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arXiv:cond-mat/9801011 · Journal of Experimental and Theoretical Physics Letters

Vortex mass in BCS systems: Kopnin and Baym-Chandler contributions

Григорий Ефимович Воловик

Published Apr 1, 1998• 12 citations
arXivHTMLPDF

Abstract

The Kopnin mass and the Baym-Chandler mass of a vortex have the same origin. Both represent the mass of the normal component trapped by the vortex. The Kopnin mass of a vortex is formed by quasiparticles localized in the vicinity of the vortex. In the superclean limit it is calculated as a linear response, in exactly the same way as the density of the normal component is calculated in a homogeneous superfluid. The Baym-Chandler mass is the hydrodynamic (associated) mass trapped by a vortex. It is analogous to the normal component formed by inhomogeneities, such as pores and impurities. Both contributions are calculated for a generic model of a continuous vortex core.

DOI
10.1134/1.567692

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The Kopnin mass and the Baym-Chandler mass of a vortex have the same origin. Both represent the mass of the normal component trapped by the vortex. The Kopnin mass of a vortex is formed by quasiparticles localized in the vicinity of the vortex. In the superclean limit it is calculated as a linear response, in exactly the same way as the density of the normal component is calculated in a homogeneous superfluid. The Baym-Chandler mass is the hydrodynamic (associated) mass trapped by a vortex. It is analogous to the normal component formed by inhomogeneities, such as pores and impurities. Both contributions are calculated for a generic model of a continuous vortex core.
The paper reassesses the old but still controversial problem of the transverse force on a vortex and the vortex mass. The transverse force from free bulk quasiparticles on the vortex, both in the Bose and the Fermi liquids, originates from the Aharonov-Bohm effect. However, in the Fermi liquid, one should take into account peculiarities of the Aharonov-Bohm effect for BCS quasiparticles described by two-component spinor wave functions. There is no connection between the transverse force (either from free bulk quasiparticles or from vortex-core bound quasiparticles) and the spectral flow in the vortex core in superfluid Fermi liquid, in contrast to widely known claims. In fact, there is no steady spectral flow in the core of the moving vortex, and the analogy with the Andreev bound states in the superconductor--normal-metal--superconductor junction, where the spectral flow is really possible, is not valid in this respect. The role of the backflow on the vortex mass is clarified. The backflow is an inevitable consequence of a mismatch between the currents inside and outside the vortex core and restores the conservation of the particle number (charge) violated by this mismatch. In the Fermi liquid, the backflow compensates the current through the core bound states, which is a source of the vortex mass (the Kopnin mass). This results in renormalization of the Kopnin vortex mass by a numerical factor.
An inertial mass of a vortex can be calculated by driving it around in a circle with a steadily revolving pinning potential. We show that in the low-frequency limit this gives precisely the same formula that was used by Baym and Chandler, but find that the result is not unique and depends on the force field used to cause the acceleration. We apply this method to the Gross-Pitaevskii model, and derive a simple formula for the vortex mass. We study both the long-range and short-range properties of the solution. We agree with earlier results that the nonzero compressibility leads to a divergent mass. From the short-range behavior of the solution we find that the mass is sensitive to the form of the pinning potential, and diverges logarithmically when the radius of this potential tends to zero.
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.
Abstract The mass (inertia) of an object is determined as the response of the momentum of the object to its velocity. This chapter discusses the inertia of an object moving in the quantum vacuum of the Galilean quantum liquid. If it is a foreign object like a 3He atom moving in the quantum vacuum of superfluid 4He, then in addition to its bare mass the object acquires an extra mass, since it involves some part of the superfluid vacuum into motion. If the object is an excitation of the vacuum, like a vortex, all its mass is provided by the liquid. In addition to the associated hydrodynamic mass, which is similar to associated mass of external body moving in the liquid, there are two more contributions to a vortex mass. The ‘relativistic’ mass is the consequence of effective metric in superfluids. It is obtained as vortex energy divided by square of speed of sound, which plays the role of speed of light. In fermionic superfluids the Kopnin mass is added, which results from the fermion zero modes in the vortex core. This mass is related to peculiar phenomenon in quantum field theory. In the limit of smooth core these bound states behave as chiral massless fermions in the presence of magnetic field, and the additional vortex mass results from the non-zero density of states in magnetic field.
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.
We derive a Galilean invariant expression for the electric field induced by a vortex moving through a charged superfluid at T=0T=0T=0, which holds for any superconductor, from the dirty to the superclean limit. The contribution of different areas around the vortex to the average electric field and to the charge distribution is analyzed. The results are extended to a neutral system, where the chemical potential takes over the role of the electrostatic potential in the charged situation. Different contributions to the vortex mass in charged and neutral superfluids are brought together for comparison and discussion.