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 $\mathrm{\Gamma }$ of the Tkachenko mode scales with its energy $E$ as $\mathrm{\Gamma }\sim E^3$ in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature.

• Received 14 January 2023
• Revised 23 September 2023
• Accepted 2 February 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.L012040

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Published by the American Physical Society