Effective charging energy of the single electron box

We present numerical results on electron tunneling in a single-electron box at low temperature. The effective action of this device is equivalent to the Hamiltonian of a classical XY spin chain with long ranged interactions. Using an efficient cluster algorithm and a new transition matrix Monte Carlo approach, we are able to compute the effective charging energy $E_C^*$ in the limit of very small tunneling resistance. While previous Monte Carlo simulations were restricted to the weak and intermediate tunneling regimes, our method extends the range of $E_C^*$-values by more than 30 orders of magnitude. This allows us to clearly observe the exponential suppression of $E_C^*$ with increasing tunneling conductance $\alpha$. For large, but fixed $\alpha$, the correction to the leading exponential behavior exhibits a crossover from an intermediate temperature behavior at $\beta E_C^* \ll 1$, to zero temperature behavior at $\beta E_C^*\gg 1$. We determine this correction in both regimes and compare the numerical results to the numerous and controversial theoretical predictions for the strong tunneling limit.