Inducing Time-Reversal-Invariant Topological Superconductivity and Fermion Parity Pumping in Quantum Wires

We propose a setup to realize time-reversal-invariant topological superconductors in quantum wires, proximity coupled to conventional superconductors. We consider a model of quantum wire with strong spin-orbit coupling and proximity coupling to two $s$-wave superconductors. When the relative phase between the two superconductors is $\varphi =\pi$ a Kramers pair of Majorana zero modes appears at each edge of the wire. We study the robustness of the phase in the presence of both time-reversal-invariant and time-reversal-breaking perturbations. In addition, we show that the system forms a natural realization of a fermion parity pump, switching the local fermion parity of both edges when the relative phase between the superconductors is changed adiabatically by $2\pi$.

Figure 1

(a) The general setup proposed for the realization of a TRI TSC. The TRI topological phase is obtained for $\varphi =\pi$. (b) Specific model considered for the nanowire. The spin orbit coupling on the two chains comprising the wire has an opposite sign as indicated by the energy dispersion curves. A finite tunneling amplitude $t_{\perp }$ between the two chains opens a gap at the crossing points.

Figure 2

Energy spectrum of a finite nanowire as a function of the relative phase between the two superconductors for (a) zero and (b) nonzero magnetic field. The length of the nanowire is $N_x=200$. The lattice model parameters are $t_{\perp }=2.5$, $t=1$, $\lambda =1$. The chemical potential is set to $\mu =0$ and the magnitude of the SC pairing is $\Delta =0.4$. (a) For $\varphi =\pi$ the system is TRI and in the topological phase as can be seen from the presence of zero-energy states. (b) Here $B=0.05$ in the $z$ direction. Energy levels crossing the bulk gap correspond to edge states at the opposite edges of the wire (red and green online). At $\varphi =\pi$ the two Majorana modes on each edge split. The crossing at zero energy does not disappear, but is shifted away from $\varphi =\pi$.

Figure 3

(a) 2D QSH model with finite tunneling probability between the edges. (b) A closed path in parameter space encircling the gapless region (cones) corresponding to a fermion parity pump. The path is not contractible due to the persistence of the gapless region for $B\ne 0$.