Low-field topological threshold in Majorana double nanowires

A hard proximity-induced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields, a soft gap emerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that in a setup of double Rashba nanowires that are coupled to an $s$-wave superconductor and subjected to an external magnetic field along the wires, the topological threshold can be significantly reduced by the destructive interference of direct and crossed-Andreev pairing in this setup, precisely down to the magnetic field regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths, which makes them ideal candidates for future braiding experiments.

Figure 1

(a) Two Rashba NWs (gray) labeled by $\tau =1,\overline{1}$ are aligned along the $x$ direction and proximity-coupled to an $s$-wave SC (red). Their separation in the $z$ direction is given by $d$. Both NWs are subjected to a magnetic field $\mathit{B}$ which points along the $x$ axis. The Rashba SOI field ${\mathbf{\alpha }}_{\tau }$ in the $\tau$-wire points along the $z$ axis. (b) Energy spectrum in the limit of strong SOI, $E_{so,\tau }\gg {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c$, with solid (dashed) lines corresponding to electron (hole) bands. The chemical potential ${\mu }_{\tau }$ is tuned to the crossing point of spin-up (blue) and spin-down (green) bands in both NWs. The proximity-induced superconductivity generates a coupling between states with opposite momenta and spins belonging to the same NW (with strength ${\mathrm{\Delta }}_{\tau }$) or to different NWs (with strength ${\mathrm{\Delta }}_c$). For $|E_{so,1}-E_{so,\overline{1}}|\gg {\mathrm{\Delta }}_c$, the crossed-Andreev pairing potential ${\mathrm{\Delta }}_c$ couples only the interior branches of the spectrum at $k=0$. Also, the magnetic field couples states of opposite spins at $k=0$ in each NW (with strength ${\mathrm{\Delta }}_{Z\tau }$).

Figure 2

Topological phase diagram as a function of the Zeeman splitting ${\mathrm{\Delta }}_Z$ and the strength of the induced crossed-Andreev pairing ${\mathrm{\Delta }}_c$ for the regime $|E_{so,1}-E_{so,\overline{1}}|\gg {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c\gg |{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_{\overline{1}}|,|{\mathrm{\Delta }}_{Z1}-{\mathrm{\Delta }}_{Z\overline{1}}|$. There are two topological phases hosting one MBS (blue) and two MBSs (red) at each end. The trivial phase (white) does not host any MBSs. To take advantage of the low-field topological threshold, the setup shown in Fig. 1 should be operated in the one-MBS phase for ${\mathrm{\Delta }}_d+{\mathrm{\Delta }}_Z>{\mathrm{\Delta }}_c>|{\mathrm{\Delta }}_d-{\mathrm{\Delta }}_Z|$.

Figure 3

(a) Top panel: Color-scale plot of the topological threshold ${\mathrm{\Delta }}_{Z,c}/{\mathrm{\Delta }}_d$ for the one-MBS phase as a function of ${\mu }_{\tau }/{\mathrm{\Delta }}_d$ for ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_d=0.5, \mathrm{\Gamma }/{\mathrm{\Delta }}_d=1$. Bottom panel: ${\mathrm{\Delta }}_{Z,c}/{\mathrm{\Delta }}_d$ as a function of ${\mu }_1/{\mathrm{\Delta }}_d$ for ${\mu }_{\overline{1}}=\mathrm{\Gamma }$ (red) and ${\mu }_{\overline{1}}/{\mathrm{\Delta }}_d=4$ (blue). The topological threshold ${\mathrm{\Delta }}_{Z,c}$ exhibits a global minimum for ${\mu }_{\tau }=\mathrm{\Gamma }$. (b) Topological phase diagram as a function of ${\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_1$ (obtained from a tight-binding diagonalization of 800 sites per NW) for finite interwire tunneling. We have $E_{so,1}/{\mathrm{\Delta }}_1=6.25, E_{so,\overline{1}}/{\mathrm{\Delta }}_1=12.25, {\mathrm{\Delta }}_{\overline{1}}/{\mathrm{\Delta }}_1=1.3, \mathrm{\Gamma }/{\mathrm{\Delta }}_1=1, {\mu }_1={\mu }_{\overline{1}}=\mathrm{\Gamma }$. Colors are the same as in Fig. 2. Black dashed lines denote the approximate phase boundaries for ${\mu }_{\tau }=0$. For ${\mu }_1={\mu }_{\overline{1}}=\mathrm{\Gamma }$ the one-MBS phase itself and its phase boundary to the trivial phase remain stable. In contrast, for ${\mu }_{\tau }=0$, the topological threshold separating the trivial and one-MBS phase is pushed to higher magnetic fields. (c) Same phase diagram as in (b) but with the two SOI vectors not being parallel to each other but still orthogonal to the magnetic field. We take ${\mu }_{\tau }=0, \mathrm{\Gamma }=0$, and $\theta =\pi /6$ for the relative angle between the SOI vectors. Unlike the one-MBS phase, the two-MBS phases are unstable against rotations of the SOI direction.

