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Progress of Theoretical Physics

The Combinatrial Method and the Two-Dimensional Ising Model

R. B. Potts, John Ward

Published Jan 1, 1955• 77 citations
Publisher

Abstract

A combinational method is used to calculate (a) the partition function of a finite rectangular Ising lattice and (b) the correlation functions of an infinite lattice.
DOI
10.1143/ptp.13.38

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A method is proposed for exactly calculating the partition function of a rectangular Ising lattice with the presence of a uniform external field. This approach is based on the method of the transfer matrix developed about seventy years ago for the rectangular Ising model in the absence of external field. The basis for the vector space is chosen as the eigenvectors of the diagonal part of the transfer matrix. The matrix elements for the non-diagonal part can be calculated very easily. Then the partition function and thermodynamical quantities can be evaluated. The limit of infinite lattice is discussed.
We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases.