We have derived an exact expression for the total nondissipative transverse force acting on a quantized vortex moving in a uniform background. The derivation is valid for neutral boson or fermion superfluids, provided the order parameter is a complex scalar quantity. This force is determined by the one-particle density matrix far away from the vortex core, and is found to be the Magnus force proportional to the superfluid density. We conclude that contributions of the localized core states do not change this force.
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.
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.
A microscopic analysis of the nondissipative force
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 of the superconducting condensate. Appropriate variation of
Shyd give
Fnd and variation of the Berry phase term is seen to contribute the Magnus or lift force of classical hydrodynamics to
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 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 in this second approach and the result obtained agrees identically with the previous Berry phase calculation. The Magnus force contribution to
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.
Adiabatic effective action for vortices in neutral and charged superfluids at zero temperature are calculated using the topological Landau-Ginzburg theory recently proposed by Hatsuda, Yahikozawa, Ao and Thouless, and vortex dynamics are examined. The Berry phase term arising in the effective action naturally yields the Magnus force in both neutral and charged superfluids. It is shown that in neutral superfluid there is only one degree of freedom, namely the center of vorticities, and the vortex energy is proportional to the sum of all vorticities so that it is finite only for the vanishing total vorticity of the system. On the other hand the effective mass and the vortex energy for a vortex in charged superfluids are defined individually as expected. The effects of the vortex core on these quantities are also estimated. The possible depinning scenario which is governed by the Magnus force and the inertial mass is also discussed.
The transverse force acting on a quantized vortex in a superfluid is a problem that has eluded a complete understanding for more than three decades. In this Letter I calculate the superfluid velocity part of the transverse force in a way closely related to Laughlin's argument for the quantization of conductance in the quantum Hall effect. A combination of this result, the vortex velocity part of the transverse force found by Thouless, Ao, and Niu [Phys. Rev. Lett. 76, 3758 (1996)], and Galilean invariance shows that there cannot be a transverse force proportional to the normal fluid velocity.