Order parameter in the A phase of superfluid 3He

Up to now we considered isotropic superfluids, in which gauge invariance was broken but they remained invariant with respect to any three-dimensional rotation. In particular, in the Fermi superfluids the order parameter, or gap ∆, was a scalar independent of the direction. This means that the wave function of Cooper pairs was in the s state with zero orbital angular momentum and spin. Superconductors with such symmetry of the order parameter are called s-wave superconductors. In superfluid 3He the Cooper pair has a total spin and a total orbital moment equal to 1 (in unit ħ). Superconductors (charged superfluids), in which Cooper pairs have orbital momentum and spin equal to 1, are called spin-triplet or p-wave superconductors. In p-wave superfluids the order parameter is a 3 × 3 matrix with complex elements (18 parameters) in general (Vollhardt and Wölfle, 1990).

We focus our attention on the A phase of superfluid 3He, for which the order parameter matrix is a direct product of two three-dimensional vectors, which correspond to wave functions with spin 1 in the spin space and with orbital moment 1 in the orbital space. The unit vector d in the spin space determines the axis along which the spin of the Cooper pair exactly vanishes, although the spin modulus is equal to 1. Spin components along any other axis also vanish but only on average. So this spin wave function has no spin polarisation, and the state is analogous to the spin state in antiferromagnets with d being an analogue of the antiferromagnetic vector. In the orbital space there are two orthogonal unit vectors m and n, which determine a complex unit vector and a unit vector l = m × n. The vector l is called the orbital vector. It delineates the axis along which the orbital moment of the Cooper pair is directed. Neutral and charged superfluids with such an order parameter are called chiral or px + ipy superfluids. So the condensate of Cooper pairs has a spontaneous angular momentum along l, which is called an intrinsic angular momentum. In charged superfluids (px + ipy-wave superconductors) the intrinsic angular momentum leads to spontaneous magnetisation.

Thermal nucleation of vortices in a uniform flow

Thermal and quantum nucleation of vortices in superfluids attracted the attention of theorists long ago (Iordanskii, 1965b; Langer and Fisher, 1967; Muirihead et al., 1984). The quantum nucleation of vortices by superflow in small orifices (Davis et al., 1992; Ihas et al., 1992) and by moving ions (Hendry et al., 1988) has been reported.

The process of vortex nucleation is crucial for onset of essential dissipation when superfluid velocities reach the critical velocity for penetration of vortices into a container. The original state is a metastable state with a persistent vortex-free superfluid flow. Vortex nucleation is necessary for transition to a state with a smaller superfluid velocity (and eventually to the stable equilibrium state with zero velocity) in the case of uniform flows in channels, or for transition to solid body rotation with an array of straight vortices parallel to the rotation axis in the case of rotating containers. In the process of vortex nucleation a small vortex loop appears, which grows in size. Eventually the vortex loop transforms to a straight vortex line in the case of rotation, or the vortex line crosses the channel crosssection decreasing the phase difference between ends of the channel by 2π (the phase slip). The latter process is illustrated in Fig. 11.1. Although vortex nucleation is a key process, which determines critical velocities, the problem of critical velocities does not reduce to the nucleation problem. The theory of critical velocities requires introduction of additional definitions and assumptions. One can find discussion of critical velocities with relevant references elsewhere (Donnelly, 1991; Varoquaux, 2015).

Vortex nucleation is possible due to either thermal or quantum fluctuations in the fluid. This section addresses the Iordanskii–Langer–Fisher theory of thermal nucleation (Iordanskii, 1965b; Langer and Fisher, 1967). The rate of thermal nucleation of vortices is governed by the Arrhenius law ∝ e −Em/T. The energetic barrier Em is determined by a maximum of the energy of a vortex loop in the process of its growth.

The motion of vortices has been an area of study for more than a century. During the classical period of vortex dynamics, from the late 1800s, many interesting properties of vortices were discovered, beginning with the notable Kelvin waves propagating along an isolated vortex line (Thompson, 1880). The main object of theoretical studies at that time was a dissipationless perfect fluid (Lamb, 1997). It was difficult for the theory to find a common ground with experiment since any classical fluid exhibits viscous effects. The situation changed after the works of Onsager (1949) and Feynman (1955) who revealed that rotating superfluids are threaded by an array of vortex lines with quantised circulation. With this discovery, the quantum period of vortex dynamics began. Rotating superfluid 4He provided the testing ground for the theories of vortex motion developed for the perfect fluid. At the same time, some effects needed an extension of the theory to include twofluid effects, and the quantum period of vortex studies was marked by progress in the understanding of vortex dynamics in the framework of the two-fluid theory. The first step in this direction was taken by Hall and Vinen (1956a), who introduced the concept of mutual friction between vortices and the normal part of the superfluid and derived the law of vortex motion in two-fluid hydrodynamics. Hall (1958) and Andronikashvili et al. (1961) were the first to study experimentally the elastic properties of vortex lines using torsional oscillators. This made it possible to observe Kelvin waves with a spectrum modified by the interaction between vortices. Elastic deformations of vortex lines were caused by pinning of vortices at solid surfaces confining the superfluid. Vortex pinning was another important concept, which emerged during the study of dynamics of quantised vortices.

