Effective charging energy of the single electron box

TL;DR

This paper introduces a novel Monte Carlo method to accurately compute the effective charging energy of a single-electron box across a wide range of tunneling conductances, revealing exponential suppression and crossover behaviors.

Contribution

The authors develop an efficient cluster algorithm and a transition matrix Monte Carlo approach, extending the calculation of $E_C^*$ by over 30 orders of magnitude and analyzing its temperature dependence.

Findings

$E_C^*$ decreases exponentially with increasing tunneling conductance $lpha$.

Identifies a crossover from intermediate to zero temperature behavior for fixed $lpha$.

Numerically confirms and compares theoretical predictions for strong tunneling limit.

Abstract

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.