Correlation functions, free energies and magnetizations in the two-dimensional random-field Ising model

Details Correlation functions, free energies and magnetizations in the two-dimensional random-field Ising model Correlation functions, free energies and magnetizations in the two-dimensional random-field Ising model

2001-05-16

arxiv.org/abs/cond-mat/0105326v1

Physical Review E 64, article # 036117 (2001)

Physics

Condensed Matter

Statistical Mechanics

RevTeX, 11 pages, 9 eps figures included, submitted to Phys. Rev. E

Scientific paper

10.1103/PhysRevE.64.036117

Transfer-matrix methods are used to calculate spin-spin correlation functions ($G$), Helmholtz free energies ($f$) and magnetizations ($m$) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature $T_{c 0}$, on long strips of width $L = 3 - 18$ sites, for binary field distributions. Analysis of the probability distributions of $G$ for varying spin-spin distances $R$ shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large $R$. Connections between field-- and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed $R/L$, the fractional widths of correlation-function distributions saturate asymptotically with $L^{-2.2}$. Considering an added uniform applied field $h$, a connection between $f(h)$, $m(h)$, the Gibbs free energy $g(m)$ and the distribution function for the uniform magnetization in zero uniform field, $P_0(m)$, is derived and first illustrated for pure systems, and then applied for non-zero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of $P_0(m)$ varies against (non-vanishing, but small) random-field intensity $H_0$ as $H_0^{-3/7}$.

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