±J Ising model on Dürer lattices

This paper tackles the problem of finding analytical expressions describing the ground state properties of Dürer lattices over which a generalized Edwards–Anderson model (±J Ising model) is defined. A local frustration analysis is performed based on representative cells for these 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 [0.00,1.00]. 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 non frustrated bonds. An analytical method is developed to cope with this analysis based on the direct probability of a plaquette being curved. Exact numerical simulations on a large number of randomly generated samples associated to Dürer lattices allow to validate previously described theoretical analysis. This analytical method gives an excellent description for the most of the range for x. A particular discussion for the point x=0.50 belonging to spin-glass phase also sheds light on the general trends of the properties described here.