Collective modes and sound propagation in a p-wave superconductor: Sr$_2$RuO$_4$

TL;DR

This paper theoretically investigates how collective modes, especially the unique clapping mode, influence sound propagation in the p-wave superconductor Sr$_2$RuO$_4$, highlighting their effects on sound velocities and attenuation near the critical temperature.

Contribution

It provides a detailed theoretical analysis of collective modes' impact on sound propagation in Sr$_2$RuO$_4$, emphasizing the role of the clapping mode and comparing collisionless and diffusive regimes.

Findings

Clapping mode couples to sound wave and affects sound velocity.

No resonance absorption due to the collective mode in metals.

Velocity change is significant in the collisionless limit, negligible in the diffusive limit.

Abstract

There are five distinct collective modes in the recently discovered p-wave superconductor Sr$_2$RuO$_4$; phase and amplitude modes of the order parameter, clapping mode (real and imaginary), and spin wave. The first two modes also exist in the ordinary s-wave superconductors, while the clapping mode with the energy $\sqrt{2} \Delta(T)$ is unique to Sr$_2$RuO$_4$ and couples to the sound wave. Here we report a theoretical study of the sound propagation in a two dimensional p-wave superconductor. We identified the clapping mode and study its effects on the longitudinal and transverse sound velocities in the superconducting state. In contrast to the case of $^3$He, there is no resonance absorption associated with the collective mode, since in metals $\omega/(v_F |{\bf q}|) \ll 1$, where $v_F$ is the Fermi velocity, {\bf q} is the wave vector, and $\omega$ is the frequency of the sound wave. However, the velocity change in the collisionless limit gets modified by the contribution from the coupling to the clapping mode. We compute this contribution and comment on the visibility of the effect. In the diffusive limit, the contribution from the collective mode turns out to be negligible. The behaviors of the sound velocity change and the attenuation coefficient near $T_c$ in the diffusive limit are calculated and compared with the existing experimental data wherever it is possible. We also present the results for the attenuation coefficients in both of the collisionless and diffusive limits at finite temperatures.