## Abstract

Except in a toroidal container, the order parameter in ^{3}He-A must have singularities at the surface. Examples of such singularities are given, and a relation between the surface topology and the kinds of singularities is derived. Quantization of circulation along surface contours is discussed, and examples of the decay of superflow through the motion of surface singularities are given.

## Keywords

Surface Singularity Geodesic Curvature Persistent Current Surface Vortex Positive Vortex
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## References

- (1).For a discussion of the special case in which the only surface singularities are in v
_{s}, see N. D. Mermin, “Games to Play with^{3}He-A,” Physica, to be published (Proceedings of the Sussex Symposium on Superfluid^{3}He).Google Scholar - (2).The analysis here is based on the discussion of v
_{s}given by N. D. Mermin and Tin-Lun Ho, Phys. Rev. Lett. 36, 594 (1976).ADSCrossRefGoogle Scholar - (3).The importance of these considerations for the stability of persistent currents was brought home to me by several very stimulating remarks of P. W. Anderson, and a ferocious lunch-time discussion with M. E. Fisher.Google Scholar
- (4).For a more detailed exposition of these points, see Ref. 2. It should be emphasized that \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) and the components of the gradient of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ell }\), together with the constraint (3) constitute the standard set of order parameter gradients for the hydro-dynamic description of a system whose order is characterized by an orthonormal triad of vectors. The constraint on the curl of \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) is simply the local integrability condition insuring that the \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\phi } ^{\left( i \right)}}\) can be reconstructed from a knowledge of (4) (cont.) \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) and the gradients of (in analogy to \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\nabla }\) x \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) = 0 in
^{4}He-II, which is the constraint necessary to permit the reconstruction of the phase). The field \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) appears in the classical differential geometry of surfaces (see, for example, the discussion of the Gauss-Bonnet theorem in Ref. 7). It has been used to discuss singularities in nematics by M. Kleman, Phil. Mag. 27, 1057 (1973), andADSCrossRefGoogle Scholar - (4a).M. Kleman, J. de Physique 34, 931 (1973).CrossRefGoogle Scholar
- (5).V. Ambegaokar, P. G. de Gennes, and D. Rainer, Phys. Rev. A 9, 2676 (1974),ADSCrossRefGoogle Scholar
- (5a).V. Ambegaokar, P. G. de Gennes, and D. Rainer, Phys. Rev. A 12, 345 (1975).ADSCrossRefGoogle Scholar
- (6).G. Toulouse and M. Kleman, J. de Physique Lettres, 37, 149 (1976).CrossRefGoogle Scholar
- (7).A result apparently first proved by Gauss himself. For a particularly clear discussion see Louis Brand, “Vector and Tensor Analysis,” Wiley, N.Y., (1947). Brand characterizes Gauss’s relation between curl \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}\) and K as “perhaps the most important result in the theory of surfaces!” Note also that the boundary condition requires \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\nabla }\) · \(\ell \) to approach the mean curvature, a+c, provided that \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\nabla }\) x \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ell } \) is non-singular at the surface.MATHGoogle Scholar
- (8).See, for example, Brand, Ref. 7, p. 308ff.Google Scholar
- (9).A very readable discussion of the Euler characteristic and the topology of two dimensional manifolds is given by William S. Massey, “Algebraic Topology: An Introduction” (Harcourt, Brace & World, N. Y., 1967).MATHGoogle Scholar
- (10).A simpler but more verbose argument leading directly to this conclusion without the differential geometry, can be constructed along the lines of the argument given in Ref. 1.Google Scholar
- (11).Ref. 9, Theorem 5.1, Chapter 1. The number of handles, n, is called the “genus” of the surface.Google Scholar
- (12).Such twistless islands are just the surface termination points of the doubly quantized coreless vortices discussed by P. W. Anderson and G. Toulouse, Phys. Rev. Lett. 38, 508 (1977).ADSCrossRefGoogle Scholar
- (13).See Lewis Carroll, “The Hunting of the Snark,” Second fit (last verse), Third fit (verses 10 and 14), and Eighth fit (verse 9).Google Scholar

## Copyright information

© Plenum Press, New York 1977