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arXiv:cond-mat/9604010 · Physical review. B, Condensed matter

Vortex dynamics in superfluids: Cyclotron-type motion

Ertugrul Demircan, Ping Ao, Q. Niu

Published Oct 1, 1996• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.54.10027

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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.
We investigate the microscopic theory of collective excitations and vortex dynamics in superfluid systems. We derive an effective Gross-Pitaevskii Lagrangian for the vortex and collective excitations and obtain the dynamical equations governing their motion. We solve these equations without assuming adiabaticity of vortex motion and show that the vortex has a natural cyclotron frequency. This frequency allows us to identify the vortex mass which turns out to be finite. We also conclude that this implies the existence of a length scale due to the Berry phase associated to the vortex and that it acts as an infrared cutoff.
We characterize the mechanisms of vortex pinning in a superfluid thin film described by the two-dimensional Gross-Pitaevskii equation. We consider a vortex "scattering experiment" whereby a single vortex in a superfluid flow interacts with a circular, uniform pinning potential. By an analogy with linear dielectrics, we develop an analytical hydrodynamic approximation that predicts vortex trajectories, the vortex fixed point and the unpinning velocity. We then solve the Gross-Pitaevskii equation to validate this model, and build a phase portrait of vortex pinning. We identify two different dynamical pinning mechanisms marked by distinctive phonon emission signatures: one enabled by acoustic radiation and another mediated by vortex dipoles nucleated within the pin. Relative to obstacle size, we find that pinning potentials on the order of the healing length are more effective for vortex capture. Our results could be useful in mitigating the negative effects of drag due to vortices in superfluid channels, in analogy to maximizing supercurrents in type-II superconductors.
In two dimensions a microscopic theory providing a basis for the naive analogy between a quantized vortex in a superfluid and an electron in a uniform magnetic field is presented. Following the variational approach developed by Peierls, Yoccoz, and Thouless, the cyclotron motion of a vortex is described by the many-body wave function, which is a linear combination of Feynman wave functions centered at different positions. An integral equation for the weighting functions of the superposition is derived by minimizing the energy functional. The matrix elements of the kernel are the overlaps between any two displaced Feynman wave functions. A numerical study is conducted for a bosonic superfluid based on a Hartree ground state. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with energy, and scales with the size of the vortex. The fluid density is finite at the vortex axis and the vorticity is distributed in the core region. The effective mass of a quantized vortex defined by the inverse of the energy-level spacing is shown to be logarithmically divergent with the size of the vortex.
Excitations in vortex cores in superconductors and other Fermi superfluids are single-particle excitations with a peculiar energy spectrum. These excitations are responsible for many important thermodynamic properties such as specific heat, London penetration length, etc. They also determine the dynamic characteristics of superconductors and superfluids through their interaction with vortices. Flux flow resistance, the Hall effect in type II superconductors and the mutual friction in superfluids are the most important phenomena which strongly depend on vortex core excitations. These phenomena determine the electromagnetic responses of type II superconductors and the hydrodynamic behaviour of superfluids and are of great significance for practical applications of superconducting devices and for understanding the most fundamental properties of correlated electrons and other Fermi particles. In this review we consider the dynamic properties of superconductors and superfluids and outline the basic ideas and results on the vortex dynamics in clean superfluid Fermi systems. The forces acting on moving vortices are discussed including the problem of the transverse force which was a matter of confusion for quite some time. We formulate the equations of the vortex dynamics, which include all the forces and the inertial term associated with excitations bound to the moving vortex.
The damping of vortex cyclotron modes is investigated within a generalized quantum theory of vortex waves. Similarly to the case of Kelvin modes, the friction coefficient turns out to be essentially unchanged under such oscillations, but it is shown to be affected by appreciable memory corrections. On the other hand, the non-equilibrium dynamics of the vortex energy, which is investigated within the framework of linear response theory, shows that its memory corrections are negligible. The vortex response is found to be of the Debye type, with a relaxation frequency whose dependence on temperature and impurity concentration reflects the complexity of the heat bath and its interaction with the vortex.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.