Magnetic flux lines in type-II superconductors and the 'hairy ball' theorem

Abstract

Many prominent phenomena originate from geometrical effects rather than from local physics. For example, the ' hairy ball ' (HB) theorem asserts that a hairy sphere cannot be combed without introducing at least one singularity, and is fulfi lled by the atmospheric circulation with the existence of stratospheric polar vortices and the fact that there is always at least one place on Earth where the horizontal wind is still. In this study, we examine the consequences of the HB theorem for the lattice of fl ux lines that form when a magnetic fi eld is applied to a type-II superconducting crystal. We fi nd that discontinuities must exist in lattice shape as a function of fi eld direction relative to the crystal. Extraordinary, ' unconventional ' fl ux line lattice shapes that spontaneously break the underlying crystal symmetry are thus remarkably likely across all type-II superconductors, both conventional and unconventional.

Rotation of the magnetic vortex lattice in Ru7B3 driven by the effects of broken time-reversal and inversion symmetry

“…The reorientation of the VL as a function of η in Fig 4(a), however, is an example of a conventional VL structure transition taking place. While the exact behavior of this transition is governed by local physics, as described by the theories discussed in the preceding paragraph, the existence of a transition as a function of η is demanded by the geometric arguments of the 'hairy ball theorem', which describes how continuous vector fields map onto the surface of objects with various topologies [11]. In this case, the vectors describing the VL are mapped onto the sphere of possible orientations of the magnetic field with respect to the crystal, and because a sphere has an Euler characteristic of χ = +2, this demands that there be singularities on the surface of the sphere which have a total winding number equal to +2.…”

Response of the Skyrmion Lattice in MnSi to Cubic Magnetocrystalline Anisotropies

“…We predict that encircling each of these points on the unit sphere as a function of field orientation, will generate a rotation of the Bragg pattern by an angle ∆ω = 2πw = π/3. This will provide a critical test of the so-called "hairy ball theorem", which has been discussed in the analogous context of the Abrikosov vortex lattice in type II superconductors [33]. A related open question to be addressed experimentally in the future concerns whether these singular points account for the degenerate multidomain configurations of the skyrmion lattice that have been observed in some of the cubic chiral magnets [4,25,27,36].…”

“…Further, cubic MCAs are also essential for the motion of skyrmion lattices under spin currents [28][29][30]. Last but not least, MCAs provide an important point of reference for the understanding of superconducting vortex lattices, where different morphologies [31,32] and the associated topological character [33] attract great current interest, while higher-order contributions in the superconducting order parameter of the underlying Ginzburg-Landau theories are still not known.…”

Probing the anisotropic vortex lattice in the Fe-based superconductor KFe2As2using small-angle neutron scattering

“…These show more complex behavior than given by anisotropic London theory, particularly for rotation of magnetic field direction between the [001] and [110] axes. However, the results can be systematized using the hairy ball theorem 26 and show signs of "nonlocal" effects. In particular, the VL phase diagram becomes more complex at low temperatures, and the difference between [100] and [110] directions may be related to the nodal structure of the gap.…”

“…26 It shows that there must be discontinuities in VL structure and orientation as the field direction is varied. The VL structure may spontaneously break the underlying crystal symmetry and give multiple VL domains; however, the resultant diffraction pattern will respect the crystal symmetries about the axis B.…”