Fourier Transform of Transfer Matrices of Plane Ising Models

Abstract

This work demonstrates the Fourier transform of the elementary transfer matrix of the generalized two-dimensional Ising model with special boundary conditions with a shift (screw type) with the form of a Hamiltonian covering the classical Ising model with an external field, as well as models equivalent to models on a triangular lattice with a chessboard type Hamiltonian (the author plans to consider the general form of interaction in the following publication). Its limit representation is obtained in the form of a sum of integral operators with the size of the system tending to infinity. This allows the actual problem of finding the maximum eigenvalue of the limiting elementary transfer matrix (its Napierian logarithm is equal to the free energy of the system) to be brought to finding the maximum eigenvalue of the sum of integral operators of a fairly simple form. This approach can help solve the problems associated with the large size of the transfer matrices.

Keywords

• generalized ising model
• hamiltonian
• elementary transfer matrix
• partition function
• free energy