Hydrodynamic equations for a U(N) invariant superfluid

TL;DR

This paper derives hydrodynamic equations for a U(N) invariant superfluid, revealing couplings between superflow and magnetization, and analyzing their effects on sound and spin wave dispersions.

Contribution

It introduces a generalized hydrodynamic framework for U(N) superfluids, including coupling effects and modifications to sound and spin wave behaviors.

Findings

Hydrodynamic equations reduce to a generalized Landau-Lifshitz form when velocities are zero.

Couplings between superflow and magnetization alter sound and spin wave properties.

Spin wave dispersion is quadratic at zero temperature, consistent with microscopic analysis.

Abstract

In this paper, we develop the appropriate set of hydrodynamic equations in a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau-Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau-Lifshitz theory. In comparison with the simple superfluid, the first sound and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with detailed microscopic analysis. In the Appendix, we show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.