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arXiv:1901.06088 · Physical Review Letters

Bosonic Superfluid on the Lowest Landau Level

Sergej Moroz, Dam Thanh Son

Published Jun 11, 2019• 19 citations
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Abstract

We develop a low-energy effective field theory of a two-dimensional bosonic superfluid on the lowest Landau level at zero temperature and identify a Berry term that governs the dynamics of coarse-grained superfluid degrees of freedom. For an infinite vortex crystal we compute how the Berry term affects the low-energy spectrum of soft collective Tkachenko oscillations and nondissipative Hall responses of the particle number current and stress tensor. This term gives rise to a quadratic in momentum term in the Hall conductivity, but does not generate a nondissipative Hall viscosity.

DOI
10.1103/physrevlett.122.235301

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We develop a low-energy effective field theory of a two-dimensional bosonic superfluid on the lowest Landau level at zero temperature and identify a Berry term that governs the dynamics of coarse-grained superfluid degrees of freedom. For an infinite vortex crystal we compute how the Berry term affects the low-energy spectrum of soft collective Tkachenko oscillations and nondissipative Hall responses of the particle number current and stress tensor. This term gives rise to a quadratic in momentum term in the Hall conductivity, but does not generate a nondissipative Hall viscosity.
Using boson-vortex duality, we formulate a low-energy effective theory of a two-dimensional vortex lattice in a bosonic Galilean-invariant compressible superfluid. The excitation spectrum contains a gapped Kohn mode and an elliptically polarized Tkachenko mode that has quadratic dispersion relation at low momenta. External rotation breaks parity and time-reversal symmetries and gives rise to Hall responses. We extract the particle number current and stress tensor linear responses and investigate the relations between them that follow from Galilean symmetry. We argue that elementary particles and vortices do not couple to the spin connection which suggests that the Hall viscosity at zero frequency and momentum vanishes in a vortex lattice.
We determine and analyze collective normal modes of a finite disk-shaped two-dimensional vortex crystal formed in a compressible bosonic superfluid in an artificial magnetic field. Using the microscopic Gross-Pitaevskii theory in the lowest Landau level approximation, we generate vortex crystal ground states and solve the Bogoliubov--de Gennes equations for small amplitude collective oscillations. We find chiral surface waves that propagate at frequencies larger than those of the bulk Tkachenko modes. Furthermore, we study low frequency bulk excitations and identify a Ruderman mode, which we find is well described by a previously developed low-energy effective field theory.
We construct, to leading orders in the momentum expansion, an effective theory of a chiral (px+ipy{p}_{x}+i{p}_{y}px​+ipy​) two-dimensional fermionic superfluid at zero temperature that is consistent with nonrelativistic general coordinate invariance. This theory naturally incorporates the parity and time-reversal violating effects such as the Hall viscosity and the edge current. The particle number current and stress tensor are computed and their linear response to electromagnetic and gravitational sources is calculated. We also consider an isolated vortex in a chiral superfluid and identify the leading chirality effect in the density depletion profile.
Hall viscosity is a nondissipative viscosity occurring in systems with broken time-reversal symmetry, such as quantum Hall phases and p+ipp+ipp+ip superfluids. Despite Hall viscosity's expected ubiquity and past observations in: classical soft matter, optical, and graphene systems, it has yet to be measured experimentally in any macroscopic quantum state of matter. Toward this end, we describe the observable effects of Hall viscosity in a simple family of rotating Bose-Einstein condensates of electrically neutral bosons, in which all of the bosons condense into a single lowest Landau level (LLL) orbital. Such phases are accessible to current cold atom experiments, and we dub them LLL superfluids. We demonstrate that LLL superfluids possess a nonuniversal Hall viscosity, leading to a range of observable consequences such as rotation of vortex-antivortex dipoles and wave-vector-dependent corrections to the speed of sound. Furthermore, using a coherent state path integral approach, we present a microscopic derivation of the Landau-Ginzburg equations of a LLL superfluid, showing explicitly how Hall viscosity enters.
A (2+12+12+1)-dimensional superfluid in a rapidly rotating trap forms an array of vortices, with collective excitations called Tkachenko modes. Du et al. [Phys. Rev. Res. 6, L012040 (2024)] argued from an effective field theory viewpoint that these excitations are described by a field theory living on a noncommutative space. We elucidate the microscopic origin of these noncommutative fields, and present a derivation of the effective field theory for this superfluid using a lowest Landau level projected coherent state path-integral approach. Aside from 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.
We have found an exact analytical solution of the Bogoliubov--de Gennes equations for the Tkachenko modes of the vortex lattice in the lowest Landau level (LLL) in the thermodynamic limit (geometry of an infinite plane) at any momenta and calculated their damping rates. At finite temperatures both Beliaev and Landau damping leads to momentum-independent damping rates in the low-energy limit, which shows that, at sufficiently low energies, Tkachenko modes become strongly damped. We then found that the mean-square fluctuations of the density grow logarithmically at large distances, which indicates that the state is ordered in the vortex lattice only on a finite (although exponentially large) distance scale and introduces a low-momentum cutoff. Using this circumstance we showed that at finite temperatures the one-body density matrix undergoes an exponential decay at large distances.