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CRC Press eBooks

Vortex Lines

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

This chapter focuses on "steady" superfluid flow, in which the superfluid velocity, vs, satisfies the equation div vs=0. Such flow can, in principle, occur only in multiply-connected systems. However, it is possible to achieve an equivalent situation in the bulk of the liquid by setting up vortex lines. The major new feature brought in by superfluidity, and a most important one indeed, is the quantization of vortex lines, arising as a consequence of the quantization of circulation. The square terms in correspond to the energy of the single vortex lines, while the cross term describes an interaction energy between two vortex lines. In the absence of vortex lines, the superfluid and normal components of a Bose liquid are completely decoupled. The velocity pattern of the vortex gives rise to a position-dependent energy, which may be used to calculate the scattering. The dynamics of superfluid flow is thus appreciably modified by the presence of vortex lines.

DOI
10.1201/9780429492679-8

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arXiv:1403.3900•Mar 16, 2014•5 citations
This chapter focuses on "steady" superfluid flow, in which the superfluid velocity, vs, satisfies the equation div vs=0. Such flow can, in principle, occur only in multiply-connected systems. However, it is possible to achieve an equivalent situation in the bulk of the liquid by setting up vortex lines. The major new feature brought in by superfluidity, and a most important one indeed, is the quantization of vortex lines, arising as a consequence of the quantization of circulation. The square terms in correspond to the energy of the single vortex lines, while the cross term describes an interaction energy between two vortex lines. In the absence of vortex lines, the superfluid and normal components of a Bose liquid are completely decoupled. The velocity pattern of the vortex gives rise to a position-dependent energy, which may be used to calculate the scattering. The dynamics of superfluid flow is thus appreciably modified by the presence of vortex lines.
In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]). Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.
Abstract This chapter presents a general discussion of the phenomenon of superfluidity, starting from the famous Landau criterion for the critical velocity. The Landau expression for the normal (non-superfluid) component of the gas, expressed in terms of the elementary excitations of the system, is derived. The relationship between Bose–Einstein condensation and the phenomenon of superfluidity is outlined, with special focus on the role played by the phase of the order parameter, whose gradient fixes the velocity of the superfluid. The chapter derives the hydrodynamic equations of superfluids at both zero and finite temperatures and discuss the solutions for first and second sound. The formalism of quantum hydrodynamics is presented, with an application to the Beliaev decay of phonons and to the calculation of the fluctuations of the phase. It also discusses the consequences for superfluidity on the rotational properties and the spatial structure of vortex lines.
In superconductors, as in Helium II (He II), the holes round which quantisation occurs can be either physical boundaries, or the cores of quantised vortex lines. For convenience we summarise here the resemblances and differences between vortex lines and vortex states in He II and in superconductors. In contrast, states in which a whole vortex is attached to the wire should be comparatively stable. The rotation of the superfluid can be satisfactorily explained by assuming that it is threaded by a series of parallel straight vortex lines. Consider a single straight vortex line in the superfluid. The energy associated with a single vortex line is mainly the kinetic energy of the circulating superfluid. The number and separation of vortex lines present for a given angular velocity clearly depends on their individual circulations. Two rival models for the ion-vortex transition have been proposed, generally known as the peeling model and the quantum transition model.
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
Owing to three conditions (namely: (a) the velocity is represented by sum of irrotational and solenoidal components; (b) the fluid is barotropic; (c) a bath with the fluid undergoes vertical vibrations) the Navier-Stokes equation admits reduction to the modified Hamilton-Jacobi equation. The modification term is the Bohmian(quantum) potential. This reduction opens possibility to define a complex-valued function, named the wave function, which is a solution of the Schrödinger equation. The solenoidal component being added to the momentum operator poses itself as a vector potential by analogy with the magnetic vector potential. The vector potential is represented by the solenoidal velocity multiplied by mass of the fluid element. Vortex tubes, rings, and balls along with the wave function guiding these objects are solutions of this equation. Motion of the vortex balls along the Bohmian trajectories gives a model of droplets moving on the fluid surface. A peculiar fluid is the superfluid physical vacuum. It contains Bose particle-antiparticle pairs. Vortex lines presented by electron-positron pairs are main torque objects. Bundles of the vortex lines can transmit a torque from one rotating disk to other unmoved disk.