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Monte-Carlo study of anisotropic scaling generated by disorder

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 $ξ_\parallel$ in the direction along defects, and a perpendicular correlation length $ξ_\perp$ in the direction perpendicular to the lines. Both $ξ_\parallel$ and $ξ_\perp$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents $ν_\parallel$ and $ν_\perp$ take different values. This property is specific for anisotropic scaling and the ratio $ν_\parallel/ν_\perp$ defines the anisotropy exponent $θ$. Estimates of quantitative characteristics of the critical behaviour for such systems were only obtained up to now 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 non-magnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $θ$ of the system are obtained, as well as an estimate of the susceptibility exponent $γ$. Our results corroborate the renormalization group predictions obtained earlier.