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arXiv:cond-mat/0509263 · Physical Review B

Quantum tunneling of vortices in two-dimensional condensates

Assa Auerbach, Daniel P. Arovas, Sankalpa Ghosh

Published Aug 24, 2006• 23 citations
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Abstract

The tunneling rate ${t}_{v}∕\ensuremath{\hbar}$ of a vortex between two pinning sites (of strength $\overline{V}$ separated by $d$) is computed using the Bogoliubov expansion of vortex wave-functions overlap. For BCS vortices, tunneling is suppressed beyond a few Fermi wavelengths. For Bose condensates, ${t}_{v}=\overline{V}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}{n}_{s}{d}^{2}∕2)$, where ${n}_{s}$ is the boson density. The analogy between vortex hopping in a superconducting film and two-dimensional electrons in a perpendicular magnetic field is exploited. We derive the variable range hopping temperature, below which vortex tunneling contributes to magnetoresistance. Using the ``quantum Hall insulator'' analogy we argue that the Hall conductivity (rather than the inverse Hall resistivity) measures the effective carrier density in domains of mobile vortices. Details of vortex wave functions and overlap calculations, and a general derivation of the Magnus coefficient for any wave function on the sphere, are provided in appendixes.

DOI
10.1103/physrevb.74.064511

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The tunneling rate tv∕\ensuremathℏ{t}_{v}∕\ensuremath{\hbar}tv​∕\ensuremathℏ of a vortex between two pinning sites (of strength V‾\overline{V}V separated by ddd) is computed using the Bogoliubov expansion of vortex wave-functions overlap. For BCS vortices, tunneling is suppressed beyond a few Fermi wavelengths. For Bose condensates, tv=V‾exp(\ensuremath−\ensuremathπnsd2∕2){t}_{v}=\overline{V}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}{n}_{s}{d}^{2}∕2)tv​=Vexp(\ensuremath−\ensuremathπns​d2∕2), where ns{n}_{s}ns​ is the boson density. The analogy between vortex hopping in a superconducting film and two-dimensional electrons in a perpendicular magnetic field is exploited. We derive the variable range hopping temperature, below which vortex tunneling contributes to magnetoresistance. Using the ``quantum Hall insulator'' analogy we argue that the Hall conductivity (rather than the inverse Hall resistivity) measures the effective carrier density in domains of mobile vortices. Details of vortex wave functions and overlap calculations, and a general derivation of the Magnus coefficient for any wave function on the sphere, are provided in appendixes.
Magnetotransport of hard-core bosons is studied using an XXZXXZXXZ quantum spin model representation, appropriately gauged on the torus to allow for an external magnetic field. We find strong lattice effects near half filling. An effective quantum mechanical description of the vortex degrees of freedom is derived. Using semiclassical and numerical analysis we compute the vortex-hopping energy tV{t}_{\mathsf{V}}tV​, which at half filling is close to magnitude of the boson hopping energy. The critical quantum melting density of the vortex lattice is estimated at $6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ vortices per unit cell. The Hall conductance is computed from the Chern numbers of the low-energy eigenstates. At zero temperature, it reverses sign abruptly at half filling. At precisely half filling, all eigenstates are doubly degenerate for any odd number of flux quanta. We prove the exact degeneracies on the torus by constructing an SU(2) algebra of point-group symmetries, associated with the center of vorticity. This result is interpreted as if each vortex carries an internal spin-half degree of freedom, which can manifest itself as a charge density modulation in its core. Our findings suggest interesting experimental implications for vortex motion of cold atoms in optical lattices and magnet transport of short coherence length superconductors.
We study hard-core lattice bosons in a magnetic field near half filling. The bare vortex hopping rate is extracted from exact diagonalizations of square clusters. We deduce a quantum melting of the vortex lattice above vortex density of 6.5x10(-3) per lattice site. The Hall conductivity reverses sign abruptly as the density crosses half filling, where its characteristic temperature scale vanishes. We prove that at precisely half filling, each vortex carries a spin-1/2 quantum number ("v spin"). Experimental implications of these results are discussed.
A vortex in a quasi two-dimensional Bose–Einstein condensate is subject to the Magnus force and can be effectively described as a planar particle in a uniform magnetic field. Quantization of this effective particle leads to the lowest Landau level where the most localized wavefunction is a Gaussian. In this Gaussian state, vortex position seems to fluctuate with an average magnitude set by the magnetic width of the Gaussian. We readdress this problem using the number-conserving version of the Bogoliubov theory. We find that the Bogoliubov mode that might be interpreted as a fluctuation of vortex position actually does not contribute to quantum depletion in the centre of vortex core. Far in the Thomas–Fermi regime the only contribution comes from phonon modes, but it is an order of magnitude less than the simple estimate, based on the magnetic width of the effective Gaussian wave packet. At the same time density of non-condensed atoms inside vortex core is a factor of 0.23 less than far from the core. The non-condensed atoms are expelled from the core by centrifugal force.
We derive an effective action for the vortex-position degree of freedom in a superfluid by integrating out condensate phase- and density-fluctuation environmental modes. When the quantum dynamics of environmental fluctuations is neglected, we confirm the occurrence of the vortex Magnus force and obtain an expression for the vortex mass. We find that this adiabatic approximation is valid only when the superfluid droplet radius RRR, or the typical distance between vortices, is very much larger than the coherence length $\ensuremath{\xi}$. We go beyond the adiabatic approximation numerically, accounting for the quantum dynamics of environmental modes and capturing their dissipative coupling to condensate dynamics. For the case of an optical-lattice superfluid, we demonstrate that vortex motion damping can be adjusted by tuning the ratio between the tunneling energy JJJ and the on-site interaction energy UUU. We comment on the possibility of realizing vortex-Landau-level physics.
Abstract Within the past decade there has been a resurgence of interest in the dynamics of superfluid and superconducting vortices. One source of motivation for this topic may be attributed to the discovery of the quantum Hall effect (von Klitzing et al., 1980), which led to the suggestion that elementary excitations in 2D systems may obey nonstandard statistics (Wilczek, 1982; Halperin, 1984). Haldane and Wu (1985) pointed out a powerful analogy between the dynamics of a vortex in a 2D superfluid film and the electron motion in a cyclotron orbit. critical value.