Monte Carlo study of anisotropic scaling generated by disorder

Abstract

We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ in the direction along defects and a perpendicular correlation length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ in the direction perpendicular to the lines. Both ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ take different values. This property is specific for anisotropic scaling and the ratio ${\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ defines the anisotropy exponent $\ensuremath{\theta}$. Until now, estimates of quantitative characteristics of the critical behavior for such systems have been obtained only within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and nonmagnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\ensuremath{\theta}$ of the system are obtained, as well as an estimate of the susceptibility exponent $\ensuremath{\gamma}$. Our results corroborate the renormalization group predictions obtained earlier.