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

TL;DR

This paper derives analytical formulas for Nambu-Goldstone modes along topological defects in superfluids, connecting finite and infinite system behaviors, and validates them with numerical simulations.

Contribution

It provides the first fully analytical formulas for dispersion relations and wavefunctions of Kelvin and ripplon modes across system sizes, unifying finite and infinite cases.

Findings

Analytical formulas match numerical simulations.

Formulas interpolate between finite and infinite system dispersion relations.

Explicit quasiparticle wavefunctions are derived.

Abstract

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.