On the surface of superfluids
Developing on a recent work on localized bubbles of ordinary relativistic fluids, we study the comparatively richer leading order surface physics of relativistic superfluids, coupled to an arbitrary stationary background metric and gauge field in $3+1$ and $2+1$ dimensions. The analysis is performed with the help of a Euclidean effective action in one lower dimension, written in terms of the superfluid Goldstone mode, the shape-field (characterizing the surface of the superfluid bubble) and the background fields. We find new terms in the ideal order constitutive relations of the superfluid surface, in both the parity-even and parity-odd sectors, with the corresponding transport coefficients entirely fixed in terms of the first order bulk transport coefficients. Some bulk transport coefficients even enter and modify the surface thermodynamics. In the process, we also evaluate the stationary first order parity-odd bulk currents in $2+1$ dimensions, which follows from four independent terms in the superfluid effective action in that sector. In the second part of the paper, we extend our analysis to stationary surfaces in $3+1$ dimensional Galilean superfluids via the null reduction of null superfluids in $4+1$ dimensions. The ideal order constitutive relations in the Galilean case also exhibit some new terms similar to their relativistic counterparts. Finally, in the relativistic context, we turn on slow but arbitrary time dependence and answer some of the key questions regarding the time-dependent dynamics of the shape-field using the second law of thermodynamics. A linearized fluctuation analysis in $2+1$ dimensions about a toy equilibrium configuration reveals some new surface modes, including parity-odd ones. Our framework can be easily applied to model more general interfaces between distinct fluid-phases.