Hydrodynamic description of collective modes in heavy-fermion superconductors

Abstract

We present a hydrodynamical theory for the collective modes of a p-wave superconductor that is described by an order parameter d which transforms as a vector under rotations generated by the spin angular-momentum operator J=L+S. Such superconducting order is possible in metals with cubic, hexagonal, or tetragonal crystal symmetry whose pairing electrons experience strong spin-orbit coupling. This is true of the known heavy-fermion superconductors. The variables J and d follow a Hamiltonian dynamics that results in an internal Josephson effect between the populations of spin-up and spin-down Cooper pairs. The effect is similar to the longitudinal spin-resonance behavior observed in the superfluid phases of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. Viscous corrections to the spin angular-momentum conservation of the pairs and to the momentum conservation of the bulk are introduced, serving to couple this Josephson mode to sound propagation. The coupling, which exists for both longitudinal and transverse sound, results in a resonant absorption contribution to the sound attenuation constant with a Lorentzian shape centered at the longitudinal spin-orbit resonance frequency. The height of this Lorentzian varies as ${\omega }^2$ with frequency, characteristic of hydrodynamics, and its width depends upon frequency and the amount of quasiparticle scattering. It is suggested that this mechanism accounts for the sound absorption peak recently observed in the heavy-fermion superconductor ${\mathrm{UBe}}_{13}$.