Superfluid transition in a correlated defect network

Motivated by recent experiments on possible supersolid behavior of $^4$He solids at low temperature, we consider a model of superfluidity in a defected solid containing a system spaning network of correlated linear dislocations, or planar grain boundaries. Using arguments based on the Harris criterion, as well as numerical simulations, we find that such correlated quenched disorder shifts the familiar superfluid lambda transition to a new disordered universality class in which the correlation length exponent $\nu\ge 1$. This results in the temperature-derivates for the superfluid density, $d\rho_{\rm s}/dT$, and for the heat capacity, $dc/dT$, remaining finite at the transition $T_{\rm c}$, and thus a less singular transition, profoundly different from the usual lambda transition.