Disorder solutions for generalized 2D Ising Model with multi-spin interaction

TL;DR

This paper derives exact disordered solutions for a generalized 2D Ising model with complex multi-spin interactions, providing formulas for free energy and eigenvalues, and explores implications for phase transitions.

Contribution

It introduces a new set of exact disordered solutions for a generalized 2D Ising model with multi-spin interactions, expanding the class of models with known solutions.

Findings

Largest eigenvalue of transfer matrix is constant across lattice sizes.

Free energy per site expressed via the largest eigenvalue.

Existence of nontrivial solutions suggests potential phase transitions.

Abstract

For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.