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Physical review. B, Condensed matter

From microscopic theory to Boltzmann kinetic equation: Application to vortex dynamics

G. Blatter, V. B. Geshkenbeǐn, Nikolai B. Kopnin

Published Jun 1, 1999• 19 citations
Publisher

Abstract

We show how to lift the problem of calculating the force acting on a topological defect in a superfluid from the microscopic to the semiclassical level: Starting from the microscopic kinetic equations for a clean superconductor, we derive a Boltzmann equation for the quasiparticle distribution function in and around the defect. The velocity $\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}$ and force $\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}$ appearing in this Boltzmann equation are given through the Hamiltonian equations $\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}={\ensuremath{\partial}}_{p}{E}_{n}(p,q)$ and $\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}=\ensuremath{-}{\ensuremath{\partial}}_{q}{E}_{n}(p,q),$ where ${E}_{n}(p,q)$ denotes the $(n\mathrm{th}$ branch in the) spectrum of the quasiparticles in the vicinity of the defect. Second, we reformulate the microscopic expression for the force acting on the defect in terms of the total momentum transfer of the quasiparticles from the heat bath to the vortex core. We illustrate our result with an application to vortices in s-wave superconductors, where we derive the vortex equation of motion and identify the Magnus, Hall, and dissipative forces.

DOI
10.1103/physrevb.59.14663

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arXiv:cond-mat/9612059•Dec 6, 1996•1 citation
We show how to lift the problem of calculating the force acting on a topological defect in a superfluid from the microscopic to the semiclassical level: Starting from the microscopic kinetic equations for a clean superconductor, we derive a Boltzmann equation for the quasiparticle distribution function in and around the defect. The velocity q\ifmmode˙\else˙\fi\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}q\ifmmode˙\else˙\fi and force p\ifmmode˙\else˙\fi\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}p\ifmmode˙\else˙\fi appearing in this Boltzmann equation are given through the Hamiltonian equations q\ifmmode˙\else˙\fi=\ensuremath∂pEn(p,q)\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}={\ensuremath{\partial}}_{p}{E}_{n}(p,q)q\ifmmode˙\else˙\fi=\ensuremath∂p​En​(p,q) and p\ifmmode˙\else˙\fi=\ensuremath−\ensuremath∂qEn(p,q),\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}=\ensuremath{-}{\ensuremath{\partial}}_{q}{E}_{n}(p,q),p\ifmmode˙\else˙\fi=\ensuremath−\ensuremath∂q​En​(p,q), where En(p,q){E}_{n}(p,q)En​(p,q) denotes the (nth(n\mathrm{th}(nth branch in the) spectrum of the quasiparticles in the vicinity of the defect. Second, we reformulate the microscopic expression for the force acting on the defect in terms of the total momentum transfer of the quasiparticles from the heat bath to the vortex core. We illustrate our result with an application to vortices in s-wave superconductors, where we derive the vortex equation of motion and identify the Magnus, Hall, and dissipative forces.
Starting from the time-dependent version of the Feynman-Hellmann theorem, the Magnus force acting on a vortex in fermion superfluid is expressed via the adiabatic curvature over the space of vortex positions. With use of the Bogoliubov--de Gennes approximation, the Magnus force in a homogeneous superfluid at T=0 is shown to originate from virtual transitions between the lowest quasiparticle core bound states. Nonadiabatic corrections to the curvature are obtained to second order in vortex velocity. The adiabatic approximation is shown to break down at a critical velocity equal to the vortex velocity in the first Landau level. The effect of elastic scattering on the Magnus force is discussed in terms of the relaxation-time approximation. It is suggested that this approximation is appropriate only for a large-scale vortex motion. In this case, the effective Magnus force is drastically reduced when the elastic-scattering rate exceeds the core excitation frequency. We conjecture that quantum vortex tunneling is governed by a local Magnus force obtained from the Berry phase approach.
