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Journal of Mathematical Physics

Solution of the Dimer Problem by the S-Matrix Method

Chungpeng Fan

Published May 1, 1969• 4 citations
Publisher

Abstract

The S-matrix approach of Hurst is applied to rederive the partition function for the dimer problem. By using every two lattice sites as a unit cell and a set of six creation and annihilation operators at each cell, the expression for the partition is reduced to a form equivalent to the vacuum expectation value of an S matrix with a quadratic interaction Lagrangian.

DOI
10.1063/1.1664915

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Journal of the Physical Society of Japan•Mar 15, 1993•3 citations
The S-matrix approach of Hurst is applied to rederive the partition function for the dimer problem. By using every two lattice sites as a unit cell and a set of six creation and annihilation operators at each cell, the expression for the partition is reduced to a form equivalent to the vacuum expectation value of an S matrix with a quadratic interaction Lagrangian.
A new method of calculating the grand partition function of many-body system is developed, adopting extensively the techniques of calculus in quantum field theory. It is shown that the grand partition function, which is a trace of the density matrix expressed in terms of field operators, can be evaluated in a way almost parallel with the evaluation of the vacuum expectation value of the S-matrix in quantum field theory, provided that appropriate modifications in notation and definitions are made. As an example, the method is applied to electron-phonon system. Further, basing on this new formalism, various non-perturbational methods are discussed.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.