±J Ising model on homogeneous Archimedean lattices

We tackle the problem of finding analytical expressions describing the ground state properties of homogeneous Archimedean lattices over which a generalized Edwards–Anderson model (±J Ising model) is defined. A local frustration analysis is performed based on representative cells for square lattices, triangular lattices and honeycomb lattices. The concentration of ferromagnetic (F) bonds x is used as the independent variable in the analysis (1−x is the concentration for antiferromagnetic (A) bonds), where x spans the range [0.0,1.0]. The presence of A bonds brings frustration, whose clear manifestation is when bonds around the minimum possible circuit of bonds (plaquette) cannot be simultaneously satisfied. The distribution of curved (frustrated) plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment, energy per bond, and fractional content of unfrustrated bonds. Two methods are developed to cope with this analysis: one based on the direct probability of a plaquette being curved; the other one is based on the consideration of the different ways bonds contribute to the particular plaquette configuration. Exact numerical simulations on a large number of randomly generated samples allow to validate previously described theoretical analysis. It is found that the second method presents slight advantages over the first one. However, both methods give an excellent description for most of the range for x. The small deviations at specific intervals of x for each lattice have to do with the self-imposed limitations of both methods due to practical reasons. A particular discussion for the point x=0.5 for each one of the lattices also shines light on the general trends of the properties described here.