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arXiv:cond-mat/0703523 · Physical Review Letters

Vortex Mass in a Superfluid at Low Frequencies

D. J. Thouless, J. R. Anglin

Published Sep 7, 2007• 39 citations
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Abstract

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.

DOI
10.1103/physrevlett.99.105301

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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 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.
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.
We apply a recently developed effective string theory for vortex lines to the case of two-dimensional trapped superfluids. We do not assume a perturbative microscopic description for the superfluid, but only a gradient expansion for the long-distance hydrodynamical description and for the trapping potential. For any regular trapping potential, we compute the spatial dependence of the superfluid density and the orbital frequency and trajectory of an off-center vortex. Our results are fully relativistic and in the nonrelativistic limit reduce to known results based on the Gross-Pitaevskii model. In our formalism, the leading effect in the nonrelativistic limit arises from two simple Feynman diagrams in which the vortex interacts with the trapping potential through the exchange of hydrodynamical modes.
Vortex motion is a complex problem due to the interplay between the short-range physics at the vortex core level and the long-range hydrodynamical effects. Here we show that the hydrodynamic equations of vortex motion in a compressible superfluid can be solved asymptotically in a model 'slab' geometry. Starting from an exact solution for an incompressible fluid, the hydrodynamic equations are solved with a series expansion in a small tunable parameter provided by the ratio of the healing length, characterising the vortex cores, to the slab width. The key dynamical properties of the vortex, the inertial and physical masses, are well defined and renormalizable. They are calculated at leading order beyond the logarithmic accuracy that has limited previous approaches. Subtracting the asymptotic solutions of the universal hydrodynamic problem from experimental observations of vortex motion exposes the physics of the vortex core and provides a window into interesting many-body phenomena that are currently poorly understood including the role of quantum pressure. Our results provide a solid framework for further detailed study of the vortex mass and vortex forces in strongly correlated and exotic superfluids.
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.
The transverse force on an isolated and moving vortex in a neutral superfluid at rest is evaluated at finite temperature in the case of noninteracting phonons. Using the Thouless, Ao, Niu (TAN) [Phys. Rev. Lett. 76, 3758 (1996)] general theory, we show that the transverse force is exactly equal to the superfluid Magnus force. We extend this theory in the case of a slowly moving vortex on a circular trajectory, and find an additional contribution coming from the centrifugal reaction. This term gives a negative vortex mass due to the phonons and diverges logarithmically at low frequency. The friction force is also evaluated for zero and finite frequencies, and compared with the scattering theory.