Vortex lattice stability and phase coherence in three-dimensional rapidly rotating Bose condensates

We establish the general equations of motion for the modes of a vortex lattice in a rapidly rotating Bose-Einstein condensate in three dimensions, taking into account the elastic energy of the lattice and the vortex line bending energy. As in two dimensions, the vortex lattice supports Tkachenko and gapped sound modes. In contrast, in three dimensions the Tkachenko mode frequency at long wavelengths becomes linear in the wavevector for any propagation direction out of the transverse plane. We compute the correlation functions of the vortex displacements and the superfluid order parameter for a homogeneous Bose gas of bounded extent in the axial direction. At zero temperature the vortex displacement correlations are convergent at large separation, but at finite temperatures, they grow with separation. The growth of the vortex displacements should lead to observable melting of vortex lattices at higher temperatures and somewhat lower particle number and faster rotation than in current experiments. At zero temperature a system of large extent in the axial direction maintains long range order-parameter correlations for large separation, but at finite temperatures the correlations decay with separation.Comment: 10 pages, 2 figures, Changes include the addition of the particle density - vortex density coupling and the correct value of the shear modulu

Tkachenko modes of vortex lattices in rapidly rotating Bose-Einstein condensates

We calculate the in-plane modes of the vortex lattice in a rotating Bose condensate from the Thomas-Fermi to the mean-field quantum Hall regimes. The Tkachenko mode frequency goes from linear in the wavevector, $k$, for lattice rotational velocities, $\Omega$, much smaller than the lowest sound wave frequency in a finite system, to quadratic in $k$ in the opposite limit. The system also supports an inertial mode of frequency $\ge 2\Omega$. The calculated frequencies are in good agreement with recent observations of Tkachenko modes at JILA, and provide evidence for the decrease in the shear modulus of the vortex lattice at rapid rotation.Comment: 4 pages, 2 figure

Vortex lattices in rapidly rotating Bose-Einstein condensates: modes and correlation functions

After delineating the physical regimes which vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate, $\Omega$, 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 $\ge 2\Omega$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\Omega$ 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.Comment: 11 pages Latex, no figure

Dislocation-Mediated Melting in Superfluid Vortex Lattices

We describe thermal melting of the two-dimensional vortex lattice in a rotating superfluid by generalizing the Halperin and Nelson theory of dislocation-mediated melting. and derive a melting temperature proportional to the renormalized shear modulus of the vortex lattice. The rigid-body rotation of the superfluid attenuates the effects of lattice compression on the energy of dislocations and hence the melting temperature, while not affecting the shearing. Finally, we discuss dislocations and thermal melting in inhomogeneous rapidly rotating Bose-Einstein condensates; we delineate a phase diagram in the temperature -- rotation rate plane, and infer that the thermal melting temperature should lie below the Bose-Einstein transition temperature.Comment: 9 pages, 2 figure

Dissipationless Phonon Hall Viscosity

We study the acoustic phonon response of crystals hosting a gapped time-reversal symmetry breaking electronic state. The phonon effective action can in general acquire a dissipationless "Hall" viscosity, which is determined by the adiabatic Berry curvature of the electron wave function. This Hall viscosity endows the system with a characteristic frequency, \omega_v; for acoustic phonons of frequency \omega, it shifts the phonon spectrum by an amount of order (\omega/\omega_v)^2 and it mixes the longitudinal and transverse acoustic phonons with a relative amplitude ratio of \omega/\omega_v and with a phase shift of +/- \pi/2, to lowest order in \omega/\omega_v. We study several examples, including the integer quantum Hall states, the quantum anomalous Hall state in Hg_{1-y}Mn_{y}Te quantum wells, and a mean-field model for p_x + i p_y superconductors. We discuss situations in which the acoustic phonon response is directly related to the gravitational response, for which striking predictions have been made. When the electron-phonon system is viewed as a whole, this provides an example where measurements of Goldstone modes may serve as a probe of adiabatic curvature of the wave function of the gapped sector of a system.Comment: 14 page

Tkachenko modes as sources of quasiperiodic pulsar spin variations

We study the long wavelength shear modes (Tkachenko waves) of triangular lattices of singly quantized vortices in neutron star interiors taking into account the mutual friction between the superfluid and the normal fluid and the shear viscosity of the normal fluid. The set of Tkachenko modes that propagate in the plane orthogonal to the spin vector are weakly damped if the coupling between the superfluid and normal fluid is small. In strong coupling, their oscillation frequencies are lower and are undamped for small and moderate shear viscosities. The periods of these modes are consistent with the observed ~100-1000 day variations in spin of PSR 1828-11.Comment: 7 pages, 3 figures, uses RevTex, v2: added discussion/references, matches published versio

Tkachenko modes in a superfluid Fermi gas at unitarity

We calculate the frequencies of the Tkachenko oscillations of a vortex lattice in a harmonically trapped superfluid Fermi gas. We use the elasto-hydrodynamic theory by properly accounting for the elastic constants, the Thomas-Fermi density profile of the atomic cloud, and the boundary conditions. Thanks to the Fermi pressure, which is responsible for larger cloud radii with respect to the case of dilute Bose-Einstein condensed gases, large vortex lattices are achievable in the unitary limit of infinite scattering length, even at relatively small angular velocities. This opens the possibility of experimentally observing vortex oscillations in the regime where the dispersion relation approaches the Tkachenko law for incompressible fluids and the mode frequency is almost comparable to the trapping frequencies.Comment: 5 pages, 1 figure; minor changes, now published as Phys. Rev. A 77, 021602(R) (2008