Single-electron box and the helicity modulus of an inverse square XY model

We calculate the average number of electrons on a metallic single-electron box as a function of the gate voltage for arbitrary values of the tunneling conductance. In the vicinity of the plateaus the problem is equivalent to calculating the helicity modulus of a classical inverse square XY model in one dimension. By a combination of perturbation theory, a two-loop renormalization group calculation, and a Monte Carlo simulation in the intermediate regime we provide a complete description of the smearing of the Coulomb staircase at zero temperature with increasing conductance.