Proximity effect between a dirty Fermi liquid and superfluid 3He

TL;DR

This paper investigates how superfluid $^3$He interacts with a dirty Fermi liquid in aerogel, revealing that odd-frequency s-wave pairs dominate the proximity effect and influence local density of states.

Contribution

It provides a fully self-consistent analysis of the proximity effect in superfluid $^3$He with aerogel, highlighting the role of odd-frequency pairs near the interface.

Findings

Odd-frequency s-wave pairs penetrate deep into the aerogel.

Proximity-induced superfluidity is dominated by odd-frequency pairs.

Local density of states shows a zero-energy peak due to odd-frequency pairs.

Abstract

The proximity effect in superfluid $^3$He partly filled with high porosity aerogel is discussed. This system can be regarded as a dirty Fermi liquid/spin-triplet p-wave superfluid junction. Our attention is mainly paid to the case when the dirty layer is in the normal state owing to the impurity pair-breaking effect by the aerogel. We use the quasiclassical Green's function to determine self-consistently the spatial variations of the p-wave order parameter and the impurity self-energy. On the basis of the fully self-consistent calculation, we analyze the spatial dependence of the pair function (anomalous Green's function). The spin-triplet pair function has in general even-frequency odd-parity and odd-frequency even-parity components. We show that the admixture of the even- and odd-frequency pairs occurs near the aerogel/superfluid $^3$He-B interface. Among those Cooper pairs, only the odd-frequency s-wave pair can penetrate deep into the aerogel layer. As a result, the proximity-induced superfluidity in a thick aerogel layer is dominated by the Cooper pair with the odd-frequency s-wave symmetry. We also analyze the local density of states and show that it has a characteristic zero-energy peak reflecting the existence of the odd-frequency s-wave pair, in agreement with previous works using the Usadel equation.