Signatures of Majorana and Weyl Fermions in confined phases of superfluid $^3$He

The B-phase of superfluid $^3$He exhibits symmetry breaking in which separate invariance under gauge-, spin- and orbital rotations is reduced to the maximal sub-group, $SO(3)_{L+S}\times{T}$. Parity is broken, but time-reversal is preserved. Broken relative spin-orbit rotational symmetry implies emergent spin-orbit coupling and non-trivial topology of the ground state, both of which are encoded in the Bogoliubov-Nambu Hamiltonian: ${\cal H} = \xi({\bf p})\tau_{3} + c\,{\bf p}\cdot\vec\sigma\,\tau_{1}$, where $c = \Delta/p_f$ is several orders of magnitude slower than the Fermi velocity. The topology of the B-phase is expressed in terms of a non-trivial winding number for the mapping between momentum space and Nambu space, $N_{\mbox{3D}} = \int\frac{d^3p}{24\pi^2}\,\epsilon_{ijk}\, \mbox{Tr}\Big\{T\,C\,({\cal H}^{-1}\partial_{p_i}{\cal H})\times ({\cal H}^{-1}\partial_{p_j}{\cal H}) ({\cal H}^{-1}\partial_{p_k}{\cal H})\Big\}\,=\, 2$, where $C$ is the particle-hole transformation. The physical consequence of $N_{\mbox{3D}}\ne 0$ is the emergence of a spectrum of Majorana fermions confined on any surface of $^3$He-B whose effective Hamiltonian is described $H = \sum_{{\bf p}_{||}}\,\psi_{-{\bf p}_{||}}^{T}{\bf p}_{||}\times\vec{\sigma}\cdot\hat{\bf s}\,\psi_{{\bf p}_{||}}$. The surface excitations are self-conjugate Majorana fermions with a gapless relativistic dispersion relation $\varepsilon({\bf p}) = c|{\bf p}_{||}|$, and their spins locked normal to the in-plane momentum and the surface normal, $\hat{\bf s}$. In this talk I describe theoretical predictions for experimental signatures based on NMR, mass flow, local ion probes and ultra-sound spectroscopy of these unique quanta that reflect the topological nature of the ground state of superfluid $^3$He.