Effective field theory for the superfluid vortex lattice from coset construction

TL;DR

The paper develops a symmetry-controlled effective field theory for a two-dimensional vortex lattice in rotating Bose–Einstein condensates trapped in a harmonic potential. Using Newton–Cartan geometry and the magnetic Bargmann algebra, it applies the coset construction to identify covariant building blocks and to implement inverse Higgs constraints, isolating a single gapless Tkachenko mode after removing the Kohn mode. The leading low-energy action is a Lifshitz-type theory with ω ∝ k^2, and the next-to-leading order action includes cubic interactions, with a new symmetry-allowed term not present in prior LLL analyses. These results provide a general, microscopics-free description of vortex-crystal dynamics, consistent with Kohn’s theorem, and extend previous frameworks beyond the strict lowest-Landau-level limit to finite Ω and m.

Abstract

Guided by symmetry principles, we construct an effective field theory that captures the long-wavelength dynamics of two-dimensional vortex crystals observed in rotating Bose-Einstein condensates trapped in a harmonic potential. By embedding the system into Newton--Cartan spacetime and analyzing its isometries, we identify the appropriate spacetime symmetry group for trapped condensates at finite angular momentum. After introducing a coarse-grained description of the vortex lattice we consider a homogeneous equilibrium configuration and discuss the associated symmetry breaking pattern. We apply the coset construction method to identify covariant structures that enter the effective action and discuss the physical interpretation of the inverse Higgs constraints. We verify that Kohn's theorem is satisfied within our construction and subsequently focus on the gapless sector of the theory. In this regime, the effective theory accommodates a single gapless excitation--the Tkachenko mode--for which we construct both the leading-order and next-to-leading-order actions, the latter including cubic interaction terms.