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

Published 16 Dec 2022 in cond-mat.str-el, cond-mat.quant-gas, and hep-th | (2212.08671v2)

Abstract: 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 $\Gamma\sim E3$ in the low-energy limit. We also discuss the width of the Tkachenko mode at a small temperature.

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  48. Notice that in Ref. [51], the contribution of the Kohn's mode is higher order in the derivative expansion since the power-counting scheme with ω∼𝐤2similar-to𝜔superscript𝐤2\omega\sim\mathbf{k}^{2}italic_ω ∼ bold_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT was used there. That counting originates from the dispersion of the low-energy Tkachenko mode.
  49. The argument we use here is nothing but saying that Xasuperscript𝑋𝑎X^{a}italic_X start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT and xisuperscript𝑥𝑖x^{i}italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT are related by a area-preserving diffeomorphism generated by a function ϕ⁢(𝐱)italic-ϕ𝐱\phi(\mathbf{x})italic_ϕ ( bold_x ). Notice that the auxiliary Poisson bracket in Eq. (S18) does not originate from the effective field theory Lagrangian.
  50. While the coupling to A0subscript𝐴0A_{0}italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT source is local, one can check that in this formulation the coupling to the 𝒜isubscript𝒜𝑖\mathcal{A}_{i}caligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT source is non-local.
  51. The quadratic in ϕitalic-ϕ\phiitalic_ϕ contribution to the canonical conjugate momentum πϕsubscript𝜋italic-ϕ\pi_{\phi}italic_π start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT does not contain temporal derivatives and thus does not affect the canonical commutation relation (S41) up to second order in ϕitalic-ϕ\phiitalic_ϕ.
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