Topological odd-parity superconductors

We investigate topological phases of full-gapped odd-parity superconductors without or with time-reversal invariance. For odd-parity superconductors, a combination of the inversion and the $U\left(1\right)$ gauge symmetry is manifestly preserved, and the combined symmetry enables us to characterize the topological phases from the knowledge of the electron dispersion. Topologically protected gapless boundary states are predicted from the Fermi-surface topology. Simple criteria for topological odd-parity superconductors, in particular, that for a non-Abelian topological phase supporting a non-Abelian anyon are provided. Implications for nodal odd-parity superconductors are also discussed.

Figure 1

(Color online) The TRI momenta ${\Gamma }_i$, and the TRI closed path ${\text{C}}_{ij}$ connecting ${\Gamma }_i$ and ${\Gamma }_j$ in the BZ. (a) 1D BZ. The solid line denotes ${\text{C}}_{12}$. (b) 2D BZ $T^2$. (c) 3D BZ and the surface BZ of a 100 face. (d) 2D band structure for a slab with a 100 face for the 3D TRI odd parity superconductor that has the Fermi surface with ${\left(-1\right)}^{\tilde{\nu }\left[{\text{C}}_{34}\right]}={\left(-1\right)}^{\tilde{\nu }\left[{\text{C}}_{56}\right]}=-1$ and the gap function $\Delta \left(\mathit{k}\right)=i\mathit{d}\left(\mathit{k}\right)\cdot \mathbf{\sigma }{\sigma }_y$ with $d_i\left(\mathit{k}\right)=\Delta \text{sin}{k}_i$. The insets show the Fermi surface (top) and the 2D surface BZ (bottom).

Figure 2

$Z_2$ classification of edge state. (a) Topologically protected zero mode. [(b) and (c)] Topologically trivial edge modes.