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Progress of Theoretical Physics

Order Parameter and Finite-Size Scaling

Hiroshi Takano, Y. Saito

Published Jun 1, 1985• 5 citations
Publisher

Abstract

On the basis of the finite-size scaling of the order parameter, a method to estimate the magnetic critical exponent is proposed. In this method, the magnitude of the order parameter for a finite system is calculated from two eigenvectors of the transfer matrix without ordering field. Applications of this method to the Ising and 3-state Potts models in two dimensions yield results in good agreement with the exact or conjectured values of the magnetic exponent.

DOI
10.1143/ptp.73.1369

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On the basis of the finite-size scaling of the order parameter, a method to estimate the magnetic critical exponent is proposed. In this method, the magnitude of the order parameter for a finite system is calculated from two eigenvectors of the transfer matrix without ordering field. Applications of this method to the Ising and 3-state Potts models in two dimensions yield results in good agreement with the exact or conjectured values of the magnetic exponent.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.
Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.
A method of obtaining the spontaneous magnetisation from finite-lattice matrix elements of the magnetic field operator, due to Yang (1952) and Uzelac (1980), is discussed. The method is demonstrated for the case of the Ising model in (1+1) dimensions, and is shown to provide smooth and rapidly convergent finite-lattice sequences. Applied to the case of the three-state Potts (Z 3 ) model in (1+1) dimensions, the method yields an estimate beta =0.111 09+or-0.000 05 for the critical exponent. This confirms Alexander's conjecture of universality with the hard hexagonal model.