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arXiv:cond-mat/0303166 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universal finite-size scaling functions with exact nonuniversal metric factors

Ming‐Chya Wu, Chin‐Kun Hu, N. Sh. Izmailian

Published Jun 26, 2003• 38 citations
arXivHTMLPDF

Abstract

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

DOI
10.1103/physreve.67.065103

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Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
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.
We use heat bath dynamics to evaluate the dynamic critical exponent zzz and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that zzz is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research.
In this paper, we briefly review our recent results in the calculation of universal scaling functions for site and bond percolation on finite square, plane triangular, and honeycomb lattices.We find that, by choosing an aspect ratio for each lattice and a very small number of nonuniversal metric factors, all scaled data of the existance probability E p and the percolation probability P fallon the same universal scaling functions.We also find that free and periodic boundary conditions share the same nonuniversal metric factors.When the aspect ratio of each lattice is reduced by the same factor, the nonouniversal metric factors remain the same.The probabilities for the "appearance of n, n = 1, 2, 3, ..., percolating clusters for bond and site percolations on various planar lattices also have universal scaling functions.The implications of such results on numerical and experimental studies of critical phenomena are discussed.