The Heat Capacity of \(^3\)He-B in Silica Aerogel

The thermodynamic potential for superfluid \(^3\)He-B embedded in a homogeneously distributed random potential is calculated from a quasiclassical reduction of the Luttinger–Ward functional to leading order in \({\mathbf s} =k_{\text {B}}T_c/E_f\). The resulting functional provides an extension of the Ginzburg–Landau free energy functional to all temperatures \(0<T\le T_c\). Theoretical predictions based on this functional for the heat capacity of superfluid \(^3\)He-B embedded in homogeneous, isotropic silica aerogel are in good agreement with experimental reports for superfluid \(^3\)He-B infused into 98.2% porous silica aerogel over the pressure range \(p=11-29\,\text{ bar }\). The analysis supports a conclusion that superfluid \(^3\)He-B infused into high-porosity silica aerogels is a gapless superfluid at all pressures.