Majorana Modes in Time-Reversal Invariant $s$-Wave Topological Superconductors

We present a time-reversal invariant $s$-wave superconductor supporting Majorana edge modes. The multiband character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the $s$-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.

Figure 1

Topological characterization of the phase structure of Hamiltonian $H$ via the partial CN sum $C_{+}$ as a function of $u_{cd}$ and $\Delta$, with $t=1$ and arbitrary $\lambda \ne 0$, for representative chemical potentials $\mu =0$ (top) and $\mu =-1$ (bottom). The black (dashed) line represents an insulator or metal phase, depending on the filling, with $\Delta =0$. CNs are calculated for $(N_x,N_y)=(100,100)$ lattice sites. Note that we may have two pairs of edge modes with $C_{+}=0$.

Figure 2

Excitation spectrum of Hamiltonian $H$ on a cylinder for $\mu =0$, $t=1$, $\lambda =1$. Panel (a), $C_{+}=1$: $\Delta =2$, $u_{cd}=3$; Panel (b), $C_{+}=-1$: $\Delta =2.5$, $u_{cd}=2$; Panel (c), $C_{+}=0$: $\Delta =2$, $u_{cd}=1$; Panel (d), $C_{+}=2$: $\Delta =0.8$, $u_{cd}=1.5$. Note that the bulk gap scales as $\min (\lambda ,\Delta )$. The number of lattice sites $(N_x,N_y)=(40,100)$.

Figure 3

Phase diagram as a function of $u_{cd}$ and $V$ with the pairing gap $\Delta$ calculated self-consistently. The magnitude of $\Delta$ is represented by a color, whose scale is indicated on the side. The number of lattice sites $(N_x,N_y)=(80,80)$.