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Physical Review E

Six-state clock model on the square lattice: Fisher zero approach with Wang-Landau sampling

Chi‐Ok Hwang

Published Oct 29, 2009• 39 citations
Publisher

Abstract

We investigate the six-state clock model with nearest-neighbor interactions on the square lattice. We obtain the density of states of the finite systems up to L=28 using the Wang-Landau sampling. With the density of states and the Fisher zero approach, we successfully find two different critical temperatures 0.632(2) and 0.997(2) for the clock model. It seems that this study supports the recent results by [Lapilli Phys. Rev. Lett. 96, 140603 (2006)] that the transitions are not of Kosterlitz and Thouless type.

DOI
10.1103/physreve.80.042103

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Critical behavior of the six-state clock model in two dimensions

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Critical behavior of two-dimensional spin systems under the random-bond six-state clock model

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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.

Journal of Applied Physics•Sep 15, 2012•8 citations
We investigate the six-state clock model with nearest-neighbor interactions on the square lattice. We obtain the density of states of the finite systems up to L=28 using the Wang-Landau sampling. With the density of states and the Fisher zero approach, we successfully find two different critical temperatures 0.632(2) and 0.997(2) for the clock model. It seems that this study supports the recent results by [Lapilli Phys. Rev. Lett. 96, 140603 (2006)] that the transitions are not of Kosterlitz and Thouless type.
Abstract The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T 1 and T 2, where T 1 < T 2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T 1 and T 2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.
We investigate the partition function zeros of the two-dimensional p-state clock model in the complex temperature plane by using the Wang-Landau method. For p=5, 6, 8, and 10, we propose a modified energy representation to enumerate exact irregular energy levels for the density of states without any binning artifacts. Comparing the leading zeros between different p's, we provide strong evidence that the upper transition at p=6 is indeed of the Berezinskii-Kosterlitz-Thouless (BKT) type in contrast to the claim of the previous Fisher zero study [Phys. Rev. E 80, 042103 (2009)10.1103/PhysRevE.80.042103]. We find that the leading zeros of p=6 at the upper transition collapse onto the zero trajectories of the larger p's including the XY limit while the finite-size behavior of p=5 differs from the converged behavior of p≥6 within the system sizes examined. In addition, we argue that the nondivergent specific heat in the BKT transition is responsible for the small partition function magnitude that decreases exponentially with increasing system size near the leading zero, fundamentally limiting access to large systems in search for zeros with an estimator under finite statistical fluctuations.
The Berezinskii-Kosterlitz-Thouless transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method.A classical analog of the entanglement entropy S(L, T ) is calculated for L × L square system up to L = 129, as a function of temperature T .The entropy exhibits a peak at T = T * (L), where the temperature depends on both L and the boundary conditions.Applying the finite-size scaling to T * (L) and assuming presence of the Berezinskii-Kosterlitz-Thouless transitions, the two distinct phase-transition temperatures are estimated to be T1 = 0.70 and T2 = 0.88.The results are in agreement with earlier studies.It should be noted that no thermodynamic functions have been used in this study.
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​.
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.
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.