Superfluid - Bose glass transition in two dimensions at T = 0: analytic solution of a mode-coupling toy model

Analytic expression for the memory function and the optical conductivity of the two-dimensional Bose gas with logarithmic interaction at T = 0 in presence of point-like impurities is obtained within the mode-coupling approximation. Depending on the value of a dimensionless combination of the model parameters proportional to the strength of the impurity potential, two different phases are distinguished, viz. the disordered superfluid and insulator (Bose glass), separated by an intermediate (quasi)metal state.