Geometric pairing of Nambu--Goldstone modes in spacetime symmetry breaking

We identify a geometric pairing mechanism in systems with spontaneously broken spacetime symmetries, whereby pairs of Goldstone fields become canonically conjugate and hybridize into a single type-B Nambu--Goldstone mode (NGM). The mechanism originates from deformations of the local spacetime volume element induced by Goldstone fluctuations, which endow the Goldstone manifold with a Berry curvature. Integrating this pairing with existing reductions, we propose a general counting formula for the number of gapless NGMs in many-body systems. We demonstrate the mechanism in a microscopic model of a two-component Bose--Einstein condensate, where the dilaton and the $U(1)$ Goldstone field combine into a single NGM with quadratic dispersion.