Ground-state properties of finite square and triangular Ising lattices with mixed exchange interactions

Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.