Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Applied Physics

Phase transitions in the six-state vector Potts model in two dimensions

Challa S. S. Murty, David Paul Landau

Published Mar 15, 1984• 15 citations
Publisher

Abstract

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.

DOI
10.1063/1.333684

Related papers

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

Murty S. S. Challa, David Paul Landau

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 ${k}_{B}$${T}_{1}$/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and ${k}_{B}$${T}_{2}$/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 ${T}_{1}$ and 0.275 at ${T}_{2}$.

Physical review. B, Condensed matter•Jan 1, 1986•145 citations

Computer simulation of phase transitions in low-dimensional Potts models

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, М. А. Магомедов

The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.

Physics of the Solid State•Jan 1, 2022•0 citations

Monte Carlo Study of the Ferromagnetic Six-State Clock Model on the Triangular Lattice

Sumiyoshi Fujiki, Tsuyoshi Horiguchi

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.

Journal of the Physical Society of Japan•Apr 15, 1995•5 citations

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

Norikazu Todoroki, Yohtaro Ueno, Seiji Miyashita

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.

arXiv:cond-mat/0207665•Physical review. B, Condensed matter•Dec 6, 2002•14 citations

Finite Size Scaling of the 2D Six-Clock model

M. Itakura, Trieste (Italy) International School for Advanced Studies

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

•Jan 1, 2001•2 citations

Phase transitions in the diluted 2D three-state Potts model on a square lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, Ataeva G. Ya. +1 more

The computer simulation method was used to study phase transitions in a two-dimensional site-diluted 3-state Potts model. Systems with linear dimensions Lx L=N, L=10/160 at a spin concentration p=1.00, 0.80 are considered. The numerical data obtained indicate that in a pure 3-state Potts model on a square lattice, a phase transition of the second order is observed in accordance with the theory. The introduction of disorder in the form of non-magnetic impurities (p=0.80) in the 3-state Potts model preserves the phase transition of the second order. Keywords: Potts model, Monte Carlo method, thermodynamic parameters, disorder.

Physics of the Solid State•Jan 1, 2022•1 citation
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.
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 Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.
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 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.
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
The computer simulation method was used to study phase transitions in a two-dimensional site-diluted 3-state Potts model. Systems with linear dimensions Lx L=N, L=10/160 at a spin concentration p=1.00, 0.80 are considered. The numerical data obtained indicate that in a pure 3-state Potts model on a square lattice, a phase transition of the second order is observed in accordance with the theory. The introduction of disorder in the form of non-magnetic impurities (p=0.80) in the 3-state Potts model preserves the phase transition of the second order. Keywords: Potts model, Monte Carlo method, thermodynamic parameters, disorder.