Topology from triviality

We show that bringing into proximity two topologically trivial systems can give rise to a topological phase. More specifically, we study a 1D metallic nanowire proximitized by a 2D superconducting substrate with a mixed $s$-wave and $p$-wave pairing, and we demonstrate both analytically and numerically that the phase diagram of such a setup can be richer than reported before. Thus apart from the two “expected” well-known phases (i.e., where the substrate and the wire are both simultaneously trivial or topological), we show that there exist two peculiar phases in which the nanowire can be in a topological regime while the substrate is trivial and vice versa.

Figure 1

A sketch of the system: a metallic nanowire (violet) on top of a 2D SC substrate with mixed singlet and triplet pairing (blue). Red arrows denote the tunneling between the substrate and the wire, and the Majorana Kramers pairs are drawn as yellow spheres at the ends of the wire.

Figure 2

The ratio of induced pairing parameters $|{\mathrm{\Delta }}_s^{\mathrm{ind}}/{\varkappa }^{\mathrm{ind}}|$ as a function of ${\mathrm{\Delta }}_s/\varkappa$ (blue lines). The black dashed vertical line marks the topological phase transition in the substrate. The red dashed lines describe the dependence of $|p_{{\sigma }_0}^{\mathrm{ind}}|$ on ${\mathrm{\Delta }}_s$ for three different values of the chemical potential in the NW. For each value of $\mu$, we have indicated the intersection points corresponding to the values of the critical ${\mathrm{\Delta }}_s$ in the substrate which would yield a topological phase transition in the NW. Note that the critical ${\mathrm{\Delta }}_s$ of the NW can be smaller, larger or equal to that of the substrate. We have set $M=1.5,\varkappa =2.5,p_F=1.1$, and $t_{\mathrm{tun}}=0.5$.

Figure 3

The phase diagram of a 1D metallic NW deposited on top of a 2D SC substrate. We plot the value of the topological index $\delta$ defined in Eq. (9) as a function of $\frac{{\mathrm{\Delta }}_s}{\varkappa }-p_F$ in the substrate, and of $\mu$ in the NW. We note that all the four possible phases corresponding to $\delta =\pm 1,\pm 2$ can form. The region with $\delta =+2$ (denoted in red) is the most interesting, and corresponds to a topological phase of the NW induced via the proximity of a trivial SC. The black dashed line marks the bulk topological phase transition for the substrate. We have set $p_F=1.1,M=m=1.5,\varkappa =2.5$, and $t_{\mathrm{tun}}=0.5$.

Figure 4

The phase diagram of a 1D NW of length $L=151$ deposited on top of a $201\times 51$ 2D SC substrate. We plot the combined topological index $\delta$, as defined in Table 1, as a function of ${\mathrm{\Delta }}_s$ and ${\mu }_{\mathrm{NW}}$ (in units of $t$, which we set to 1). The topological character of the NW is obtained by calculating the MP of the lowest-energy states (with a cutoff at 0.9). The dashed line at ${\mathrm{\Delta }}_s\sim 0.86t$ marks the bulk topological phase transition. We set ${\mu }_{\mathrm{SC}}=3,{\mathrm{\Delta }}_t=0.5$, and $t_{\mathrm{tun}}=2$.

Figure 5

The phase diagrams of a 1D metallic NW deposited on top of a 2D SC substrate with a finite Rashba SO coupling ($\lambda =0.2$ and $\lambda =0.4$ in the middle and lower panels, respectively) and in its absence (the upper panel). We plot the MP as a function of ${\mathrm{\Delta }}_s$ and the chemical potential ${\mu }_{\mathrm{NW}}$ of the NW (in units of the hopping parameter $t$ which we set to 1 in our simulations). An MP value of 1 corresponds to the formation of Majorana Kramers pairs at the ends of the wire, and 0 to a trivial phase. The black dashed lines at ${\mathrm{\Delta }}_s\sim 0.86t$ (upper), ${\mathrm{\Delta }}_s\sim 0.74t$ (middle), and ${\mathrm{\Delta }}_s\sim 0.63t$ (lower) mark the bulk topological phase transition. We set ${\mu }_{\mathrm{SC}}=3,{\mathrm{\Delta }}_t=0.5$, and $t_{\mathrm{tun}}=2$.

Figure 6

The same phase diagram as in the upper panel of Fig. 5 in the presence of disorder. The values of parameters are the same as in Fig. 4.

Figure 7

The Fermi momentum of the NW plotted as a function of the Fermi momentum of the substrate. As we see from this plot, it is almost independent on the Fermi momentum of the substrate. We took ${\mathrm{\Delta }}_s=\varkappa (0.1+p_F)$, and we have set $M=m=1.5,\varkappa =2.5,\mu =2$, and $t_{\mathrm{tun}}=0.5$.

Figure 8

The phase diagram of a 1D metallic NW deposited on top of a 2D SC substrate. We plot the value of the topological index $\delta$ defined in Eq. (9) of the main text as a function of ${\mathrm{\Delta }}_s$, and of the chemical potential $\mu$, which is taken to be the same for the NW and the substrate. We note that all the four possible phases corresponding to $\delta =\pm 1,\pm 2$ can form due to the Fermi momenta mismatch occurring because of different masses of quasiparticles in the SC and the NW. We have set $M=1.5,m=9,\varkappa =2.5$, and $t_{\mathrm{tun}}=0.5$.

Figure 9

The phase diagram of a 1D metallic NW deposited on top of a 2D SC substrate with a mixed $s$-wave and TRS-breaking $p$-wave pairing. Axes and color codes are the same as in Fig. 5. The black dashed line at ${\mathrm{\Delta }}_s\sim 0.86t$ marks the bulk topological phase transition. We set ${\mu }_{\mathrm{SC}}=3,{\mathrm{\Delta }}_t=0.5$, and $t_{\mathrm{tun}}=2$.