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In our previous studies of the antiferromagnetic Potts and Q =6 clock models in three dimensions, we have argued the existence of an incompletely ordered phase (IOP) which is characterized by soft rigidity with a nonintegral stiffness exponent and an incomplete order (though long ranged) such that some of the spin states are exclusively dominant. To investigate the IOP s further, we study Q-state general clock models with one varying energy parameter which can include the ferromagnetic Potts models, by means of our Monte Carlo twist method and analytically in the pair approximation. The twist method gives the following results for Q =6. There are two kinds of IOP's (IOP1 and IOP2) on the opposite side of the ferromagnetic Potts model. Their stiffness exponents are about 1.2 less than 2 (for the highest rigidity) in good agreement with those previously obtained. A pair of adjacent clock states are dominantly well mixed in the IOP1, as are a set of three adjacent ones in the IOP2. Thermal fluctuations in the IOP s can be characterized by the spin configurations of the soft structures with buffers that prevent spins in different states by more than the least angle from getting with each other, which is consistent with the nonintegral stiffness exponent. Large entropy contributions to the IOP s are revealed in both approaches. The phase boundary, which extends from the ferromagnetic Potts point, is clearly of first order, while among those relevant to the IOP's one is discontinuous and the other is continuous but the last two are not evident. It is strongly suggested that there is a transition without symmetry breaking between the IOP2 and the rigid phase. In the pair approximation applied for Q =4-12, various properties for Q =6 are consistent with those obtained by the simulation, except some other properties. Peculiar Q dependence of the highest critical temperature of the IOP in the extreme case is also found.

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We study the three-dimensional generalized six-state clock model at values of the energy parameters, at which the system is considered to have the The high temperature phase transition is investigated by using nonequilibrium relaxation method (NERM). We estimate the critical exponents β = 0.34(1) and ν = 0.66(4). These values are consistent with the 3D-XY universality class. The low temperature phase transition is found to be of 1 first-order by using MCTM and the finite-size-scaling analysis.

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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...

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