A microscopic analysis of the nondissipative force Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ acting on a line vortex in a type-II superconductor at T=0 is given. All work presented assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS superconducting ground state by movement of the vortex and show how this phase enters into the hydrodynamic action Shyd{\mathit{S}}_{\mathrm{hyd}}Shyd​ of the superconducting condensate. Appropriate variation of Shyd{\mathit{S}}_{\mathrm{hyd}}Shyd​ give Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ and variation of the Berry phase term is seen to contribute the Magnus or lift force of classical hydrodynamics to Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​. This analysis, based on the BCS ground state of a charged superconductor, confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e\ensuremath{\rightarrow}0, is seen to reproduce the Berry phase determined by these authors for a neutral superfluid. We also provide a second, independent, determination of Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ through a microscopic derivation of the continuity equation for the condensate linear momentum. This equation yields the acceleration equation for the superflow and shows that the vortex acts as a sink for the condensate linear momentum. The rate at which momentum is lost to the vortex determines Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ in this second approach and the result obtained agrees identically with the previous Berry phase calculation. The Magnus force contribution to Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ is seen in both calculations to be a consequence of the vortex topology and motion. A preliminary discussion is given regarding finite temperature extensions of the Berry phase calculation, with emphasis on its relevance for the sign anomaly occurring in Hall-effect experiments on type-II superconductors in the flux flow regime.
The forces on the vortex, transverse to its velocity, are considered. In addition to the superfluid Magnus force from the condensate (superfluid component), there are transverse forces from thermal quasiparticles and external fields violating the Galilean invariance. The forces between quasiparticles and the vortex originate from interference of quasiparticles with trajectories on the left and on the right from the vortex like similar forces for electrons interacting with the thin magnetic-flux tube (the Aharonov-Bohm effect). These forces are derived for phonons from the equations of superfluid hydrodynamics, and for BCS quasiparticles from the Bogolyubov--de Gennes equations. The effect of external fields breaking Galilean invariance is analyzed for vortices in the two-dimensional Josephson junction array. The symmetry analysis of the classical equations for the array shows that the total transverse force on the vortex vanishes. Therefore the Hall effect which is linear in the transverse force is absent also. This means that the Magnus force from the superfluid component exactly cancels with the transverse force from the external fields. The results of other approaches are also brought together for discussion.
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.
In a recent article, Kwon et al. [Nature (London) 600, 64 (2021)NATUAS0028-083610.1038/s41586-021-04047-4] revealed nonuniversal dissipative dynamics of quantum vortices in a fermionic superfluid. The enhancement of the dissipative process is pronounced for the Bardeen-Cooper-Schrieffer interaction regime, and it was suggested that the effect is due to the presence of quasiparticles localized inside the vortex core. We test this hypothesis through numerical simulations with time-dependent density-functional theory: a fully microscopic framework with fermionic degrees of freedom. The results of fully microscopic calculations expose the impact of the vortex-bound states on dissipative dynamics in a fermionic superfluid. Their contribution is too weak to explain the experimental measurements, and we identify that thermal effects, giving rise to mutual friction between superfluid and the normal component, dominate the observed dynamics.
An alternative approach to the derivation of the force on a vortex based in an adiabatic approximation in the action of the superfluid system is developed. Assuming that the vortex motion is relatively slow compared with the characteristic times involved in the microscopic degrees of freedom, the effect of the superfluid and its excitations is reduced to a gauge potential, and the associated transverse force. The vortex velocity part of the transverse force is found in terms of the thermal expectation of the angular momentum of the fluid around the vortex. The excitations countercirculate the vortex reducing the effective density to that of the superfluid part only. Non-adiabatic contributions appear in this formalism as a non-Abelian gauge potential connecting different microscopic states, in particular the renormalization of the vortex mass is given by the lowest order diagonal correction to the adiabatic theory, and the long wavelength contributions found to depend only on the static structure factor for the fluid.