Locations of multicritical points for spin glasses on regular lattices

TL;DR

This paper introduces a refined systematic method to accurately locate multicritical points of spin glasses on various regular lattices, improving precision over previous techniques and aligning well with numerical estimates.

Contribution

The study develops an improved, systematic approach based on duality and partition functions to precisely determine multicritical points for spin glasses on different lattices.

Findings

Precise multicritical point on square lattice: p_c = 0.890813

Precise multicritical point on triangular lattice: p_c = 0.835985

Precise multicritical point on hexagonal lattice: p_c = 0.932593

Abstract

We present an analysis leading to precise locations of the multicritical points for spin glasses on regular lattices. The conventional technique for determination of the location of the multicritical point was previously derived using a hypothesis emerging from duality and the replica method. In the present study, we propose a systematic technique, by an improved technique, giving more precise locations of the multicritical points on the square, triangular, and hexagonal lattices by carefully examining relationship between two partition functions related with each other by the duality. We can find that the multicritical points of the $\pm J$ Ising model are located at $p_c = 0.890813$ on the square lattice, where $p_c$ means the probability of $J_{ij} = J(>0)$, at $p_c = 0.835985$ on the triangular lattice, and at $p_c = 0.932593$ on the hexagonal lattice. These results are in excellent agreement with recent numerical estimations.