Frustration in mixed two-dimensional ±J Ising lattices

Method of the sublattice previously introduced for homogeneous lattices is adapted here to characterize ground state properties of two inhomogeneous lattices: Kagomé lattice with coordination 4 and Five-points-star lattice with coordination 5. A representative cell must be chosen in each case in such way that main geometrical and topological properties of the lattice are well represented. Ferromagnetic interactions (in concentration x) and antiferromagnetic interactions (in concentration 1−x) define different possible configurations in the cell. By means of combinatorial and probability analysis weight functions are obtained allowing to calculate properties such as frustration length, energy per bond, and fractional content of unfrustrated interactions for the entire ground manifold. We report analytic expressions as functions of x which are later compared to average results coming from numerically solving 40 000 randomly prepared samples in an exact way. The good agreement between numerical and theoretical analysis validates the use of the method for inhomogeneous lattices and helps to interpret results. Values for these topological and physical parameters are also compared to those available for homogeneous two-dimensional lattices, looking for general trends. Roles of coordination number, plaquette shape and topology in the previously mentioned properties are discussed and established.