Cosmiclike domain walls in superfluidB3: Instantons and diabolical points in (k,r) space

The possible planar superfluid B-B boundaries between inequivalent B-phase vacua are considered; such B-B interfaces provide an analogy with the cosmic domain walls that are believed to have precipitated in the phase transitions of the early Universe. Several of them display nontrivial structure in (k,r) space (i.e., the union of the momentum and real spaces). Such a wall represents an instanton connecting two B-phase vacua with different k-space topology. The transition between the vacua occurs through the formation of a pointlike defect either in the (k,r) space, or in the (k,t) space. These defects are so-called diabolical points of codimension 4, at which the fermionic energy tends to zero, thus providing the fermionic zero modes. Such points are new examples (within condensed-matter physics) of the peculiar diabolical points, which are characterized by the occurrence of a contact between the different branches of the quasiparticle spectra; in the present case, the branches of particles and holes, respectively. These points are here discussed for the case of the superfluid phases of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in close analogy with the quantum field theory of fermions interacting with classical bosonic fields. The cosmiclike domain walls in superfluid ${}_{}{}^3{}_{}{}^{}\mathit{B}$ are observable in principle; in particular, the motion of the superfluid A-B interface is governed at low temperatures by the periodical emission of these topological excitation planes. The edges of B-B interfaces serve to generate fractionally quantized pure and mixed mass and spin supercurrent vortices in ${}_{}{}^3{}_{}{}^{}\mathit{B}$, while holes in these surfaces may give rise to the corresponding vortex rings and combined vortex and/or spin-disclination rings.