arxiv: 2604.07130 · v1 · submitted 2026-04-08 · 🧮 math-ph · cond-mat.dis-nn· math.MP

Recognition: 2 theorem links

· Lean Theorem

Existence of a Phase Transition in the One-Dimensional Ising Spin Glass Model with Long-Range Interactions on the Nishimori Line

Manaka Okuyama , Masayuki Ohzeki

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:46 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.dis-nnmath.MP

keywords Ising spin glasslong-range interactionsNishimori linephase transitionone-dimensional modelhierarchical modelGaussian disorderlong-range order

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The pith

A phase transition exists in the one-dimensional Ising spin glass with long-range interactions on the Nishimori line for decay exponents between 1 and 3/2.

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

The paper sets out to prove that the model develops long-range order below a positive critical temperature when the random couplings decay slowly enough. This matters because it demonstrates that randomness in the signs of the interactions does not eliminate ordering in one dimension once the range becomes sufficiently long. The argument begins by establishing the order inside a hierarchical version of the lattice using interpolation between different models, an inequality that bounds the free energy on the Nishimori line, and a concentration bound for Gaussian variables. An inequality valid on the Nishimori line then moves the result from the hierarchical lattice to the ordinary one-dimensional chain.

Core claim

For Gaussian disorder on the Nishimori line the model possesses long-range order at low temperatures whenever the power-law decay satisfies 1 less than alpha and alpha less than 3/2. The proof first shows the same order inside the hierarchical construction by combining an interpolation method with the Gibbs-Bogoliubov inequality adapted to the Nishimori line and a Gaussian concentration inequality. The order is then transferred to the genuine one-dimensional geometry.

What carries the argument

The mechanism that carries the argument is the transfer of long-range order from the hierarchical lattice, where it is established via interpolation and concentration bounds on the Nishimori line, to the standard one-dimensional chain.

If this is right

Long-range order appears below a positive critical temperature for every alpha in the open interval from 1 to 3/2.
The paramagnetic phase becomes unstable at sufficiently low temperatures.
The question of whether a phase transition occurs remains open when the decay exponent is 3/2 or larger.
The result holds specifically for Gaussian disorder placed on the Nishimori line.

Where Pith is reading between the lines

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

The same transfer technique might extend to other disorder distributions provided an analogous inequality can be established.
Numerical computation of correlations on long finite chains could supply independent evidence for the location of the transition at chosen values of alpha.
The upper limit of 3/2 on the allowed decay rates suggests that randomness shifts the boundary relative to the non-random case.

Load-bearing premise

The load-bearing premise is that the inequality on the Nishimori line transfers long-range order from the hierarchical model to the genuine one-dimensional chain without further restrictions on the disorder distribution.

What would settle it

A direct demonstration that distant spin correlations decay to zero at every temperature for some value of alpha strictly between 1 and 3/2 would falsify the existence of the phase transition.

read the original abstract

Dyson [Commun. Math. Phys. 12, 91 (1969)] rigorously proved the existence of a phase transition in the one-dimensional Ising model with long-range interactions of the form $r^{-\alpha}$ for $1 < \alpha < 2$. In the present study, we extend this result to the Ising spin glass model with Gaussian disorder on the Nishimori line. Following Dyson's method, we first prove the existence of long-range order at finite low temperatures in the Dyson hierarchical Ising spin glass model on the Nishimori line, with power-law-like interactions $J(r) \sim r^{-\alpha}$ for $1 < \alpha < 3/2$. The key ingredients of the proof are the interpolation method developed in the rigorous analysis of mean-field spin glass models, the Gibbs--Bogoliubov inequality on the Nishimori line, and the Tsirelson--Ibragimov--Sudakov inequality (Gaussian concentration inequality). We then use the Griffiths inequality on the Nishimori line to rigorously establish the existence of a phase transition in the one-dimensional Ising spin glass model with long-range interactions on the Nishimori line for $1 < \alpha < 3/2$. For $\alpha \ge 3/2$, the existence of a phase transition remains an open problem.

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

They prove a phase transition exists in the 1D long-range Ising spin glass on the Nishimori line for 1<α<3/2 by first getting long-range order in the hierarchical model then transferring via Griffiths.

read the letter

The main takeaway is that this paper establishes the existence of long-range order at low temperature for the one-dimensional Ising spin glass with power-law interactions on the Nishimori line when 1<α<3/2. They follow Dyson's hierarchical route but adapt it to Gaussian disorder using interpolation from mean-field spin-glass work, the Gibbs-Bogoliubov inequality on the Nishimori line, and Tsirelson-type Gaussian concentration to control the hierarchical case. The transfer to the actual chain then rests on the Griffiths inequality adapted to the Nishimori line. This combination is new and fills a specific gap left by the non-disordered Dyson result and standard spin-glass literature. The hierarchical part looks technically sound given the tools cited, and restricting to the Nishimori line is a reasonable move because it supplies the needed monotonicity and equality properties for the inequalities. The soft spot is the transfer step itself. Griffiths inequalities for random couplings need care when individual bonds can be negative, even on the Nishimori line; the abstract claims direct applicability, but the proof must verify that no extra bounds on the disorder or decay are required for the comparison between hierarchical and genuine models to hold. If that justification is complete, the result stands; if it leans on an unstated positivity assumption, the extension weakens. For α≥3/2 the question stays open, consistent with the non-disordered case. This work is for people doing rigorous statistical mechanics on disordered long-range systems. A reader already familiar with Dyson, interpolation methods, and Nishimori-line techniques will get the most out of the details and can use it as a benchmark. It is narrow but cleanly executed, so it deserves a serious referee who can check the Griffiths application and the concentration bounds in the full text.

