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 $f_p=-{\mathrm{\Theta }}_{CW}/T_f$ 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.

Figure 1

(a) Triangular and (b) tetrahedral clusters.

Figure 2

Marginal distributions for each component of a triangular cluster, for $T/J=0.1, c=4$, and $J_0/J=-5.0$, where there are strong frustration effects (see discussion below). $\mathcal{N}={10}^5$ is the size of the population of fields. (a) Marginal along the $h_1$ field, (b) marginal along the $h_2$ field, and (c) marginal along the $h_3$ field.

Figure 3

$T/J$ versus $J_0/J$ phase diagrams in clusters of equilateral triangles. The thin line represents the SL to PM crossover temperature $T^{*}/J$.

Figure 4

(a) Specific heat $C_m$ vs $T/J$ for intracluster couplings $J_0/J=-5.0$ and $J_0/J=-3.0$, for $c=4$. The inset shows the corresponding entropy. (b) Specific heat $C_m$ versus $T/J$ for $c=4$ and $c=8$, for $J_0/J=-3.0$.

Figure 5

(a) Inverse susceptibility ${\chi }^{-1}$ versus $T/J$ profiles for fixed intracluster couplings $J_0/J=-5.0$ and connectivity values $c=2, c=4, c=8$, and $c=12$. The inset shows the cusps in detail. (b) Inverse susceptibility ${\chi }^{-1}$ vs $T/J$ profiles for fixed $c=8$ and two values of the intracluster couplings: $J_0/J=-3.0$ and $J_0/J=-5.0$. The inset shows the cusps in detail. The thin straight lines represent the fit of the Curie-Weiss law to the linear region of ${\chi }^{-1}$.

Figure 6

Frustration parameter $f_p$ vs $c^{-1}$.

Figure 7

(a) Total spin of the cluster $Q$, CSG order parameter $q$, and free energy $f$ vs $J_0/J$ for $T/J=0.10$ and $c=8$. (b) The same, but for $T/J=0.15$. Solid (dashed) lines indicate heating (cooling). The arrows indicate the loci of the discontinuous transitions.

Figure 8

$T/J$ vs $J_0/J$ phase diagrams in clusters of regular tetrahedrons for $c=4, c=8$, and $c=12$. Thin lines represent first-order phase transitions.