Incompletely ordered phase in the three-dimensional six-state clock model: Evidence for an absence of ordered phases ofXYcharacter

Abstract

By employing our interfacial method, which uses Monte Carlo simulations, we show in various ways that the three-dimensional six-state clock model has an incompletely ordered phase (IOP) due to entropy gains where two nearest clock-spin states are dominant with equal weight. Our obtained results strongly imply its equivalence with the three-state antiferromagnetic Potts model, confirming the absence of ordered phases of XY'character and the existence of a difterent universality class for the upper phase transition of the IOP. The disordered flat phase recently found in the restricted solid-on-solid model is pointed out to be an IOP. Two-dimensional (2D) clock models have been intensively studied so far, ' but there are few studies on 3D ones. One seems to expect that high-state clock models undergo a phase transition from a disordered state to an intermediate state which shows the same critical behavior as expected in the XY model, like the 2D models with the state number larger than four. ' However, there are no grounds for this. Besides the above question, recent studies on related models such as antiferromagnetic (AF)Potts models have also stimulated our interest in the 3D six-state clock (6CL) model with the following questions.Is it equivalent to the three-state AF Potts (3AFP) model with ferromagnetic next-nearest neighbor interactions in three dimensions in the same way as it holds in two dimensions?If the answer is yes, then does it undergo a phase transition belonging to a distinct universality class as Ueno et al. argued or a phase transition of the XY universality class as Banaver et al. and Wang, Swendson, and Kotecky argued in the 3AFP model? Recently, we studied the 3D q-state AF Potts models by developing an interfacial approach by use of Monte Carlo (MC) simulation.The MC interfacial approach has been found to have various advantages in studying properties of ordered phases as well as critical behaviors. In fact we obtained that each of these Potts models with q =3-5 undergoes a second-order phase transition while the q=6 model does not. Further we got strong suggestions that the q= 3 and 4 models are in different universality classes; the q=5 model is also suggested to be in some distinct. universality class, which is only a suggestion. Our results are contrary to the theoretical results obtained by Banavar et al. who argue that equivalence between the q-state AF Potts model and the n ( =tI -1) vector model. Very recently, Wang, Swendson and Kotecky studied the 3AFP model and obtained critical exponents v and y which are close to the corresponding values of the n=2 vector model obtained by Le Guillou and Zinn-Justin. 'The present model is also attractive according to the suggestion from the studies in the 2D case that it is closely related to the 3D stacked triangular AF Ising model with ferromagnetic next-nearest-neighbor interactions. We also studied the latter 2D model, as well as the 3D model by our interfacial method. Our results clearly revealed the existence of the Kosterlitz-Thouless ph...

Scaling relation for dangerously irrelevant symmetry-breaking fields

“…The vertical axis is also scaled by L −ω with ω = 1.983 estimated from the exponents ν, y λ and x h = 0.519( [11]) by (8). The data seem to be almost converged for the larger sizes and also its scaling function is well fitted by the expected function (9). These observations strongly support the the new scaling relation (3).…”

“…Previously, a scaling relation was proposed by Ueno et al [9] and by Oshikawa [2]. Their argument is based on the basic assumption that there is a well defined domain wall splitting the whole system and the excess freeenergy caused by the domain walls is the scaling variable.…”

Ordered phase and scaling inZnmodels and the three-state antiferromagnetic Potts model in three dimensions

“…6 for the AFP model, or the Incompletely Ordered Phase (IOP) proposed in Ref. 13 for the 6CL model. In the vicinity of the critical point, the temperature dependence of the bare λ n is not essential because the leading dependence on the temperature is determined by the critical effect, as shown in eq.…”

Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models

“…Here the lattice itself imposes the discreteness of the microscopic order parameter (translating to tripled or quadrupled monopoles in the field theory [33]), but macroscopically the order appears U(1)like when coarse grained on length scales below ξ . DI perturbations of this kind are better known from classical models, with a prototypic example being threedimensional (3D) q-state clock models with q ≥ 4 [34][35][36][37][38][39][40][41][42][43][44][45][46][47]. As illustrated in Fig.…”