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arXiv:cond-mat/0207665 · Physical review. B, Condensed matter

Ordered phase and phase transitions in the three-dimensional generalized six-state clock model

Norikazu Todoroki, Yohtaro Ueno, Seiji Miyashita

Published Dec 6, 2002• 14 citations
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Abstract

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 same behavior as the stacked triangular antiferromagnetic Ising model and the three-state antiferromagnetic Potts model. First, we investigate ordered phases by using the Monte Carlo twist method (MCTM). We confirmed the existence of an incompletely ordered phase (IOP1) at intermediate temperature, besides the completely ordered phase at low temperature. In this intermediate phase, two neighboring states of the six-state model mix, while one of them is selected in the low-temperature phase. We examine the fluctuation the mixing rate of the two states in IOP1 and clarify that the mixing rate is very stable around 1:1. The high-temperature phase transition is investigated by using a nonequilibrium relaxation method. We estimate the critical exponents \ensuremath{\beta}=0.34(1) and \ensuremath{\nu}=0.66(4). These values are consistent with the $3\mathrm{D}\ensuremath{-}XY$ universality class. The low temperature phase transition is found to be of first-order by using MCTM and finite-size-scaling analyses.

DOI
10.1103/physrevb.66.214405

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•Jan 1, 2001•2 citations
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 same behavior as the stacked triangular antiferromagnetic Ising model and the three-state antiferromagnetic Potts model. First, we investigate ordered phases by using the Monte Carlo twist method (MCTM). We confirmed the existence of an incompletely ordered phase (IOP1) at intermediate temperature, besides the completely ordered phase at low temperature. In this intermediate phase, two neighboring states of the six-state model mix, while one of them is selected in the low-temperature phase. We examine the fluctuation the mixing rate of the two states in IOP1 and clarify that the mixing rate is very stable around 1:1. The high-temperature phase transition is investigated by using a nonequilibrium relaxation method. We estimate the critical exponents \ensuremath{\beta}=0.34(1) and \ensuremath{\nu}=0.66(4). These values are consistent with the 3D\ensuremath−XY3\mathrm{D}\ensuremath{-}XY3D\ensuremath−XY universality class. The low temperature phase transition is found to be of first-order by using MCTM and finite-size-scaling analyses.
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 close with 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\ensuremath{\sim}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.
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 different 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.
We have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at kB{k}_{B}kB​T1{T}_{1}T1​/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and kB{k}_{B}kB​T2{T}_{2}T2​/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at T1{T}_{1}T1​ and 0.275 at T2{T}_{2}T2​.
The static critical properties of the six-state vector Potts model (clock model) on L×L square lattices are studied in zero field using Monte Carlo simulations. Data are obtained for L≤48 keeping between 2.4 and 4.0×104 MCS (Monte Carlo steps/site) for computing averages. The data are analyzed using a block distribution technique in which the critical temperature and exponents may be estimated from the size dependence of the cumulants of the order parameter. We see clear evidence for two phase transitions but our estimates for the transition temperatures are different from results obtained through other methods. For the lattice sizes considered the intermediate phase does not appear to be XY-like.
The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.
We investigate the isotropic-anisotropic phase transition of the twodimensional XY model with six-fold anisotropy, using Monte Carlo renormalization group method. The result indicates difficulty of observing asymptotic critical behavior in Monte Carlo simulations, owing to the marginal flow at the fixed point. PACS numbers: 02.70.Lq, 75.10.Hk Typeset using REVTEX 1 The XY model with Z6 symmetry breaking field represents not only planar spin magnets with Z6 symmetric crystal field [1], but also models with Z2 × Z3 symmetry (which is isomorphic to Z6) such as three-state antiferromagnetic Potts model on the square and the cubic lattice [2] and Ising antiferromagnet on the triangular and the stacked-triangular lattice [3–8]. The critical behavior of the model in two dimension is well understood by scaling argument [1]: The model undergoes two distinct phase transitions, both of which being