±J Ising model on mixed Archimedean lattices: (33,42), (32,4,3,4), (3,122), (4,6,12)

This paper addresses the problem of finding analytical expressions describing the ground state properties of mixed 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 (33,42), (32,4,3,4), (3,122) and (4,6,12) lattices, following the notation proposed by Grünbaum and Shephard. 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 [0.00,1.00]. 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 associated to (33,42) and (32,4,3,4) lattices allow to validate the previously described theoretical analysis. It is found that the first method presents slight advantages over the second 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 are relevant to the self-imposed limitations of both methods due to practical reasons. A particular discussion for the point x=0.50 for each one of the four lattices also sheds light on the general trends of the properties described here.