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Dynamics and Quantum Tunneling of Vortices

Eugen Šimánek

Published Sep 8, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780195078282.003.0008

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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.
Vortex dynamics in superfluids is investigated in the framework of the nonlinear Schr\"odinger equation. The natural motion of the vortex is of cyclotron type, whose frequency is found to be on the order of the phonon velocity divided by the coherence length, and may be heavily damped due to phonon radiation. Trapping foreign particles into the vortex core can reduce the cyclotron frequency and make the cyclotron motion underdamped. The density fluctuations can follow the vortex motion adiabatically within the phonon wavelength at the cyclotron frequency, which results in a further downward renormalization of the cyclotron frequency. We have also discussed applications on the dynamics of vortices in superconducting films. \textcopyright{} 1996 The American Physical Society.
The tunneling rate ${t}_{v}∕\ensuremath{\hbar}$ 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, ${t}_{v}=\overline{V}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}{n}_{s}{d}^{2}∕2)$, 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.
Vortex mass, which is the inertia of a quantum vortex, has never been observed in superfluids and is a long-standing problem in low-temperature physics. The impact of the mass is considered negligible in typical experiments with superfluid 4He^{4}\mathrm{He}4He. Recent developments of experimental techniques for manipulating quantum vortices in superfluid atomic gases have enabled us to test this problem more accurately. By introducing the vortex mass time and length as universal scales to many-body problems of massive quantum vortices, the theoretical description is formulated in the simplest manner and is universally applicable to different quantum fluids, including fermionic and multicomponent superfluids. There are two branches, the cyclotron and massless branches, for the circular motion of a pair of like-sign vortices. Finding a stable cyclotron branch for the motion of vortices is a clear evidence of vortex mass and superfluid $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{B}$ is the specific example of a system where this phenomena could be observed. The impact of the mass on the massless branch is small but can be enhanced by taking the difference in the two-body dynamics of point vortices with different initial conditions. Our results imply that the vortex mass is a direct cause of the splitting instability of a doubly quantized vortex at absolute zero and that the vortex mass length characterizes the final state after the instability. It is also demonstrated that a pair of massive vortices with opposite circulations has a critical distance characterized by the vortex mass length, below which they are spontaneously annihilated without thermal fluctuations.