Correlated cluster mean-field theory for spin-glass systems

The competition between cluster spin glass (CSG) and ferromagnetism or antiferromagnetism is studied in this work. The model considers clusters of spins with short-range ferromagnetic or antiferromagnetic (FE-AF) interactions ($J_0$) and long-range disordered couplings ($J$) between clusters. The problem is treated by adapting the correlated cluster mean-field theory of D. Yamamoto [Phys. Rev. B 79, 144427 (2009)]. Phase diagrams $T/J\times J_0/J$ are obtained for different cluster sizes $n_s$. The results show that the CSG phase is found below the freezing temperature $T_f$ for lower intensities of $J_0/J$. The increase of short-range FE interaction can favor the CSG phase, while the AF one reduces the CSG region by decreasing the $T_f$. However, there are always critical values of $J_0$ where AF or FE orders become stable. The results also indicate a strong influence of the cluster size in the competition of magnetic phases. For AF cluster, the increase of $n_s$ diminishes $T_f$ reducing the CSG phase region, which indicates that the cluster surface spins can play an important role in the CSG arising.

Figure 1

Schematic representation for a square lattice divided into clusters with $n_s=4$. The mean fields are pointed by arrows that represent (a) the interactions between cluster $\nu$ with its neighbors and (b) the interactions on ${\nu }^{\prime }$ used to evaluate the $m^{s{s}^{\prime }}$. For the AF case it is considered two sublattices that are represented by solid and open circles.

Figure 2

Phase diagram of $T/J$ vs. $J_0/J$ for square-lattice clusters with four spins. The FE order is found for large, short-range interactions $J_0/J>0$, while the AF can appear at negative values of $J_0/J$. Solid and dashed lines indicate continuous transition and spinodal of CSG phase, respectively. The cluster SG phase occurs for small interactions $J_0/J$.

Figure 3

Magnetic susceptibility $\chi$ as a function of the temperature for several values of $J_0/J$. The inset exhibits $\chi$ vs. $J_0/J$ for low temperatures. The increase of ferromagnetic short-range interactions destroys the CSG phase introducing the FE order.

Figure 4

$\chi$ versus $T/J$ for AF $J_0/J$ values. The inset exhibits the behavior of $1/\chi$.

Figure 5

Schematic representation for a cluster with $n_s=16$. It is used in the same convention as described in the legend of Fig. 1.

Figure 6

Phase diagrams $T/J$ vs. $J_0/J$ for several cluster sizes: 4, 6, 8, 9, 12, and 16. The inset exhibits $T_f/J$ as a function of $n_s$ for $J_0/J=0.1$. The $T_f$ increases with $n_s$ and $J_0$ at the same time that the enhancement of $J_0$ can stabilize the FE phase.

Figure 7

Phase diagrams $T/J$ vs. $J_0/J$ for antiferromagnetic interactions and several cluster size. The inset exhibits the behavior of $T_f$ when $n_s$ increases for a constant $J_0/J=-0.1$.