Time-Reversal-Invariant Topological Superconductors and Superfluids in Two and Three Dimensions

We construct time-reversal invariant topological superconductors and superfluids in two and three dimensions. These states have a full pairing gap in the bulk, gapless counterpropagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. The superfluid ${}^3\mathrm{He}$ $B$ phase provides a physical realization of the topological superfluidity, with experimentally measurable surface states protected by the time-reversal symmetry. We show that the time-reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex and connects each vortex with its superpartner.

Figure 1

(Top row) Schematic comparison of $2d$ chiral superconductor and the QH state. In both systems, TR symmetry is broken and the edge states carry a definite chirality. (Bottom row) Schematic comparison of $2d$ TRI topological superconductor and the QSH insulator. Both systems preserve TR symmetry and have a helical pair of edge states, where opposite spin states counterpropagate. The dashed lines show that the edge states of the superconductors are Majorana fermions so that the $E<0$ part of the quasiparticle spectra are redundant.

Figure 2

(a) The fusion of two vortices with opposite vorticity in a conventional superconductor. The two vortices are time-reversal partners, and they fuse into a boson. (b) The fusion of two vortices with opposite vorticity in a $Z_2$ topological superconductor. The two vortices are superpartners, and they fuse into a fermion. The insets indicate schematic Feynman diagrams of these two processes.

Figure 3

Illustration of a $3d$ TRI topological superconductor with two TRI vortex rings which are (a) linked or (b) unlinked. The $E-k_{\parallel }$ dispersion relations show schematically the quasiparticle levels confined on the red vortex ring in both cases. “○” and “×” stand for the quasiparticle levels that are Kramers’ partners of each other. Only case (a) has a pair of Majorana zero modes located on each vortex ring since the rings are linked.