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Physical Review Letters

New numerical method to study phase transitions

Jooyoung Lee, J. Michael Kosterlitz

Published Jul 9, 1990• 324 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.65.137

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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 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 unambigously a weak first order transition even when accessible system sizes are L/ξ<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.
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.
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.