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Chinese Journal of Physics

Au Effkient Monte Carlo Technique for Continuous Functions and Universal Scaling Functions

Koo-Chul Lee, Y. K. Kwon

Published Oct 1, 1992• 0 citations
OpenAlex

Abstract

The functional Monte Carlo method devised by one of the authors for the efficiency and accuracy can be used to calculate universal scaling functions. In order to demonstrate the technique we take 3-state Potts model for which many of the exact critical properties are known. Various thermodynamic functions of Potts model on square lattice of sizes ranging from 6 X 6 to 33 X 33 are calculated and the two scale factor universality (or hyperuniversality) theory put forward by Privman and Fisher is tested and verified. We also estimate the higher order derivatives of universal function using hyperuniversality hypothesis.

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The functional Monte Carlo method devised by one of the authors for the efficiency and accuracy can be used to calculate universal scaling functions. In order to demonstrate the technique we take 3-state Potts model for which many of the exact critical properties are known. Various thermodynamic functions of Potts model on square lattice of sizes ranging from 6 X 6 to 33 X 33 are calculated and the two scale factor universality (or hyperuniversality) theory put forward by Privman and Fisher is tested and verified. We also estimate the higher order derivatives of universal function using hyperuniversality hypothesis.
We present a method which allows calculation of whole universal finite-size-scaling functions from Monte Carlo data. The crux of the new method is a technique of isolating the singular part from the total free energy. We apply this method to the 3-state Potts model on a square lattice and find the normalized scaling function Y(x) in the form 1+x+5.31x2{\mathit{x}}^{2}x2-1.2x3{\mathit{x}}^{3}x3-0.67x4{\mathit{x}}^{4}x4 . . . near the bulk critical point x=0, together with the normalizing universal amplitude E=1.053 and a nonuniversal metric factor D=-0.387. It also behaves as Y(x)\ensuremath{\approxeq}(\ifmmode\pm\else\textpm\fi{}x${)}^{\mathit{d}\ensuremath{\nu}}$${\mathit{Q}}_{\ifmmode\pm\else\textpm\fi{}}$ for large x in agreement with the hyperuniversality hypothesis with Q+{\mathit{Q}}_{+}Q+​/${\mathit{Q}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}1.
The idea of universal finite-size-scaling functions of the Ising model is tested by Monte Carlo simulations for various lattices. Not only regular lattices such as the square lattice but quasiperiodic lattices such as the Penrose lattice are treated. We show that the finite-size-scaling functions of the order parameter for various lattices are collapsed on a single curve by choosing two nonuniversal scaling metric factors. We extend the idea of the universal finite-size-scaling functions to the order-parameter distribution function. We pay attention to the effects of boundary conditions.
The idea of universal finite-size-scaling functions of the Ising model is tested by Monte Carlo simulations for various lattices. Not only regular lattices such as the square lattice but quasiperiodic lattices such as the Penrose lattice are treated. We show that the finite-size-scaling functions of the order parameter for various lattices are collapsed on a single curve by choosing two nonuniversal scaling metric factors. We extend the idea of the universal finite-size-scaling functions to the order-parameter distribution function. We pay attention to the effects of boundary conditions.
Abstract The critical behavior of the ( n + 1 ) -states Potts model in d -dimensions is studied with functional renormalization group techniques. We devise a general method to derive β -functions for continuous values of d and n and we write the flow equation for the effective potential (LPA’) when instead n is fixed. We calculate several critical exponents, which are found to be in good agreement with Monte Carlo simulations and ϵ-expansion results available in the literature. In particular, we focus on Percolation ( n → 0 ) and Spanning Forest ( n → − 1 ) which are the only non-trivial universality classes in d = 4,5 and where our methods converge faster.
Universality is a fundamental concept in modern physics. For the q-state Potts model, the critical exponents are merely determined by the order-parameter symmetry S_{q}, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment, i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of 0<q<2. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyperuniversality that the critical exponents are the same for 0<q<2 and, moreover, the Ising system (q=2) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be q dependent. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.
The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.