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arXiv:cond-mat/0108369 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model

Щур Лев Николаевич, Oleg A. Vasilyev

Published Dec 14, 2001• 22 citations

Abstract

We present a different way of probing the universality class of the site-diluted two-dimensional Ising model. We analyze Monte Carlo data for the magnetic susceptibility, introducing a fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude, and the sample-dependent pseudocritical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration q of the empty sites for all investigated values of q < or =0.25. At the same time the average effective exponent gamma(eff) is found to vary with the concentration q, which may be argued to be due to logarithmic corrections to the power law of the pure system. These corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.

DOI: 10.1103/physreve.65.016107

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We present a different way of probing the universality class of the site-diluted two-dimensional Ising model. We analyze Monte Carlo data for the magnetic susceptibility, introducing a fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude, and the sample-dependent pseudocritical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration q of the empty sites for all investigated values of q < or =0.25. At the same time the average effective exponent gamma(eff) is found to vary with the concentration q, which may be argued to be due to logarithmic corrections to the power law of the pure system. These corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.

We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization

M

per spin, energy

E

per spin, magnetic susceptibility

χ

per spin, and specific heat

C

per spin are then calculated after the system has equilibrated. At small dilution concentrations

d<0.1

, we find that the value of the critical exponent

β

does not deviate from its pure Ising value. At higher dilution concentrations

d>0.1

, however, we find

β

to strongly depend on the value of

d

. We are able to locate a critical temperature

T_c

and a critical dilution concentration

d_c

where the phase transition occurs. We find

T_c

to depend linearly on

d

. In the phase diagrams of

M

,

E

,

χ

, and

C

, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility

χ

at the low temperature and high dilution region.

We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio

Γ^+/Γ^-

for the magnetic susceptibility critical amplitudes. Our estimate

Γ^+/Γ^- = 1.67 \pm 0.15

is in a good agreement with the recent MC analysis of the random-bond Ising model giving further support that both random-site and random-bond dilutions lead to the same universality class.

Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections and of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.