Failure of the Goldstone Theorem for Vector Fields and Boundary-Mode Proliferation in Hyperbolic Lattices

Hyperbolic lattices extend crystallinity into curved space, where negative curvature and exponentially large boundaries reshape collective excitations beyond Euclidean intuition. In this Letter, we push the study beyond scalar fields by exploring vector fields on hyperbolic lattices. Using phonons as an example, we show that the Goldstone theorem breaks down for vector fields in hyperbolic lattices. In stark contrast to Euclidean crystals, where the Goldstone theorem ensures that acoustic phonon modes are gapless, hyperbolic lattices with coordination number $z > 2d$ exhibit a finite bulk phonon gap. We identify the origin of this breakdown: the Goldstone modes here belong to nonunitary representations of the translation group and therefore cannot form gapless excitation branches. We further show that when boundaries are included, this bulk spetrum gap is filled by an extensive number of low-frequency boundary modes.