Effective theory of surface oscillations in self-bound superfluid droplets

We investigate the low-energy dynamics of small-amplitude surface oscillations of spherical superfluid droplets in vacuum. Starting from the effective field theory of superfluid phonons, we derive an effective action governing the surface oscillations under a fixed particle-number constraint. The normal-mode eigenfrequencies $\omega_{\ell}$ for each angular momentum quantum number $\ell$ are determined and shown to depend on a dimensionless parameter measuring the ratio of surface tension to bulk compressibility energy. We identify a critical value of this parameters at which the breathing mode ($\ell = 0$) becomes mechanically unstable, and show that all multipole surface modes with $\ell \geq 2$ enter the low-energy regime when the surface tension is sufficiently small. Within this regime, we further quantize the surface oscillations, whose quanta correspond to ripplons, allowing the construction of general multi-ripplon states obeying angular-momentum selection rules. We also apply our formalism to a concrete example: a weakly interacting two-component Bose mixture realizing a self-bound superfluid droplet. The resulting description is universal in the sense that it applies to surface dynamics of generic nonrelativistic superfluids with a free interface, independent of microscopic details.