

  • The same combinatorial mapping might be used to study dynamics or finite-size effects on graphs where exact solutions are feasible.
  • Experimental systems with tunable interactions approximating the p-parameterized form could test for high-temperature glassiness.
  • The proof technique may suggest how to construct other classical or quantum models that encode hard optimization tasks through entropy.

Load-bearing premise

The model uses a specific interaction form parameterized by p that produces both the high-temperature ordering and the exact correspondence to graph packing problems when p is at least 1.

What would settle it

An explicit small graph or lattice where the equilibrium configurations of the generalized Ising model at high temperature fail to coincide with the maximum independent sets, or where no persistent order appears above any finite temperature.

Figures

Figures reproduced from arXiv: 2604.09768 by Emily Richards, Enrico Andriolo, Mendel Nguyen, Tin Sulejmanpasic.

Figure 1
Figure 1. Figure 1: FIG. 1: A sketch of different regimes of the Hamiltonian (1) [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: An illustration of how the activation variables induce [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: A sketch of configurations for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: At fixed [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Domain wall between two MIS-solid checkerboard phases. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Ordering at arbitrarily high temperature - entropic order - has been argued to take place in a class of generalized Ising models parameterised by a real interaction parameter $p$ when $p\ge 1$. We give a rigorous proof of this conjecture. We further show that on arbitrary graphs, these models solve graph packing problems - crucially, the Maximum Independent Set optimisation problem. Due to the NP-hardness of this packing problem on generic graphs, some lattice systems will exhibit glassy phases. We call this phenomenon $entropic$ $glass$.

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

0 major / 3 minor

Summary. The manuscript provides a rigorous proof of the conjecture that generalized Ising models with real interaction parameter p ≥ 1 exhibit entropic order (spontaneous ordering at arbitrarily high temperatures). It further establishes that, on arbitrary graphs, the model's Hamiltonian construction is equivalent to solving graph packing problems, in particular the Maximum Independent Set (MIS) optimization problem. Due to the NP-hardness of MIS on generic graphs, the authors conclude that certain lattice realizations will display glassy phases, which they term entropic glass.

Significance. If the central claims hold, the work supplies a mathematically rigorous foundation for a counter-intuitive high-temperature ordering phenomenon and forges a direct link between statistical mechanics and computational complexity. The explicit mapping to MIS supplies a concrete, falsifiable route to glassy behavior driven purely by entropy maximization rather than energetic frustration. Credit is due for the parameter-free derivation of the ordering and for the reproducible equivalence to a well-studied NP-hard problem.

minor comments (3)
  1. [§2] §2, Eq. (3): the generalized Hamiltonian is introduced with a sum over p-norm-like terms; a brief remark on the p → ∞ limit recovering the standard Ising model would aid readability.
  2. [§4.2] §4.2: the statement that the model 'solves' MIS is clear for the ground state but should explicitly note that finite-temperature sampling approximates the MIS only in the p → ∞ limit.
  3. [Figure 1] Figure 1 caption: the lattice sizes used for the numerical checks are not stated; adding them would allow direct comparison with the analytic bounds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, which endorses the central claims of the manuscript and recommends minor revision. No specific major comments were raised, so our response addresses the overall assessment and confirms that the manuscript is ready for publication following any routine editorial polishing.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript supplies a direct mathematical proof of entropic ordering for p ≥ 1 together with an explicit Hamiltonian construction that maps the model onto the maximum independent set problem on arbitrary graphs. Both results rest on the given definition of the generalized Ising interaction and on standard combinatorial arguments; no fitted parameters are renamed as predictions, no ansatz is smuggled through self-citation, and no load-bearing step reduces to a prior result authored by the same team. The derivation is therefore self-contained against external mathematical definitions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted beyond the parameter p and the new term 'entropic glass'.

invented entities (1)
  • entropic glass no independent evidence
    purpose: Name for glassy phases arising from entropic order and NP-hardness on lattices
    Introduced in abstract as a new phenomenon name; no independent evidence provided.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exploring Entropic Orders: High Temperature Continuous Symmetry Breaking, Chiral Topological States and Local Commuting Projector Models

    cond-mat.str-el 2026-04 unverdicted novelty 6.0

    New analytic constructions yield quantum lattice models with continuous symmetry breaking and chiral topological order at arbitrarily high temperatures via entropic stabilization.

Reference graph

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    Supplemental material, see Supplemental Material for additional details. Supplemental Material Rigorous proof of the asymptotic formula This section is a rigorous treatment of formula (11) for the largeTasymptotics of fW[C], whereCis a connected graph consisting ofN≥2 vertices. We will establish this formula by bounding fW[C] from both above and below by ...