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arXiv:0809.0139 · Chinese Physics B

Critical behaviour of the ferromagnetic Ising model on a triangular lattice

Luo Zhi-Huan, Loan Mushtaq, Liu Yan, Lin Jian-Rong

Published Jul 1, 2009• 11 citations
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Abstract

We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L = 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T c = 3.6403(2). A convincing finite-size scaling analysis of the model yields ν = 0.9995(21), β/ν = 0.12400(17), γ/ν = 1.75223(22), γ'/ν = 1.7555(22), α/ν = 0.00077(420) (scaling) and α/ν = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.

DOI
10.1088/1674-1056/18/7/012

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EPJ Web of Conferences•Jan 1, 2018•0 citations
We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L = 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T c = 3.6403(2). A convincing finite-size scaling analysis of the model yields ν = 0.9995(21), β/ν = 0.12400(17), γ/ν = 1.75223(22), γ'/ν = 1.7555(22), α/ν = 0.00077(420) (scaling) and α/ν = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
Phase transitions in the two-dimensional ferromagnetic Potts model with the number of spin states q = 3 on a triangular lattice were studied using Monte-Carlo method. The systems of linear size L = 20–120 were considered. Using the method of the fourth-order Binder cumulants, it was demonstrated that the second order phase transition is observed in the two-dimensional ferromagnetic Potts model. The static critical exponents for the heat capacity α, susceptibility γ, magnetization β and correlation radius ν were calculated based on the finite-dimensional scaling theory.
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.
The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.
In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic two-dimensional Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\ensuremath{\le}0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\ensuremath{\le}1.0$ regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ depend on the entropic index qqq in the range $0.5<q\ensuremath{\le}1.0$ in the form $\ensuremath{\alpha}(q)=(10{q}^{2}\ensuremath{-}33q+23)/20$, $\ensuremath{\beta}(q)=(2q\ensuremath{-}1)/8$, and $\ensuremath{\gamma}(q)=({q}^{2}\ensuremath{-}q+7)/4$. On the other hand, the critical exponent $\ensuremath{\nu}$ does not depend on qqq. This suggests a violation of the scaling relations $2\ensuremath{\beta}+\ensuremath{\gamma}=d\ensuremath{\nu}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.
By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.