Perturbational Approach to the Two-Dimensional Next Nearest Neighbor Ising Problem

Abstract

The Feynman diagram technique is applied to the Ising model with the next nearest neighbor interaction according to the suggestions by Hurst and Gibberd. The partition func­ tion in the lowest order approximation is evaluated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The rules for perturbation calculation are discussed and some simple diagrams are investigated. neighbor Ising model,5) and Slater's model of ferroelectric,7) the exponent has additional quartic terms. The partition function of the last case cannot be solved exactly, but the technique of many-body theory is applicable to obtaining an approximate solution. In this paper, we apply their method to the rectangular next nearest neighbor Ising model. We derive an expression for the partition function in § 2. In § 3, the partition function in the lowest order approximation is calculated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The physical meaning of this approximation is also discussed. Correction to this approximation can be obtained by expanding the partition function in the pertur­ bation series by using the Feynman diagram technique. In § 4, we evaluate the Green function which is necessary for this perturbation calculation. In § 5, we present the rules for calculating the terms of the perturbation series, and in­ vestigate some simple diagrams.