We present a powerful method of identifying the nature of transitions by numerical simulation of finite systems. By studying the finite size scaling properties of free energy barrier between competing states, we can identify unambiguously a weak first order transition even when accessible system sizes are L/{xi} < 0.05 as in the five state Potts model in two dimensions. When studying a continuous phase transition we obtain quite accurate estimates of critical exponents by treating it as a field driven first order transition. The method has been successfully applied to various systems.We present a powerful method of identifying the nature of transitions by numerical simulation of finite systems. By studying the finite size scaling properties of free energy barrier between competing states, we can identify unambiguously a weak first order transition even when accessible system sizes are L/{xi} < 0.05 as in the five state Potts model in two dimensions. When studying a continuous phase transition we obtain quite accurate estimates of critical exponents by treating it as a field driven first order transition. The method has been successfully applied to various systems.

Jan 1, 19910 citations

New numerical method to study phase transitions

Jooyoung Lee, J. Michael Kosterlitz

We present a new numerical method to identify the nature of a phase transition. When combined with finite-size scaling, the method can identify unambiguously a weak first-order transition even when accessible system sizes are L/\ensuremath{\xi}0.05 as in the five-state Potts model. At a continuous transition, exponents can be determined surprisingly accurately and the computational effort required is moderate. The method is tested on the 2D Potts and 3D Ising models, but should be quite generally applicable.We present a new numerical method to identify the nature of a phase transition. When combined with finite-size scaling, the method can identify unambiguously a weak first-order transition even when accessible system sizes are L/\ensuremath{\xi}0.05 as in the five-state Potts model. At a continuous transition, exponents can be determined surprisingly accurately and the computational effort required is moderate. The method is tested on the 2D Potts and 3D Ising models, but should be quite generally applicable.

Physical Review LettersJul 9, 1990324 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

The crossover from first to second-order finite-size scaling: a numerical study

Christian Borgs, Paul E.L. Rakow, Stefan Kappler

We consider a particular case of the two dimensional Blume-Emery-Griffiths model to study the finite-size scaling for a field driven first-order phase transition with two coexisting phases not related by a symmetry. For low temperatures we verify the asymptotic (large volume) predictions of the rigorous theory of Borgs and Kotecky. Near the critical temperature we show that all data fit onto a unique curve, even when the correlation length ξ becomes comparable to or larger than the size of the system, provided the linear dimension L of the system is rescaled by ξ.We consider a particular case of the two dimensional Blume-Emery-Griffiths model to study the finite-size scaling for a field driven first-order phase transition with two coexisting phases not related by a symmetry. For low temperatures we verify the asymptotic (large volume) predictions of the rigorous theory of Borgs and Kotecky. Near the critical temperature we show that all data fit onto a unique curve, even when the correlation length ξ becomes comparable to or larger than the size of the system, provided the linear dimension L of the system is rescaled by ξ.

Journal de Physique IJul 1, 19946 citations

Finite-Size Scaling and Critical Exponents. A New Approach and Its Application to Anderson Localisation

Etienne Hofstetter, Michael Schreiber

Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.

Europhysics Letters (EPL)Mar 20, 199341 citations

Locating fixed points in the phase plane

Yanhua Zhang, Ye-Yin Zhao, Lizhu Chen +5 more

The critical point is a fixed point in finite-size scaling. To quantify the behavior of such a fixed point, we define, at a given temperature and scaling exponent ratio, the width of scaled observables for different sizes. The minimum of the width reveals the position of the fixed point, its corresponding phase transition temperature, and scaling exponent ratio. The value of this ratio tells the nature of the fixed point, which can be a critical point, a point of the first-order phase transition line, or a point of the crossover region. To demonstrate the effectiveness of this method, we apply it to three typical samples produced by the three-dimensional three-state Potts model. Results show the method to be more precise and effective than conventional methods. Finally, we discuss a possible application at the Beam Energy Scan program of the Relativistic Heavy Ion Collider.The critical point is a fixed point in finite-size scaling. To quantify the behavior of such a fixed point, we define, at a given temperature and scaling exponent ratio, the width of scaled observables for different sizes. The minimum of the width reveals the position of the fixed point, its corresponding phase transition temperature, and scaling exponent ratio. The value of this ratio tells the nature of the fixed point, which can be a critical point, a point of the first-order phase transition line, or a point of the crossover region. To demonstrate the effectiveness of this method, we apply it to three typical samples produced by the three-dimensional three-state Potts model. Results show the method to be more precise and effective than conventional methods. Finally, we discuss a possible application at the Beam Energy Scan program of the Relativistic Heavy Ion Collider.

arXiv:1905.11388Physical review. ENov 27, 20192 citations

{\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.