The third important theoretical framework, invented to describe vortex motion in rotating superfluids, was so-called macroscopic hydrodynamics. This relied on a coarse-graining procedure of averaging hydrodynamical equations over scales much larger than the intervortex spacing. Such hydrodynamics was used in the pioneering work on dynamics of superfluid vortices by Hall and Vinen (1956a) and further developed by Hall (1960) and Bekarevich and Khalatnikov (1961). It was a continuum theory similar to the elasticity theory. However, it only included bending deformations of vortex lines and ignored the crystalline order of the vortex array.

A tour to classical turbulence: scaling arguments, cascade and Kolmogorov spectrum

The theory of classical turbulence starts from the Navier–Stokes equation (1.87) for an incompressible fluid. But it is a long way from this starting point to a full picture of turbulent flows. The linearised Navier–Stokes equation can be solved more or less straightforwardly for various geometries of laminar flows. If the velocity of the flow grows, the non-linear inertial term (v · ∇)v in the Navier–Stokes equation becomes relevant and eventually leads to instability of the laminar flow. The instability threshold is controlled by the Reynolds number (1.88). After the instability threshold is reached, the flow becomes strongly inhomogeneous in space and time in a chaotic manner. This is despite the fully deterministic character of the Navier–Stokes equation. The emergence of chaos from the deterministic description is a fundamental problem in physics and mathematics, but we mostly skip this transient process of turbulence evolution except for a few comments in Section 14.8. We are interested in a discussion of developed turbulence, which arises at rather high Reynolds numbers of the order of a few thousands or more. Usually it is assumed that in the state of developed turbulence the fluid is infinite and homogeneous in space and time, but only on average. At scales smaller than the size L of the fluid there are intensive temporal and spatial fluctuations of the velocity field. The natural description of such a chaotic field is in terms of probability distributions and random correlation functions. The velocity ⟨v⟩ averaged over a scale comparable with the fluid size L is not so important since it can be removed by the Galilean transformation. More important are fluctuations of velocity v = v − ⟨v⟩. Further we omit the ‘prime’, assuming that ⟨v⟩ = 0. The amplitude of velocity fluctuations depends on the scale at which it is considered. Following Landau and Lifshitz (1987, Chapter III) let us introduce the scale-dependent Reynolds number Rel = lv(l)/ν for the velocity fluctuation v(l) at the scale l. The scale-dependent Reynolds number characterises the effectiveness of the viscosity, which becomes important at small scales with Rel ∼ 1.

Thermodynamics of a one-component perfect fluid

In the strict sense of the word, hydrodynamics describes the dynamical behaviour of a fluid. But sometimes the hydrodynamical approach refers to phenomenological theories dealing with various types of condensed media, such as solids, liquid crystals, superconductors, magnetically ordered systems and so on. Two important and interconnected features characterise the hydrodynamical description.

1. • It refers to spatial and temporal scales much longer than any relevant microscopical scale of the medium under consideration.

2. • It does not need the microscopical theory for derivation of dynamical equations but uses as a starting point a set of conservation laws and thermodynamical and symmetry properties of the medium under consideration.

The latter feature gives us the possibility to study condensed matter without waiting for the moment when a closed self-consistent microscopical theory is developed. Sometimes it can be a long time to wait for such a moment. For example, one may recall the microscopical theory of fluid with strong interactions, or as the latest example the microscopical theory of high-Tc superconductivity. In fact, the cases when the hydrodynamical description can be derived rigorously from the ‘first-principle’ theory are more the exceptions rather than the rule. Such exceptions include, for example, weakly non-ideal gases and weak-coupling superconductors. Even if it is possible to derive the hydrodynamical description from the microscopical theory, the former as based on the most global properties (conservation laws and symmetry) is a reliable check of the microscopical theory. If hydrodynamics does not follow from a microscopical theory this is an alarming signal of potential problems with the microscopical theory.

Impressive evidence of the fruitfulness of the hydrodynamical (phenomenological) approach to condensed matter physics is provided by the volumes of Landau and Lifshitz's course addressing continuous media: Electrodynamics of Continuous Media, Theory of Elasticity, and Fluid Mechanics (Landau and Lifshitz, 1984, 1986, 1987). The hydrodynamical approach was very fruitful also for studying properties of rotating superfluids, as will be demonstrated in this book. The hydrodynamical description always deals with the continuous medium even if the medium under consideration is a lattice (an atomic lattice in elasticity theory, for example). Indeed, the lattice constant is a microscopical scale which should be ignored in accordance with the nature of the hydrodynamical description.