Decorated Clusters and Geometrical Frustration in Cluster Spin Glass: A Random Graph Approach

We develop a theory to investigate how geometrically frustrated clusters that become decorated affect the Cluster Spin Glass phase. The cluster structure is assumed to be a tetrahedron composed of Ising spins with z-anisotropy placed at its vertices that interact antiferromagnetically. We consider the probability $1-p_J$ of finding an impurity at a vertex of the tetrahedron that interacts ferromagnetically with the remaining elements inside the tetrahedron. An intercluster disorder is added as a random Gaussian interaction. The order parameters are obtained using the sparse random graph technique, which introduces the connectivity of the network of clusters as a controllable parameter in the theory. We examine changes that occur in the Cluster Spin Glass phase as a function of $p_J$ and $c$, in addition to the antiferromagnetic intracluster couplings $J_1$. For intermediate values of $p_J$, unexpected results appear. Even when some clusters contain a ferromagnetic impurity, there will still be robust geometric frustration effects in the cluster network. However, the $p_J$ threshold for this to occur depends on connectivity. Conversely, below this threshold, reduced GF effects favor the reappearance of the CSG phase. Furthermore, the Curie-Weiss temperature $\Theta_W$ has a gradual change of signal, indicating that the effects of the impurities extend to the paramagnetic phase.