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

TL;DR

This paper proves the existence of a phase transition in a one-dimensional long-range Ising spin glass model on the Nishimori line for certain interaction decay rates, extending Dyson's classical result.

Contribution

It extends Dyson's phase transition proof to the Ising spin glass with Gaussian disorder on the Nishimori line for $1<\alpha<3/2$, using advanced inequalities and interpolation methods.

Findings

Proves long-range order at low temperatures for $1<\alpha<3/2$

Establishes phase transition existence on the Nishimori line for the specified range of $\alpha$

Open problem remains for $\alpha \ge 3/2$.

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.