Abstract
Except in a toroidal container, the order parameter in 3He-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.
Work supported in part by the National Science Foundation under Grant No. DMP 74-24394 and through the Materials Science Center of Cornell University (Technical Report No. 2775).
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References
For a discussion of the special case in which the only surface singularities are in vs, see N. D. Mermin, “Games to Play with 3He-A,” Physica, to be published (Proceedings of the Sussex Symposium on Superfluid 3He).
The analysis here is based on the discussion of vs given by N. D. Mermin and Tin-Lun Ho, Phys. Rev. Lett. 36, 594 (1976).
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.
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 4He-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), and
M. Kleman, J. de Physique 34, 931 (1973).
V. Ambegaokar, P. G. de Gennes, and D. Rainer, Phys. Rev. A 9, 2676 (1974),
V. Ambegaokar, P. G. de Gennes, and D. Rainer, Phys. Rev. A 12, 345 (1975).
G. Toulouse and M. Kleman, J. de Physique Lettres, 37, 149 (1976).
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.
See, for example, Brand, Ref. 7, p. 308ff.
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).
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.
Ref. 9, Theorem 5.1, Chapter 1. The number of handles, n, is called the “genus” of the surface.
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).
See Lewis Carroll, “The Hunting of the Snark,” Second fit (last verse), Third fit (verses 10 and 14), and Eighth fit (verse 9).
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© 1977 Plenum Press, New York
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Mermin, N.D. (1977). Surface Singularities and Superflow in 3He-A. In: Trickey, S.B., Adams, E.D., Dufty, J.W. (eds) Quantum Fluids and Solids. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-2418-8_2
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DOI: https://doi.org/10.1007/978-1-4684-2418-8_2
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