Correlated cluster mean-field theory for spin systems

A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of “correlations” between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.

Figure 1

Schematic representation of the honeycomb lattice divided into hexagonal clusters. The arrows indicate the effective fields acting on the sites (a) in cluster $C$ and (b) in cluster $C^{\prime }$ when the spin at site 1 is fixed to be $+1$ or $-1$.

Figure 2

Schematic representation of the square lattice divided into square-shaped clusters. The arrows indicate the effective fields acting on the sites (a) in cluster $C$ and (b) in cluster $C^{\prime }$ when the spins at site 1 and site 2 are fixed to be $+1$ or $-1$, respectively.

Figure 3

Schematic representation of the triangular lattice divided into triangle-shaped clusters. (a) The effective fields acting on site 1. (b) Positional relation between three clusters ($C$, $C^{\prime }$, and $C^{″}$).

Figure 4

Schematic representation of eight-site clusters for the simple cubic lattice.

Figure 5

The temperature dependence of the magnetization of an isotropic Heisenberg ferromagnet at $h/J=0.1$. Comparison of the result obtained from the CCMF approach with those of the MFT, BPW, RPA, Kikuchi’s square approximation, and the QMC calculations. The error bar of the QMC results are smaller than the linewidth.

Figure 6

Schematic representation of the square lattice divided into pairs of neighboring sites. The arrows indicate the effective fields acting on the sites in cluster $C$.