Nodes of the Gap Function and Anomalies in Thermodynamic Properties of Superfluid $^3$He

TL;DR

This paper investigates deviations in thermodynamic properties of superfluid $^3$He from BCS theory, attributing them to momentum-dependent gap functions with nodes, and derives a parameter-free relation matching experimental data.

Contribution

It introduces a model linking gap function nodes and pairing interaction sign to thermodynamic deviations, providing a new understanding of superfluid $^3$He behavior.

Findings

Deviations are due to momentum-dependent gap functions with nodes.

Sign of the pairing interaction at the Fermi surface influences deviations.

Derived a parameter-free relation consistent with experimental data.

Abstract

Departures of thermodynamic properties of three-dimensional superfluid $^3$He from the predictions of BCS theory are analyzed. Attention is focused on deviations of the ratios $\Delta(T=0)/T_c$ and $[C_s(T_c)-C_n(T_c)]/C_n(T_c)$ from their BCS values, where $\Delta(T=0)$ is the pairing gap at zero temperature, $T_c$ is the critical temperature, and $C_s$ and $C_n$ are the superfluid and normal specific heats. We attribute these deviations to the momentum dependence of the gap function $\Delta(p)$, which becomes well pronounced when this function has a pair of nodes lying on either side of the Fermi surface. We demonstrate that such a situation arises if the P-wave pairing interaction $V(p_1,p_2)$, evaluated at the Fermi surface, has a sign opposite to that anticipated in BCS theory. Taking account of the momentum structure of the gap function, we derive a closed relation between the two ratios that contains no adjustable parameters and agrees with the experimental data. Some important features of the effective pairing interaction are inferred from the analysis.