On the Replica Symmetry of a Variant of the Sherrington-Kirkpatrick Spin Glass

TL;DR

This paper investigates a modified Sherrington-Kirkpatrick spin glass model with a quartic correction, demonstrating that the conjectured high-temperature phase boundary does not accurately describe the model's behavior.

Contribution

It proves that the proposed replica symmetry boundary condition does not hold for certain magnetization values, challenging previous conjectures.

Findings

The limiting free energy is negative for certain parameters satisfying the conjectured boundary.

The second moment method fails to establish replica symmetry in the full high temperature region.

The high temperature phase boundary condition is not universally valid for the modified model.

Abstract

We consider $N$ i.i.d. Ising spins with mean $m\in (-1,1)$ whose interactions are described by a Sherrington-Kirkpatrick Hamiltonian with a quartic correction. This model was recently introduced by Bolthausen in \cite{Bolt2} as a toy model to understand whether a second moment argument can be used to derive the replica symmetric formula in the full high temperature regime if $m\neq 0$. In \cite{Bolt2}, Bolthausen suggested that a natural analogue of the de Almeida-Thouless condition for the toy model is \begin{equation}\label{eq:conj} \beta^2(1-m^2)^2\leq 1. \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (1)\end{equation} Here, $\beta \geq 0$ corresponds to the inverse temperature. While the second moment method implies replica symmetry for $\beta $ sufficiently small, Bolthausen showed that the method fails to prove replica symmetry in the full region described by (1). A natural question that was left open in \cite{Bolt2} is whether (1) correctly characterizes the high temperature phase of the toy model. In this note, we show that this is indeed not the case. We prove that if $|m| \geq m_*$, for some $m_* \in (0,1)$, the limiting free energy of the toy model is negative for suitable $\beta $ that satisfy (1).