Anisotropic-planar-rotor model ordering

The first Monte Carlo simulation [244] (see also the broad discussion in Section 5.3 of Ref. 246) focusing on the order of the herringbone transition was based on the strictly two-dimensional anisotropic-planar-rotor model (2.5), which is similar to the planar model used in Ref. 274 except that the quadrupole interactions are treated only approximately (see Section II.B for forther details). The instrument of diagnostics was the three-component herringbone order parameter [244, 246] defined as... 

Figure 32. Herringbone order parameter for the anisotropic-planar-rotor model (2.5) as a function of the reduced temperature T = TIK. Circles Monte Carlo results [244]. Dotted line mean-field approximation [62, 141]. Solid line triangular cluster-variational method [62]. Arrow first-order transition temperature obtained from a real-space renormalization group treatment of a planar quadrupolar six-state model [345]. (Adapted from Fig. 2 of Ref. 345.)...

Figure 33. Herringbone orientational correlation functions F (3.15) in the inset and the logarithmic derivatives 62 In (3.18) as a function of distance I in units of the lattice constant a - 4.26 A in the disordered phase of the anisotropic-planar-rotor model (2.5) from Monte Carlo simulations at 7" = 25.5 K and a linear system size of L = 180. The different symbols distinguish the three symmetty axes a, and the dashed line marks the plateau 2/. In the inset all three F fall on top of each other and the different symbols denote here the two oscillating parts of the antiferromagnetic-like ordering pattern. (Adapted from Fig. 1 of Ref. 273.)...

It is concluded [217] that an interpretation of the ideal herringbone transition within the anisotropic-planar-rotor model (2.5) as a weak first-order transition seems most probable, especially since previous assignments [56, 244] can be rationalized. This phase transition is fluctuation-driven in the sense of the Landau theory because the mean-field theory [141] yields a second-order transition. Assuming that defects of the -v/3 lattice and additional fluctuations due to full rotations and translations in three dimensions are not relevant and only renormalize the nonuniversal quantities, these assignments should be correct for other reasonable models and also for experiment [217]. 

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