Grand canonical partition function of a serial metallic island system

TL;DR

This paper calculates the grand canonical partition function for a serial metallic island system using path integral formalism, providing insights into Coulomb blockade effects and stability diagrams.

Contribution

It introduces a novel path integral approach to compute the partition function and electron number in metallic island systems, including Coulomb interactions and tunneling effects.

Findings

Analytical expression for the partition function using phase fields

Method to calculate average electron number via winding numbers

Application to Coulomb blockade and stability diagram construction

Abstract

We present a calculation of the grand canonical partition function of a serial metallic island system by the imaginary-time path integral formalism. To this purpose, all electronic excitations in the lead and island electrodes are described using Grassmann numbers. Coulomb charging energy of the system is represented in terms of phase fields conjugate to the island charges. By the large channel approximation, the tunneling action phase dependence can also be determined explicitly. Therefore, we represent the partition function as a path integral over phase fields with a path probability given in an analytically known effective action functional. Using the result, we also propose a calculation of the average electron number of the serial island system in terms of the expectation value of winding numbers. Finally, as an example, we describe the Coulomb blockade effect in the two-island system by the average electron number and propose a method to construct the quantum stability diagram.