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Brazilian Journal of Physics

Recursive-search method for ferromagnetic ising systems: combination with a finite-size scaling approach

PATRICK CARVALHO DA SILVA, Umberto Laino Fulco, Fernando Dantas Nobre, L. R. da Silva, LIACIR S. LUCENA

Published Jun 1, 2002• 9 citations
Publisher

Abstract

A method for obtaining critical properties of physical systems is presented. Based on a recursive relation involving a physical parameter of the system, it drives the system spontaneously to the critical point, providing an efficient way to estimate critical properties. The method is illustrated for several ferromagnetic Ising systems on well-known Bravais lattices. A finite-size scaling approach is performed, by applying the method on lattices of different sizes. The efficiency of the method is confirmed by evaluating critical temperatures, as well as critical exponents, that turn up to be in good agreement with those available in the literature, with a relatively small computational effort.

DOI
10.1590/s0103-97332002000300024

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A method for obtaining critical properties of physical systems is presented. Based on a recursive relation involving a physical parameter of the system, it drives the system spontaneously to the critical point, providing an efficient way to estimate critical properties. The method is illustrated for several ferromagnetic Ising systems on well-known Bravais lattices. A finite-size scaling approach is performed, by applying the method on lattices of different sizes. The efficiency of the method is confirmed by evaluating critical temperatures, as well as critical exponents, that turn up to be in good agreement with those available in the literature, with a relatively small computational effort.
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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.
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.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2- and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-dimensional Ising Model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.