Spin and charge signatures of topological superconductivity in Rashba nanowires

We consider a Rashba nanowire with a proximity gap which can be brought into the topological phase by tuning external magnetic field or chemical potential. We study spin and charge of the bulk quasiparticle states when passing through the topological transition for open and closed systems. We show, analytically and numerically, that the spin of bulk states around the topological gap reverses its sign when crossing the transition due to band inversion, independent of the presence of Majorana fermions in the system. This spin reversal can be considered as a bulk signature of topological superconductivity that can be accessed experimentally. We find a similar behavior for the charge of the bulk quasiparticle states, also exhibiting a sign reversal at the transition. We show that these signatures are robust against random static disorder.

Figure 1

A semiconducting nanowire proximity-coupled to an $s$-wave superconductor. The Rashba SOI vector $\alpha$ points in the $y$ direction and an external magnetic field $B_x$ is applied in the $x$ direction. In the topological phase, spin, and chargeless MFs (orange ovals) are localized at the ends, with corresponding probability density indicated by black lines.

Figure 2

The spectrum of Eq. (3) in the trivial (a) and topological (b) phases for closed systems. The blue/red color of dispersion lines $E_{\lambda \eta }\left(k\right)$ indicates the negative/positive sign of the spin component $S_{\lambda \eta }^x\left(k\right)$ for states of a given $\lambda \eta$ band. Around $k=0$, the spin $S_{\lambda \overline{1}}^x\left(k\right)$ of states with energies closest to the Fermi level $\mu =0$ in the trivial phase is opposite to the one in the topological phase. The insets show the spectrum around $k=0$ where such a sign reversal of $S_{\lambda \overline{1}}^x\left(k\right)$ occurs. We used the following parameters: $\mu =0, {\mathrm{\Delta }}_{\mathrm{sc}}=0.5E_{\mathrm{so}}$, and ${\mathrm{\Delta }}_Z=0.3E_{\mathrm{so}}$ (${\mathrm{\Delta }}_Z=0.7E_{\mathrm{so}}$) in the trivial (topological) phase, so that the interior gap ${\mathrm{\Delta }}_i=2|{\mathrm{\Delta }}_Z-{\mathrm{\Delta }}_{\mathrm{sc}}|$ remains the same.

Figure 3

Phase diagram as function of chemical potential $\mu$ and magnetic field ${\mathrm{\Delta }}_Z$ by measuring (a) the $x$ component of the quasiparticle spin $S_{\overline{1}\overline{1}}^x$ and (b) the quasiparticle charge $Q_{\overline{1}\overline{1}}$ at fixed momentum $k=0.05k_{\mathrm{so}}$ and fixed superconducting pairing ${\mathrm{\Delta }}_{\mathrm{sc}}=0.5E_{\mathrm{so}}$. Indeed, the phase boundary at the topological transition line ${\mathrm{\Delta }}_Z=\sqrt{{\mathrm{\Delta }}_{\mathrm{sc}}^2+{\mu }^2}$ (dashed line) is very well visible. A similar phase diagram is found for the states above $\mu$ ($S_{1\overline{1}}^x$ and $Q_{1\overline{1}}$), differing only in a global minus sign due to particle-hole symmetry.

Figure 4

Spin projection $S_{\overline{1}\overline{1}}^x$ [(a) and (b)] and charge $Q_{\overline{1}\overline{1}}$ [(a') and (b')] of bulk states as a function of the chemical potential $\mu$ for two values of momenta [(a) and (a')] $k=3.3 \mu {\mathrm{m}}^{-1}$ and [(b) and (b')] $k=0.6 \mu {\mathrm{m}}^{-1}$ for various values of the SOI strength: $\alpha =0.1,0.2,0.4,\cdots ,2.2,2.28,2.4$ eVÅ. The parameters of the system are ${\mathrm{\Delta }}_{\mathrm{sc}}=0.5$ meV and ${\mathrm{\Delta }}_Z=0.7$ meV. The sign of charge and spin reverses as the system undergoes the topological phase transition at $\mu =\pm 0.5$ meV (denoted by gray dashed vertical lines). The boundary between phases is most pronounced for the states close to $k=0$ and almost independent of the SOI.

Figure 5

The energy spectrum and spin $S^x$ in the trivial (a) and topological (b) phase for open and closed systems. We find excellent agreement between $E_n$ (red circles) and $S^x\left(E_n\right)=S_n^x$ (blue circles) obtained from the tight-binding model for open systems supporting MFs [see Eq. (1)] and $E_{\lambda \overline{1}}\left(k\right)$ (orange solid line) and $S_{\lambda \overline{1}}^x\left(k\right)$ obtained for closed systems without MFs [see Eq. (2)]. The topological phase transition is clearly indicated by the sign reversal of $S^x$ also in open systems. The trivial phase is plotted for ${\mathrm{\Delta }}_Z=0.4$ meV while the topological one for ${\mathrm{\Delta }}_Z=0.6$ meV. The other parameters are fixed as ${\mathrm{\Delta }}_{\mathrm{sc}}=0.5$ meV, $\mu =0, \alpha =0.9$ eV Å($E_{\mathrm{so}}\approx 1$ meV), $N=1200$, and $m=0.015m_e$, with $m_e$ being the bare electron mass, for InSb nanowires ($t=10$ meV, $a=15$ nm). In addition, we consider a random on-site disorder potential of strength $|\delta {\mu }_j|<1$ meV. The spin $S^x\left(E_n\right)$ (green circles) corresponding to $E_n$ (brown circles) of the disordered nanowire undergoes the same reversal of sign as in the clean case, demonstrating its robustness.