Geometry-induced flocking and topological sound on a defect-free curved surface

Abstract

We study an ordered polar active flock on a torus and show that topological sound persists on a compact curved surface without topological defects or physical boundaries. Using the covariant Toner Tu theory, we derive an effective nonHermitian Dirac operator whose curvature-induced mass changes sign across the outer and inner equators, producing two Jackiw Rebbi domain walls. These support co-propagating but distinct chiral edge excitations: a density mode localized on the positively curved outer equator and a Goldstone mode localized on the negatively curved inner equator. The bulk bands possess opposite half-integer Chern numbers whose jumps across the domain walls are determined by the sign of the Gaussian curvature. We further show that the localised modes are protected by a one-dimensional Callias index theorem, while the sum of the local indices obeys the Poincare Hopf theorem on the compact surface. Our results establish that curvature alone, independent of defects and boundaries, is sufficient to generate and protect topological sound in active matter, providing a unified connection between non-Hermitian topology, differential geometry, and hydrodynamic theory of collective motion.