Disorder Solutions for Generalized Ising and Potts Models with Multispin Interaction
In this work, a special elementary transfer matrix is constructed for generalized Ising models and Potts models with the general form of a finite Hamiltonian with a multi-spin interaction in a space of arbitrary dimensionality, the Napierian logarithm of its maximum eigenvalue is equal to the free energy of the system. In some cases, it was possible to obtain an explicit form of the eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix. On this basis we obtained systems of nonlinear equations for the interaction coefficients of the Hamiltonian for finding the exact value of the free energy on a set of disorder solutions. Using the Levenberg-Marquardt method, the existence of nontrivial solutions of the resulting systems of equations for plane and three-dimensional Ising models was shown. In some special cases (the 2D Ising model, the interaction potential, including the interaction of the next nearest neighbors and quadruple interactions; the 3D model with a special Hamiltonian symmetric relative to the change of all spin signs, for which it is possible to reduce the system of equations to the system for a planar model) three parameters are written in explicit form. The domain of existence of these solutions is described.