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Canadian Journal of Physics

A reformulation of the dimer problem

R. W. Gibberd

Published Aug 1, 1968• 4 citations
Publisher

Abstract

It is shown that the partition function of the generalized dimer problem can be formulated in terms of a vacuum-to-vacuum expectation value of time-ordered operators. This expression is then evaluated by using Green's function technique, which has already been used in conjunction with the Ising model and ferroelectric problem.
DOI
10.1139/p68-499

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General Information

CERN

The possibility of calculating the partition function for Ising models on certain finite groups through group representation theory and the Dimer method is discussed.

•Jan 1, 1981•0 citations
The partition function (ZZZ) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, ZZZ, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.
In quantum field theory the Green function is usually calculated as the expectation value of the time-ordered product of fields over the vacuum. In some cases, especially in degenerate systems, expectation values over general states are required. The corresponding Green functions are essentially more complex than in the vacuum, because they cannot be written in terms of standard Feynman diagrams. Here a method is proposed to determine the structure of these Green functions and to derive nonperturbative equations for them. The main idea is to transform the cumulants describing correlations into interaction terms.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.