  We study the effect of uncorrelated random disorder on the temperature
dependence of the superfluid stiffness in the two-dimensional classical XY
model. By means of a perturbative expansion in the disorder potential,
equivalent to the T-matrix approximation, we provide an extension of the
effective-medium-theory result able to describe the low-temperature stiffness,
and its separate diamagnetic and paramagnetic contributions. These analytical
results provide an excellent description of the Monte Carlo simulations for two
prototype examples of uncorrelated disorder. Our findings offer an interesting
perspective on the effects of quenched disorder on longitudinal phase
fluctuations in two-dimensional superfluid systems.
