Topology of gap nodes in superfluid³He: π₄ Homotopy group for³He-B disclination

Abstract

The topologically stable zeros in the energy spectrum of Fermi excitations in superfluid³He both in uniform phases and in textures are classified. This generalizes the classification of the defects of the order parameter in real coordinate space to the classification of zeros in the gap, which are the more general defects in coherent superfluid or superconducting states both in real space and momentumk space. The zeros are described by classes of mappings of the spherical surfacesS ⁿ, embracing the (6-n-1)-dimensional manifold of zeros in six-dimensional (k, r) space, into the space of the Bogolyubov-Nambu matrices, which describe the Fermi excitations. The examples of topologically nontrivial manifolds of zeros are discussed, including the closed line of zeros in five-dimensional space, which is described by the π₄ homotopy groups and exists in the core of the³He-B disclination. This object demonstrates the coupling between the real space topology of disclination and the extended space topology of zeros in the disclination core.