Geometrical frustration and cluster spin glass with random graphs

We develop a based on a sparse random graph to account for the interplay between geometric frustration and disorder in cluster magnetism. Our theory allows introduction of the cluster network connectivity as a controllable parameter. Two types of inner cluster geometry are considered: triangular and tetrahedral. The theory was developed for general, nonuniform intracluster interactions, but in the present paper the results presented correspond to uniform, antiferromagnetic (AF) intraclusters interaction J_{0}/J. The clusters are represented by nodes on a finite connectivity random graph, and the intercluster interactions are randomly Gaussian distributed. The graph realizations are treated in replica theory using the formalism of order parameter functions, which allows one to calculate the distribution of local fields and, as a consequence, the relevant observable. In the case of triangular cluster geometry, there is the onset of a classical spin liquid state at a temperature T^{*}/J and then, a cluster spin glass (CSG) phase at a temperature T_{/}J. The CSG ground state is robust even for very weak disorder or large negative J_{0}/J. These results does not depend on the network connectivity. Nevertheless, variations in the connectivity strongly affect the level of frustration f_{p}=-Θ_{CW}/T_{f} for large J_{0}/J. In contrast, for the nonfrustrated tetrahedral cluster geometry, the CSG ground state is suppressed for weak disorder or large negative J_{0}/J. The CSG boundary phase presents a reentrance which is dependent on the network connectivity.