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Journal of Physics A Mathematical and General

Phase transitions of the 6-clock model in two dimensions

A Yamagata, Ikuo Ono

Published Jan 7, 1991• 38 citations
Publisher

Abstract

It is confirmed that there exists a Kosterlitz-Thouless (KT) like phase in the ferromagnetic 6-clock model on the square lattice through studies of the interfacial free energy estimated by Monte Carlo simulations. The authors find that lower and upper transition temperatures are T1=0.75 and T2=0.90 respectively, and that the critical exponent eta for the correlation function in the KT-like phase varies from 0.15 at T1 to 0.26 at T2. The correlation length zeta just above T2 is estimated directly from the magnetization profile, and it is shown to behave as zeta approximately exp(bt-12/), t=(T-T2)/T2 approaching the KT-like phase.

DOI
10.1088/0305-4470/24/1/033

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It is confirmed that there exists a Kosterlitz-Thouless (KT) like phase in the ferromagnetic 6-clock model on the square lattice through studies of the interfacial free energy estimated by Monte Carlo simulations. The authors find that lower and upper transition temperatures are T1=0.75 and T2=0.90 respectively, and that the critical exponent eta for the correlation function in the KT-like phase varies from 0.15 at T1 to 0.26 at T2. The correlation length zeta just above T2 is estimated directly from the magnetization profile, and it is shown to behave as zeta approximately exp(bt-12/), t=(T-T2)/T2 approaching the KT-like phase.
A Monte Carlo study is carried out on the six-state clock model on the triangular lattice. Physical quantities which characterize the Kosterlitz-Thouless (KT) phase transition are calculated. The temperature dependence of the exponent of the spin-pair correlation function, η( T ), in the KT phase is obtained precisely. The exponential divergence of the correlation length, ξ∼ exp ( a t - b ), with the theoretical value, b =0.5, gives a good fit to the data by the present MC simulations by choosing a =1.6∼1.8, η=0.23∼0.25 at the upper critical temperature T U =1.46∼1.49. However, the choice of critical parameters, a ∼3.1, b ∼0.3 and η∼0.28 at T U ∼1.53 gives the most consistent result by a finite-size scaling analysis on several physical quantities. The lower critical temperature, T L , is estimated to be 1.1.
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 critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the 8-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ = 1 and different values of variance: from σ2 = 0.1 to σ2 = 3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. Our results show that, a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
Using Monte Carlo simulation, the one dimensional N-state clock model with long range inverse square interaction has been studied. In the case of N = 6, the temperature dependence of the specific heat shows two peaks at T1 and T2 (< T1). The critical exponent η calculated from the system size dependence of magnetization changes with temperature in the region between T1 and T2. The behaviors are very similar to those of the two-dimensional 6-state clock model with the nearest neighbor interaction which shows two steps of phase transition, Berezinskii-Kosterlitz-Thouless transition and ordinary long range order. We have also confirmed that the temperature T2 shift toward zero with the increase of N.
A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.