Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate

We construct an effective field theory describing the collective Tkachenko oscillation mode of a vortex lattice in a two-dimensional rotating Bose-Einstein condensate in the long-wavelength regime. The theory has the form of a noncommutative field theory of a Nambu-Goldstone boson, which exhibits a noncommutative version of dipole symmetry. From the effective field theory, we show that, at zero temperature, the decay width $\Gamma$ of the Tkachenko mode scales with its energy $E$ as $\Gamma\sim E^3$ in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature.