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arXiv:cond-mat/9806065 · Physical Review Letters

Dynamic Vortex Mass in Clean Fermi Superfluids and Superconductors

Nikolai B. Kopnin, V. M. Vinokur

Published Nov 2, 1998• 111 citations
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Abstract

We calculate the dynamic vortex mass for clean Fermi superfluids including both $s$- and $d$-wave superconductors as a response to a vortex acceleration. Assuming a finite quasiparticle mean free time, the vortex mass appears to be a tensor with components depending on the quasiparticle relaxation time. The diagonal component dominates in the limit of long mean free time while the off-diagonal mass takes over in the moderately clean regime.

DOI
10.1103/physrevlett.81.3952

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We calculate the dynamic vortex mass for clean Fermi superfluids including both sss- and ddd-wave superconductors as a response to a vortex acceleration. Assuming a finite quasiparticle mean free time, the vortex mass appears to be a tensor with components depending on the quasiparticle relaxation time. The diagonal component dominates in the limit of long mean free time while the off-diagonal mass takes over in the moderately clean regime.
Vortex mass in Fermi superfluids and superconductors and its influence on quantum tunneling of vortices are discussed. The vortex mass is essentially enhanced due to the fermion zero modes in the core of the vortex: the bound states of the Bogoliubov quasiparticles localized in the core. These bound states form the normal component, which is nonzero even in the low-temperature limit. In the collisionless regime ω 0 τ≫1 the normal component trapped by the vortex is unbound from the normal component in a bulk superfluid/superconductor and adds to the inertial mass of the moving vortex. In a d-wave superconductor the vortex mass has an additional factor of (B c2/B)1/2 due to the gap nodes.
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.
Quantum vortices are commonly described as funnel-like objects around which the superfluid swirls, and their motion is typically modeled in terms of massless particles. Here we show that in Fermi superfluids the normal component confined in the vortex core provides the vortex with a finite inertial mass. This inertia imparts an unambiguous signature to the dynamic behavior of vortices, specifically manifesting as small-amplitude transverse oscillations which remarkably follow the prediction of a simple point-like model supplemented by an effective mass. We demonstrate this phenomenon through large-scale time-dependent simulations of Fermi superfluids across a wide range of interaction parameters, at both zero and finite temperatures, and for various initial conditions. Our findings pave the way for the exploration of inertial effects in superfluid vortex dynamics.
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 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.
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.
Vortex mass is a key concept in the study of superfluid dynamics, referring to the inertia of vortices in a superfluid, which affects their motion and behavior. Despite being an important quantity, the vortex mass has never been observed experimentally, and remains an unresolved issue in this field. As of now, a large body of research assumes that the vortex mass is a local parameter. In contrast, we present a calculation that suggests a logarithmic dependence on the system size, agreeing with some earlier predictions in the context of Bose gases. We analyze the problem using an effective field theory that describes ultracold atomic Fermi gases over the BEC-BCS crossover at both zero and nonzero temperatures. Our study reveals a strong dependence of the vortex mass on the scattering length; in particular, the vortex mass grows rapidly when moving toward the BCS side. Furthermore, we find that the system-size dependence of the vortex mass results in values an order of magnitude larger than those predicted by other models for realistic system sizes. This implies that the vortex mass could be observable in a wider parameter range than was previously expected. This is particularly relevant considering recent advances in experimental techniques that place the observation of vortex mass in superfluid Fermi gases within reach.