Disorder Solutions for Generalized Ising Model with Multispin Interaction

Abstract

This study demonstrates a development of convenient formulae for obtaining the value of the free energy in the thermodynamic limit on a set of exact disorder solutions depending on four parameters for a 2D generalized Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the four interactions for the planar model, and for the 3D generalized Ising model in an external magnetic field with all kinds of interactions in the tetrahedron formed by four spins: at the origin of the coordinates and the closest to it along three coordinate axes in the first coordinate octant. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and a cyclic closure of the set of all points (in natural ordering). For both the planar model and the 3D model, elementary transfer matrices with non-negative matrix elements are constructed, while the free energy in the thermodynamic limit is equal to the Napierian logarithm of the maximum eigenvalue of the transfer matrix. This maximum eigenvalue can be found for a special kind of eigenvector with positive components. The region of existence of these solutions is described. The examples show the existence of nontrivial solutions of the resulting systems of equations for plane and three-dimensional generalized Ising models. The system of equations and the value of free energy in the thermodynamic limit will remain the same for plane and three-dimensional models with Hamiltonians, in which the value of the maximum in the natural ordering of the spin is replaced by the value of the spin at almost any other point in the lattice, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and repeatability of the components of the found eigenvectors, disappearing when we go beyond the framework of the obtained set of exact solutions, are the reason for the search for phase transitions in the vicinity of this set of disordered solutions.