Nambu-Goldstone Modes Propagating along Topological Defects: Kelvin and Ripple Modes from Small to Large Systems

Nambu-Goldstone modes associated with (topological) defects such as vortices and domain walls in (super)fluids are known to possess quadratic/non-integer dispersion relations in finite/infinite-size systems. Here, we report interpolating formulas connecting the dispersion relations in finite- and infinite-size systems for Kelvin modes along a quantum vortex and ripplons on a domain wall in superfluids. Our method can provide not only the dispersion relations but also the explicit forms of quasiparticle wavefunctions $(u,v)$. We find a complete agreement between the analytical formulas and numerical simulations. All these formulas are derived in a fully analytical way, and hence not empirical ones. We also discuss common structures in the derivation of these formulas and speculate on the general procedure.