Solutions of several ³He-A textures bearing surface singularities

Exact solutions are found of three textures for superfluid helium-3 in the A phase: two cylindrically symmetric textures in half-spaces each bearing a point singularity on the surface (which are categorized as the circular and hyperbolic boojum textures by Mermin), and a vortex sheet texture in a slab geometry, which is singular on two straight lines, one on each wall. The dipole interaction between the spin axis % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFKbGbaK% aaaaa!387B!\[{\hat d}\] and the orbital axis Î is taken into account to leading order by letting % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFKbGbaK% aaaaa!387B!\[{\hat d}\] = Î, assuming temperature T is near the transition temperature T _c, and the container size is ≫10^\s-3 cm (the dipole characteristic length). Guided by these exact solutions, we then construct an approximate solution for a coreless vortex texture, which was first introduced by Anderson and Toulouse and later modified by Mermin in order for it to fit between two parallel surfaces. It was proposed by Anderson and Toulouse that such coreless vortex textures should replace usual vortex lines in mediating the decay of superfluidity and in transferring angular momentum from the container wall to the superfluid ³He-A. This is found to be true here only if D158.0(ξR)^1/2, where D is the thickness of the slab geometry assumed R is the lateral size of this geometry, and ξ is the core size of a usual vortex line in ³He-A. This analysis is good for R ≫ D, but it should remain semiquantitatively valid down to R ~(2—3)D.