A small superconducting grain in the canonical ensemble

TL;DR

This paper investigates superconducting correlations in ultrasmall grains with fixed particle number, revealing parity dependence and universal behavior of key properties as functions of level spacing and BCS gap.

Contribution

It introduces a detailed analysis of superconductivity in small grains within the canonical ensemble using the Lanczos method, highlighting universal scaling behaviors.

Findings

Ground state and excitation gap are parity dependent.

Both quantities are universal functions of the ratio δ/Δ.

Superconductivity characterized by energy space correlation functions.

Abstract

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing $\delta$. Both quantities turn out to be parity dependent and universal functions of the ratio $\delta/\Delta$ ($\Delta$ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.