Geometrical Frustration and Cluster Spin Glass with Random Graphs

TL;DR

This paper introduces a new theoretical approach using sparse random graphs to study the effects of geometric frustration and disorder on cluster magnetism, revealing distinct behaviors for triangular and tetrahedral clusters.

Contribution

It develops a novel method to analyze cluster spin systems on random graphs, accounting for geometric frustration and disorder, with detailed results for triangular and tetrahedral geometries.

Findings

Triangular clusters exhibit a spin liquid and cluster spin glass phases.

The cluster spin glass state is robust against weak disorder and large negative intra-cluster interactions.

Tetrahedral clusters do not support a spin glass state under weak disorder or large negative interactions.

Abstract

We develop a novel method based in the sparse random graph to account the interplay between geometric frustration and disorder in cluster magnetism. Our theory allows to introduce the cluster network connectivity as a controllable parameter. Two types of inner cluster geometry are considered: triangular and tetrahedral. The theory was developed for a general, non-uniform intra-cluster interactions, but in the present paper the results presented correspond to uniform, anti-ferromagnetic (AF) intra-clusters interactions $J_{0}/J$. The clusters are represented by nodes on a finite connectivity random graph, and the inter-cluster interactions are random Gaussian distributed. The graph realizations are treated in replica theory using the formalism of order parameter functions, which allows 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 $T_{f}/J$. 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 $f_{p}=-\Theta_{CW}/T_{f}$ for large $J_{0}/J$. In contrast, for the non-frustrated tetrahedral cluster geometry, the CSG ground state is suppressed for weak disorder or large negative $J_{0}/J$. The CSG boundary phase presents a re-entrance which is dependent on the network connectivity.