Parity effect in a small superconducting grain: A rigorous result

TL;DR

This paper rigorously proves that the parity parameter in ultra-small superconducting grains is always non-zero, establishing a fundamental property of such quantum systems using advanced mathematical techniques.

Contribution

It introduces a rigorous proof that the parity parameter remains non-zero in small superconducting grains, using a generalized Lieb's spin-reflection positivity method.

Findings

Parity parameter is non-zero in small superconducting grains

A positive lower bound for the parity parameter is established

The proof employs a generalized Lieb's spin-reflection positivity technique

Abstract

The parity effect in an ultra-small superconducting grain is examined. By applying a generalized version of Lieb's spin-reflection positivity technique, we show rigorously that the parity parameter $\Delta_P$ is nonvanishing in such a system. A positive lower bound for $\Delta_P$ is derived.