Exact solution Ising-like $2 \times 2 \times \infty$ models with multispin interactions

TL;DR

This paper derives exact solutions for the free energy and heat capacity of generalized Ising models with multispin interactions on a 2x2 infinite strip, using transfer matrix methods and analyzing special cases like the gonigendrial model.

Contribution

It provides exact analytical solutions for complex Ising-like models with multispin interactions, including special boundary conditions and the gonigendrial model, expanding the understanding of their thermodynamic properties.

Findings

Exact free energy and heat capacity for the generalized Ising model on a 2x2x∞ strip.

Analytical solutions for the gonigendrial model with different boundary conditions.

Comparison of numerical and analytical results for the 3x3x∞ gonigendrial model.

Abstract

The paper considers the generalized Ising model in the $2\times2\times\infty$ strip with a Hamiltonian invariant with respect to the central axis of rotation through the angle $\pi/2$, which includes all possible multiplicative interactions of an even number of spins in the unit cube.The exact value of free energy and heat capacity in the thermodynamic limit is found. For percolation invariant with respect to rotation about the central axis of rotation through the angle $\pi/2$, a limit relation for the non-percolation probability is derived. The solution was obtained by the transfer matrix method. In the general case, finding the largest eigenvalue of a $16 \times 16$ transfer matrix reduces to solving a quartic equation using the Ferrari method. In special cases, switching the transfer matrix of models with auxiliary matrices made it possible to reduce the problem to quadratic and linear equations. A separate chapter is devoted to the gonigendrial model in the $2\times2\times\infty$ strip, the exact values of the free energy and heat capacity in the thermodynamic limit are found for the case of free boundary conditions and an analogue of cyclically closed boundary conditions in both directions perpendicular to the central axis of rotation. The numerical solution for the $3\times3\times\infty$ gonigendrial model in the case of free boundary conditions and in the case of cyclic closure is compared with the corresponding analytical solutions in the volume $2\times2\times\infty$. The closeness of free energy and heat capacity in the thermodynamic limit for the gonigendrial model on the entire three-dimensional lattice and for the analog of cyclic closure in the $2\times2\times\infty$ strip is shown. The exact value of the free energy and heat capacity in the thermodynamic limit in the $2\times2\times\infty$ strip is obtained for the Antiferromagnetic layered Ising model.