Figure 4

(a) Topological phase diagram as a function of ${\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_d$ and ${\mu }_1/{\mathrm{\Delta }}_d$ (obtained from a tight-binding diagonalization of 800 sites per NW) for finite interwire tunneling, $\mathrm{\Gamma }/{\mathrm{\Delta }}_d=1$. We have set $E_{so,\tau }/{\mathrm{\Delta }}_d=6.25, {\mathrm{\Delta }}_c/{\mathrm{\Delta }}_d=0.5$, and ${\mu }_{\overline{1}}/{\mathrm{\Delta }}_d=1$. Colors are the same as in Fig. 2 of the main text. We see that without tuning the chemical potentials to the sweet spot ${\mu }_{\tau }=\mathrm{\Gamma }$ the topological threshold ${\mathrm{\Delta }}_{Z,c}$ is shifted to significantly higher magnetic fields and not much is gained in a double-NW setup compared to single NWs. (b) Same topological phase diagram as in (a) but for ${\mu }_{\overline{1}}/{\mathrm{\Delta }}_d=4$. Once more, we see that without the tuning to the sweet spot ${\mu }_{\tau }=\mathrm{\Gamma }$ no low-field topological threshold is observed.

Figure 5

(a) Topological phase diagram as a function of ${\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_d$ and ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_d$ for the regime of weakly detuned SOI energies, $|E_{so,1}-E_{so,\overline{1}}|\ll {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c$. The color coding scheme is the same as in Fig. 2 in the main text. The dashed black line denotes a gap closing at finite momentum. We have chosen $E_{so,\tau }/{\mathrm{\Delta }}_d=2$ and ${\mu }_1={\mu }_{\overline{1}}=0$. The two-MBS phase which appeared for ${\mathrm{\Delta }}_c>{\mathrm{\Delta }}_d+{\mathrm{\Delta }}_Z$ when $|E_{so,1}-E_{so,\overline{1}}|\gg {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c$ turns into a trivial phase. All other topological phases remain unchanged. (b) Topological phase diagram as a function of ${\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_1$ for the regime, $|E_{so,1}-E_{so,\overline{1}}|\sim {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c$. We have chosen $E_{so,1}/{\mathrm{\Delta }}_1=2, E_{so,\overline{1}}/{\mathrm{\Delta }}_1=4.5, {\mu }_1={\mu }_{\overline{1}}=0, {\mathrm{\Delta }}_{\overline{1}}/{\mathrm{\Delta }}_1=1.3$. We note that both two-MBS phases disappear for ${\mathrm{\Delta }}_c>{\mathrm{\Delta }}_c^{*}$; see Eq. (B10).

Figure 6

(a) Topological phase diagram as a function of ${\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_c/{\mathrm{\Delta }}_1$ for the regime of strongly detuned SOI energies, $|E_{so,1}-E_{so,\overline{1}}|\gg {\mathrm{\Delta }}_{Z\tau },{\mathrm{\Delta }}_{\tau },{\mathrm{\Delta }}_c$, and a rotation of the magnetic field in the $\overline{1}$-wire by $\varphi =0.2$. The color-coding scheme is the same as in Fig. 2 in the main text. We have chosen $E_{so,1}/{\mathrm{\Delta }}_1=6.25, E_{so,\overline{1}}/{\mathrm{\Delta }}_1=12.25, N=800, {\mathrm{\Delta }}_{\overline{1}}/{\mathrm{\Delta }}_1=1.3, {\mu }_1=0$, and ${\mu }_{\overline{1}}=0$. While the one-MBS remains stable, the two-MBS phases are unstable against rotations of the magnetic field with $sin\left(\varphi \right)\ne 0$. (b) Same topological phase diagram as in (a) but with finite interwire tunneling, $\mathrm{\Gamma }/{\mathrm{\Delta }}_1=1$. Moreover, we also set $\varphi =0$ and ${\mu }_{\tau }=0$. Consequently, the effects of interwire tunneling are [unlike in Fig. 3 of the main text] not compensated and the topological threshold from the trivial to the one-MBS phase is pushed to substantially higher magnetic fields. Thus, to get the maximum advantage of the double-nanowire setup, it is crucial to compensate for these shifts due to interwire tunneling.