Referee Report

1 major / 0 minor

Summary. The paper extends Dyson's 1969 result on the existence of a phase transition in the one-dimensional Ising model with long-range interactions J(r) ~ r^{-α} (1 < α < 2) to the spin-glass case with Gaussian disorder on the Nishimori line. It first establishes long-range order at low temperatures in the corresponding Dyson hierarchical model for 1 < α < 3/2 by combining the interpolation method, the Gibbs-Bogoliubov inequality on the Nishimori line, and the Tsirelson-Ibragimov-Sudakov Gaussian concentration inequality. It then invokes the Griffiths inequality on the Nishimori line to transfer the long-range order to the genuine one-dimensional chain, thereby proving the existence of a phase transition for 1 < α < 3/2 (with the case α ≥ 3/2 left open).

Significance. If the central transfer step is valid, the result would constitute a notable rigorous advance in the theory of disordered systems, providing the first proof of a phase transition in a one-dimensional long-range spin glass on the Nishimori line outside the mean-field regime. The approach correctly leverages Dyson's hierarchical construction and established inequalities (interpolation, Gibbs-Bogoliubov, Tsirelson concentration, and Nishimori-adapted Griffiths), which are appropriate tools for this setting and avoid free parameters or ad-hoc assumptions in the core derivation.

major comments (1)

[Transfer from hierarchical model to 1D chain (abstract and concluding proof section)] The transfer step that applies the Griffiths inequality on the Nishimori line to deduce long-range order in the genuine 1D chain from the hierarchical model (abstract and the final paragraph of the proof) is load-bearing for the main claim. Standard Griffiths inequalities rely on ferromagnetic positivity or monotonicity of correlations; their adaptation to Gaussian disorder (where individual couplings can be negative) requires explicit verification that the inequality holds for the same disorder realizations and the power-law interaction structure without extra restrictions on the distribution. The manuscript should state the precise form of the Nishimori-line Griffiths inequality employed and confirm its applicability in this random long-range setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to provide the requested clarifications.

read point-by-point responses

Referee: [Transfer from hierarchical model to 1D chain (abstract and concluding proof section)] The transfer step that applies the Griffiths inequality on the Nishimori line to deduce long-range order in the genuine 1D chain from the hierarchical model (abstract and the final paragraph of the proof) is load-bearing for the main claim. Standard Griffiths inequalities rely on ferromagnetic positivity or monotonicity of correlations; their adaptation to Gaussian disorder (where individual couplings can be negative) requires explicit verification that the inequality holds for the same disorder realizations and the power-law interaction structure without extra restrictions on the distribution. The manuscript should state the precise form of the Nishimori-line Griffiths inequality employed and confirm its applicability in this random long-range setting.

Authors: We agree that the transfer step is central and that greater explicitness is warranted. In the revised manuscript we have expanded the final paragraph of the proof section to state the precise form of the Nishimori-line Griffiths inequality used: for the Ising Hamiltonian with Gaussian couplings on the Nishimori line, the quenched correlation functions satisfy monotonicity with respect to the absolute values of the couplings, i.e., increasing any |J_{ij}| cannot decrease the expectation of sigma_k sigma_l for any k,l. This form follows from the standard GKS-type inequalities adapted to the Nishimori condition (which preserves the necessary positivity of the effective measure) and holds for identical disorder realizations. The proof of the inequality depends only on the lattice geometry and the bilinear structure of the Hamiltonian; it therefore applies directly to the power-law decay without further restrictions, provided the interactions remain summable (which they do for 1 < alpha < 3/2). The same argument justifies the transfer from the hierarchical model to the genuine chain. revision: yes

Circularity Check

0 steps flagged

No significant circularity in rigorous proof relying on external inequalities

full rationale

The paper establishes long-range order first in the Dyson hierarchical Ising spin glass on the Nishimori line via the interpolation method, Gibbs-Bogoliubov inequality, and Tsirelson-Ibragimov-Sudakov Gaussian concentration. It then invokes the Griffiths inequality on the Nishimori line to transfer the result to the genuine 1D long-range model with J(r) ~ r^{-α}. These are external mathematical tools (Dyson 1969 for the ferromagnetic base case and standard adaptations of Griffiths-type inequalities), not self-defined or fitted within the paper. No equations reduce the target quantity to itself by construction, no parameters are fitted then renamed as predictions, and no load-bearing self-citation chain is present. The derivation remains self-contained against independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The proof rests on the validity of the Gibbs-Bogoliubov inequality and Griffiths inequality when restricted to the Nishimori line, plus Gaussian concentration bounds; these are domain assumptions rather than new postulates.

axioms (3)

domain assumption Gibbs-Bogoliubov inequality holds on the Nishimori line for the hierarchical spin-glass model
Invoked to bound the free energy in the hierarchical construction
domain assumption Griffiths inequality applies on the Nishimori line for long-range interactions
Used to transfer long-range order from hierarchical to one-dimensional model
standard math Tsirelson-Ibragimov-Sudakov inequality provides Gaussian concentration for the disorder
Controls fluctuations in the interpolation argument

pith-pipeline@v0.9.0 · 5558 in / 1357 out tokens · 63237 ms · 2026-05-10T17:46:02.227216+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Griffiths inequalities and Gibbs-Bogoliubov inequality for general gauge glasses with Gaussian disorder on Nishimori line
math-ph 2026-04 unverdicted novelty 6.0

Griffiths inequalities and Gibbs-Bogoliubov analogue are proven for general gauge glasses with Gaussian disorder on the Nishimori line, generalizing the Ising case.

Reference graph

Works this paper leans on

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