Monte Carlo Study of Pure-Phase Cumulants of 2D q-State Potts Models

. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.

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Monte-Carlo Study of Pure-Phase Cumulants of 2D q-State Potts Models

Wolfhard Janke, Stefan Kappler

We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point $β_t$ . By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point $β_t$. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.

arXiv:hep-lat/9701011Journal de Physique IMay 1, 19978 citations

Large- $q$ expansion of the energy and magnetization cumulants for the two-dimensional $q$ -state Potts modelLarge-$q$ expansion of the energy and magnetization cumulants for the two-dimensional $q$-state Potts model

Arisue, H, Tabata, K

We have calculated the large- $q$ expansion for the energy cumulants and the magnetization cumulants at the phase transition point in the two-dimensional $q$ -state Potts model to the 21st or 23rd order in $1/\\sqrt{q}$ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for $q>4$ on the first order transition point. The result confirms us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior not only of the energy cumulants but also of the magnetization cumulants for $q \\to 4_+$ .We have calculated the large-$q$ expansion for the energy cumulants and the magnetization cumulants at the phase transition point in the two-dimensional $q$-state Potts model to the 21st or 23rd order in $1/\\sqrt{q}$ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for $q>4$ on the first order transition point. The result confirms us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior not only of the energy cumulants but also of the magnetization cumulants for $q \\to 4_+$.

Jan 1, 19992 citations

FIRST ORDER PHASE TRANSITION OF THE Q -STATE POTTS MODEL IN TWO DIMENSIONS

H. Arisue, Kanehito Tabata

We have calculated the large- $q$ series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the $q$ -state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of $10^2 - 10^4$ than the Monte Carlo simulations. From the large- $q$ series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase transition point.We have calculated the large-$q$ series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the $q$-state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of $10^2 - 10^4$ than the Monte Carlo simulations. From the large-$q$ series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase transition point.

arXiv:hep-lat/0007025International Workshop on Non-Perturbative Methods and Lattice QCDApr 1, 20011 citation

Large-q expansion of the energy cumulants for the two-dimensional q-state Potts model

H. Arisue, Kanehito Tabata

We have calculated the large-q expansion for the energy cumulants at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in $1/\sqrt{q}$ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for $q>4$ on the first order transition point. The result confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the cumulants for $q \to 4_+$ .We have calculated the large-q expansion for the energy cumulants at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in $1/\sqrt{q}$ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for $q>4$ on the first order transition point. The result confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the cumulants for $q \to 4_+$.

arXiv:hep-lat/9810029Oct 14, 19981 citation

2D Potts-Model Correlation Lengths: Numerical Evidence for ξ o = ξ d at β t

W Janke, S Kappler

We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.

arXiv:hep-lat/9509056Europhysics Letters (EPL)Sep 1, 199515 citations

Finite-size scaling and Monte Carlo simulations of first-order phase transitions

Jooyoung Lee, J. Michael Kosterlitz

We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)= ${\mathit{U}}_{0}$ ( ${\mathit{tL}}^{\mathit{d}}$ )+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$ ${\mathit{U}}_{1}$ ( ${\mathit{tL}}^{\mathit{d}}$ ) with t=1- ${\mathit{T}}_{\mathit{c}}$ /T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-( ${\mathit{e}}_{1}$ / ${\mathit{e}}_{2}$ - ${\mathit{e}}_{2}$ / ${\mathit{e}}_{1}$ ${)}^{2}$ /12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature ${\mathit{T}}_{\mathit{c}}^{(2)}$ (L)- ${\mathit{T}}_{\mathit{c}}$ =O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at ${\mathit{T}}_{\mathit{c}}$ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=${\mathit{U}}_{0}$(${\mathit{tL}}^{\mathit{d}}$)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{U}}_{1}$(${\mathit{tL}}^{\mathit{d}}$) with t=1-${\mathit{T}}_{\mathit{c}}$/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(${\mathit{e}}_{1}$/${\mathit{e}}_{2}$-${\mathit{e}}_{2}$/${\mathit{e}}_{1}$${)}^{2}$/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature ${\mathit{T}}_{\mathit{c}}^{(2)}$(L)-${\mathit{T}}_{\mathit{c}}$=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at ${\mathit{T}}_{\mathit{c}}$ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.

Physical review. B, Condensed matterFeb 1, 1991316 citations

\\beta_t

β_t

Large-

q

expansion of the energy and magnetization cumulants for the two-dimensional

q

-state Potts model

We have calculated the large-

q

expansion for the energy cumulants and the magnetization cumulants at the phase transition point in the two-dimensional

q

-state Potts model to the 21st or 23rd order in

1/\\sqrt{q}

using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for

q>4

on the first order transition point. The result confirms us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior not only of the energy cumulants but also of the magnetization cumulants for

q \\to 4_+

We have calculated the large-

q

series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the

q

-state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of

10^2 - 10^4

than the Monte Carlo simulations. From the large-

q

series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase transition point.

We have calculated the large-q expansion for the energy cumulants at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in

1/\sqrt{q}

using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for

q>4

on the first order transition point. The result confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the cumulants for

q \to 4_+

{\mathit{U}}_{0}

(

{\mathit{tL}}^{\mathit{d}}

)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$

{\mathit{U}}_{1}

(

{\mathit{tL}}^{\mathit{d}}

) with t=1-

{\mathit{T}}_{\mathit{c}}

/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(

{\mathit{e}}_{1}

{\mathit{e}}_{2}

{\mathit{e}}_{2}

{\mathit{e}}_{1}

{)}^{2}

/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature

{\mathit{T}}_{\mathit{c}}^{(2)}

(L)-

{\mathit{T}}_{\mathit{c}}

=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at

{\mathit{T}}_{\mathit{c}}

using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.