Pairbreaking effect of correlated impurities in a superfluid Fermi liquid

The conventional theory of superconducting alloys does not take into account a discreet character of impurities. Experimental data for superfluid $^3$He in aerogel and for some of high-$T_c$ superconductors reveal a significant discrepancy between the observed temperatures $T_c$ of their transitions in the superfluid or superconducting state and that predicted by the theory. Here a theoretical scheme is presented for finding corrections to the $T_c$ originating from spatial correlations between impurities. Analysis is limited to the Ginzburg and Landau temperature region. The shift of $T_c$ with respect to the pure material is represented as a series in concentration of the impurities $x$. In the first order on $x$ the conventional mean-field result for the lowering of $T_c$ is recovered. Contribution of correlations enters the second order term. It is expressed via the structure factor of the ensemble of impurities. For superfluid $^3$He in silica aerogel the sign of the correction corresponds to an enhancement of the $T_c$ so that the resulting pairbreaking effect of impurities is weakened. When correlation radius of impurities $R$ exceeds the coherence length of the superfluid $\xi_0$ the contribution of correlations to the shift of $T_c$ acquires a factor $\sim(R/\xi_0)^2$ and the weakening of the pairbreaking effect becomes appreciable. The presented scheme is applied to the superfluid $^3$He in aerogel.