Order Parameters and Percolation for Finite Square Ising Lattices with Mixed Exchange Interactions

Abstract

Square Ising lattices with equal amount and equal magnitude of ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) are considered. The size is given by the total number of spins N, which is varied. The way in which the N spins are distributed (array) represents the shape of the lattice. We look here for size and shape dependence of parameters characterizing the ground level of these systems: ground state energy per bond e g and site order parameters q, p and h. The results presented below correspond to exact solutions for averages over 500 samples for each kind of array. The computer techniques to perform the calculations are based on partial (or ‘intelligent’) enumeration of the configuration space. We concentrate here on p and h due to their novelty. We present evidence showing that p goes to zero in the thermodynamic limit due to a bimodal distribution, such that the component centered at p = 0 begins to dominate as size increases. On the other hand the single mode distribution for values of h is asymmetric maximizing at about 0.6 and average value of 0.5. The properties of these two distributions become more evident as size grows. The lattices defined only by those bonds that never show frustration (diluted lattices) present a clear tendency to percolate. About 2/3 of the diluted lattices percolate independent of size.