Disorder solutions for the free energy of the Ising-like models

TL;DR

This paper derives exact free energy solutions for a broad class of Ising-like models with multispin interactions, providing explicit formulas and solutions for models with specific symmetries and supports.

Contribution

It introduces a method to obtain exact free energy values for arbitrary-dimensional Ising-like models with multispin interactions and boundary conditions, including explicit formulas and special cases.

Findings

Exact free energy formulas for models with finite support and multispin interactions.

Simplified formulas for models invariant under spin reversal.

Solutions provided for models on simplices, cubes, and ANNNI supports in multiple dimensions.

Abstract

For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.