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arXiv:1803.10934 · SciPost Physics

Effective field theory of a vortex lattice in a bosonic superfluid

Sergej Moroz, Carlos Hoyos, Claudio Benzoni, Dam Thanh Son

Published Oct 26, 2018• 35 citations
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Abstract

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.

DOI
10.21468/scipostphys.5.4.039

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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 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.
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.
We study Tkachenko modes of the square vortex lattice of a two-component Bose-Einstein condensate (BEC) in the mean-field quantum Hall regime, considering the coupling of these modes with density excitations. We derive the hydrodynamic equations and obtain the dispersion relations of the excitation modes. We find that there are two types of excitations, gapped inertial modes and gapless Tkachenko modes. These modes have two branches which we call acoustic and optical modes in analogy with phonons. The former has quadratic while the latter has linear wave-number dependence in both inertial and Tkachenko modes. Acoustic Tkachenko mode is found to be anisotropic while the other three modes are isotropic. The anisotropy of the acoustic Tkachenko mode reflects the four-fold symmetry of the square lattice.
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.
After delineating the physical regimes that vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate $\ensuremath{\Omega}$ increases, we derive the normal modes of the vortex lattice in two dimensions at zero temperature. Taking into account effects of the finite compressibility, we find an inertial mode of frequency $\ensuremath{\geqslant}2\ensuremath{\Omega}$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\ensuremath{\Omega}$ is small compared with the lowest compressional mode frequency, to quadratic in the wave vector in the opposite limit. We calculate the correlation functions of vortex displacements and phase, density and superfluid velocities, and find that the zero-point excitations of the soft quadratic Tkachenko modes lead in a large system to a loss of long-range phase correlations, growing logarithmically with distance, and hence lead to a fragmented state at zero temperature. The vortex positional ordering is preserved at zero temperature, but the thermally excited Tkachenko modes cause the relative positional fluctuations to grow logarithmically with separation at finite temperature. The superfluid density, defined in terms of the transverse velocity autocorrelation function, vanishes at all temperatures. Finally we construct the long-wavelength single-particle Green's function in the rotating system and calculate the condensate depletion as a function of temperature.
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.