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Journal of Physics Conference Series

Monte Carlo simulation of the one-dimensionalN-state clock model with long range inverse square interaction

Masahiro Takamoto, Takehiro Kinoshita, Yoshinori Muraoka, Toshihiro Idogaki

Published Mar 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/150/4/042200

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By using the Monte Carlo simulation, we investigate the critical behavior of the 1-dimensional spin system with infinite long-range (LR) interaction r^{-(1+σ)}. By changing the value σ, we compare the results with those of the d-dimensional spin system with the nearest-neighbor(NN) interaction only. In the case of XY spin model, we obtain three different type phase transitions, i.e. the mean field type for 0 < σ < 0.5 , the σ-dependent non-trivial one for 0.5 < σ < 1 and 'Berezinskii-Kosterlitz-Thouless (BKT)-like’transition at σ = 1 as in the NN model of d > 4 , 2 < d < 4 and d = 2 , respectively. In the case of q-state clock spin model with σ = 1, we also confirm the BKT-like transition together with the similar q-dependence of the critical behavior to that of the 2-dimensional NN model. These results suggest that the infinite long-range interaction can partly compensate the role of the lattice dimensionality by increasing the effective value d from d = 1 to d = 2.
We discuss the critical behaviour of the 1-dimensional classical XY model with power law decay long range interactions by using Monte Carlo simulation. It has been predicted by renormalization group calculation that the critical behaviour changes with the power of interactions. From the simulation result of temperature dependence of the specific heat, we confirm that the system shows the critical behaviour of the mean field type and the non-trivial long range order. We also estimate the critical exponent η for the system with the inverse square long range interactions, and by using the phenomenological renormalization group plot, we confirm the 'Berezinskii-Kosterlitz-Thouless (BKT)-like' transition which has been predicted by the spin wave calculations.
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.
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​.
We study the phase transition properties of the two-dimentional q-state clock model by an extensive Monte Carlo simulation. Several independent physical quantities, including the specific heat, the magnetization and the helicity modulus had been calculated for q > 5. The specific heat and the magnetization shows the behavior of two distinct phase transitions, including the higher phase transition, T2, between the disordered phase and the quasi-long-range ordered phase and the lower phase transition, T1, between the quasi-long-range ordered phase and the quasi-long-range ordered phase. The helicity modulus not only show that T2 is the Kosterlitz-Thouless phase transition but also has signature of phase transition around T1.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.