Exact Ground-State Energy of the Ising Spin Glass on Strips

TL;DR

This paper introduces an exact analytical method to compute the ground-state energy of the Ising spin glass on strips, revealing complex probability singularities similar to Lee-Yang zeros, and providing insights into the model's behavior.

Contribution

The authors develop a novel analytical approach that surpasses numerical methods by calculating energies at complex probabilities for Ising spin glasses on strips.

Findings

Identified singularities in the complex probability plane for the models.

Found potential limiting points near p ≈ 0.9 for the ±J model.

Demonstrated the method's applicability to various strip sizes.

Abstract

We propose a new method for exact analytical calculation of the ground-state energy of the Ising spin glass on strips. An outstanding advantage of this method over the numerical transfer matrix technique is that the energy is obtained for complex values of the probability describing quenched randomness. We study the $\pm J$ and the site-random models using this method for strips of various sizes up to $5\times\infty$. The ground-state energy of these models is found to have singular points in the complex-probability plane, reminiscent of Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The $\pm J$ Ising model has a series of singularities which may approach a limiting point around $p \sim 0.9$ on the real axis in the limit of infinite width.