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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="normal">Γ</a:mi></a:math> of the Tkachenko mode scales with its energy <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>E</c:mi></c:math> as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi mathvariant="normal">Γ</d:mi><d:mo>∼</d:mo><d:msup><d:mi>E</d:mi><d:mn>3</d:mn></d:msup></d:mrow></d:math> in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature. Published by the American Physical Society 2024