Existence of Majorana Fermions and Topological Order in Nodal Superconductors with Spin-Orbit Interactions in External Magnetic Fields

We demonstrate that Majorana fermions exist in edges of systems and in a vortex core even for superconductors with nodal excitations such as the $d$-wave pairing state under a particular but realistic condition in the case with an antisymmetric spin-orbit interaction and a nonzero magnetic field below the upper critical field. We clarify that the Majorana fermion state is topologically protected in spite of the presence of bulk gapless nodal excitations, because of the existence of a nontrivial topological number. Our finding drastically enlarges target systems where we can explore the Majorana fermion state.

Figure 1

Energy spectra for systems with open boundaries for the $x$ direction and the periodic boundary condition for the $y$ direction. $\mu =-4t$, $\lambda =0.5t$, ${\Delta }_0=t$, and $h=2t$. (a),(b) $d_{x^2-y^2}$-wave pairing. (d),(e) $d_{xy}$-wave pairing. The distance between two edges is $L=90$ [(a) and (d)] and $L=30$ [(b) and (e)]. In (a) and (d), chiral gapless edge modes at $x=0$ and $x=L$ are depicted, respectively, in green and red curves. (c),(f) Chiral edge modes counterpropagating on two opposite edges for $d_{x^2-y^2}$-wave pairing (c) and $d_{xy}$-wave pairing (f).