Nematic order by thermal disorder in a three-dimensional lattice spin model with dipolarlike interactions

At low temperatures, some lattice spin models with simple ferromagnetic or antiferromagnetic interactions (for example, nearest-neighbor interaction being isotropic in spin space on a bipartite three-dimensional lattice) produce orientationally ordered phases exhibiting nematic (second-rank) order, in addition to the primary first-rank one; on the other hand, in the literature, they have been rather seldom investigated in this respect. Here we study the thermodynamic properties of a three-dimensional model with dipolar-like interaction. Its ground state is found to exhibit full orientational order with respect to a suitably defined staggered magnetization (polarization), but no nematic second-rank order. Extensive Monte Carlo simulations, in conjunction with finite-size scaling analysis, have been used for characterizing its critical behavior; on the other hand, it has been found that nematic order does indeed set in at low temperatures, via a mechanism of order by disorder.

Figure 1

The present figure shows the cubic unit cell for the model under investigation, together with the three ground-state configurations discussed in the text [Eq. (13)], with spin orientations represented by appropriate arrows. Meaning of symbols: (red) continuous line: $D_1$ configuration; (green) dashed line: $D_2$ configuration; (blue) dotted line: $D_3$ configuration; black lines mark cell edges, and black dots identify the vertices.

Figure 2

Simulation results for the configurational specific heat, obtained with different sample sizes; the statistical errors (not shown) range between 1% and 5%.

Figure 3

Simulation estimates for the mean staggered magnetization $M$, obtained with different sample sizes; here and in the following figures, the errors fall within symbol size.

Figure 4

Simulation estimates for the susceptibility $\chi$ associated with the staggered magnetization $M$, obtained with different sample sizes.

Figure 5

Scaling behavior of the magnetization $M$.

Figure 6

Simulation results for the fourth-order cumulant (27) obtained with different sample sizes.

Figure 7

Simulation results for the nematic second-rank order parameter ${\overline{P}}_2$, obtained with different sample sizes.

Figure 8

Simulation results for the nematic second-rank order parameter ${\overline{P}}_4$, obtained with different sample sizes.

Figure 9

Simulation results for the quantity $\varphi$, as defined in the text [Eq. (24)] obtained with different sample sizes.