Combinatorial Approach to the Ground-State Energy of Square and Triangular +/- J Spin Glasses

TL;DR

This paper introduces a new combinatorial analytical method to accurately compute the ground-state energy of square and triangular spin glasses with varying +/- J interactions, aligning well with numerical simulations.

Contribution

It develops a novel combinatorial approach using series expansion and diagram stability to determine ground-state energies in spin glasses with different bond concentrations.

Findings

Accurately predicts ground-state energy as a function of negative bond concentration.

Provides a converging series expansion for energy calculation.

Achieves excellent agreement with numerical simulations.

Abstract

A new combinatorial, analytical approach to the ground-state energy problem of spin glasses with different concentrations of +/- J interactions is developed. The energy e_0 is expressed in terms of the fraction of broken bonds mu_0 and expanded into a fast converging series of the average length of a segment between two frustrated plaquettes of a minimum-weighted perfect matching Lambda_mean. The concept of so called diagram stability s is introduced in order to calculate coefficients of this expansion. Finally, the fraction mu_0 as a function of the concentration p of negative bonds is obtained for triangular and square infinite lattices in excellent accordance with numerical simulations of large adequate systems.