Critical behavior ofm-component magnets with correlated impurities

We study the critical behavior of an $m$-component classical spin system with quenched impurities correlated along an ${\epsilon }_d$-dimensional "line" and randomly distributed in $d-{\epsilon }_d$ dimensions ($d=4-\epsilon$). The presence of this line of impurities makes the system anisotropic and the interactions highly nonlocal. The renormalization group (RG) is used to approach the critical region and the quantities of interest are calculated in a double $\epsilon$, ${\epsilon }_d$ expansion. A two-loop calculation is needed to expose fully the divergent structure, and the theory is proved to be renormalizable up to this order. A consequence of the double $\epsilon$, ${\epsilon }_d$ expansion is the fact that the RG functions and consequently the critical exponents depend on the ratio $\frac{{\epsilon }_d}{(\epsilon +{\epsilon }_d)}$. The solution of the RG equations leads to the existence of two correlation lengths: parallel to the "line" and perpendicular to it, with critical exponents ${\nu }_{\parallel }$ and ${\nu }_{\perp }$, respectively, with the relation ${\nu }_{\parallel }=z{\nu }_{\perp }$. The exponent $z$ results from the presence of anisotropy in the system. New scaling laws are found for the critical exponents: $\gamma ={\nu }_{\perp }(2-\eta )$ and $\alpha =2-(d-{\epsilon }_d){\nu }_{\perp }-{\epsilon }_d{\nu }_{\parallel }$. We establish a relation between our model and a quantum model in one less dimension with random pointlike impurities. For this system we predict a quantum-to-classical crossover at finite temperature with cross-over exponent ${\nu }_{\parallel }^{-1\mathrm{}}$.