Statistics of Two-Dimensional Lattices with Many Components

Abstract

The statistics of the two-dimensional Ising lattice, for which each lattice site may take one of two states, has been partly generalized to the following model: each lattice site may take one of s states and each pair of neighboring sites in different states has a common excess energy, J , compared with a pair of neighboring sites in a same state. The partition function for the square lattice has a temperature symmetry and the Curie temperature, T c , is given by \begin{aligned} (s-1)\exp(-2J/kT_{c})+2\exp(-J/kT_{c})-1{=}0, \end{aligned} ( k is the Boltzmann constant). Although the partition function itself has only been obtained in the form of power series valid at low and high temperature regions, it is highly probable that for any finite s the specific heat is infinite at the Curie temperature. The transition is not the normal phase transition of the first order which the Bragg-Williams approximation predicts for our lattice with s ≧3.