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.

Abstract

Introduction and Mathematical Motivation

About 25 years ago we became interested in studying the effects of small scale features of a topography on long waves propagating on the water surface for large distances, such as a tsunami. Our goal was to study solitary waves over rapidly varying disordered topographies. At the time the theory for linear acoustic waves over rapidly varying (one dimensional-1D) layered media was maturing and becoming quite sophisticated [1, 2]. The theory developed by George Papanicolaou and collaborators considered linear hyperbolic systems with disordered rapidly varying coefficients as a model to understand linear pulse-shaped waves travelling in a medium with a random propagation speed. This setup is very useful for understanding the effect of uncertainty on a travelling wave, in direct and inverse problems related to the Earth's subsurface. Mathematically it called for new technology showing that random modelling produced some universal results of interest. These included effective wave propagation properties that were independent of a specific realisation. To study linear travelling waves in the presence of random multiple scattering, Papanicolaou and collaborators developed an asymptotic theory for stochastic ordinary differential equations (SODEs) regarding the transmitted and reflected signals. Limit theorems for randomly forced oscillators characterised asymptotically the expected value for the transmitted and reflected signals after the wave had propagated for large distances. Three scales are involved in this analysis: the medium's microscale ε2, the pulse's characteristic width ε (the mesoscale) and the large propagation distance O(1) (the macroscale).

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.

Abstract

Introduction

The irrotational water wave is a special case of the irrotational vortex sheet. For the vortex sheet problem, two fluids whose motions are described by the incompressible, irrotational Euler equations meet at an interface. This interface, the vortex sheet, is free to move, and moves according to the velocities of the two fluids restricted to the interface. Each fluid has its own non-negative, constant density. Different geometries are possible, but to be definite, at present we consider the case in which the fluids are two-dimensional and such that each fluid region has one component, which is of infinite vertical extent and horizontally periodic. Thus, we may say that we have an upper fluid and a lower fluid. In the water wave case, the density of the upper fluid is equal to zero.

Without surface tension, if each of the two fluids has positive density, then the vortex sheet is known to have an ill-posed initial value problem; this has been demonstrated by several authors. We note that when discussing ill-posedness of a problem, to be precise, one should mention the function spaces under consideration; for example, Caflisch and Orellana have shown that the vortex sheet initial value problem is ill-posed in Sobolev spaces [1]. In analytic function spaces, however, solutions of the vortex sheet problem have been shown to exist by a Cauchy-Kowalewski argument [2].

The ill-posedness of the vortex sheet initial value problem (when the two fluids have positive densities) is caused by the presence of the Kelvin-Helmholtz instability.