Non-commutative effective field theory of the lowest Landau level superfluid
Nandagopal Manoj
TL;DR
This work provides a manifestly lowest-Landau-level effective field theory for a 2D Bose superfluid in a rapidly rotating trap by deriving a non-commutative field theory from a LLL-projected coherent-state path integral. The approach connects NC degrees of freedom to microscopic parameters, enabling quantitative predictions such as the Tkachenko-mode dispersion $\omega^2 = \frac{\lambda}{2} \mu^2 l_B^4 k^4$ with $\lambda \approx 0.29$ (triangular lattice) and a renormalized interaction $g^* \approx 1.160\, g$. It clarifies how magnetic translation symmetry is encoded in the NC description, discusses UV-IR aspects, and sets the stage for extensions to LLL hydrodynamics and trap-induced inhomogeneities relevant to rotating Bose-Einstein condensates. The framework provides a bridge between microscopic lattice-scale physics and long-wavelength universal behavior in LLL superfluids.
Abstract
A 2+1D superfluid in a rapidly rotating trap forms an array of vortices, with collective excitations called Tkachenko modes. Du et al. (2024) argued from an effective field theory viewpoint that these excitations are described by a field theory living on a non-commutative space. We elucidate the microscopic origin of these non-commutative fields, and present a novel derivation of the effective field theory for this superfluid using a lowest Landau level projected coherent state path integral approach. Besides conceptual clarity, this approach makes quantitative predictions about the long-wavelength, low-energy behavior in terms of the microscopic parameters of the short-range interacting lowest Landau level superfluid -- relevant to trapped Bose-Einstein condensate experiments.