Majorana Bound States and Nonlocal Spin Correlations in a Quantum Wire on an Unconventional Superconductor

We study theoretically the proximity effect of a one-dimensional metallic quantum wire (in the absence of spin-orbit interaction) lying on top of an unconventional superconductor. Three different material classes are considered as a substrate: (i) a chiral superconductor in class D with broken time-reversal symmetry and a class DIII superconductor (ii) with and (iii) without a nontrivial ${\mathbb{Z}}_2$ number. Interestingly, we find degenerate zero energy Majorana bound states at both ends of the wire for all three cases. They are unstable against spin-orbit interaction in case (i), while they are topologically protected by time-reversal symmetry in cases (ii) and (iii). Remarkably, we show that nonlocal spin correlations between the two ends of the wire can be simply controlled by a gate potential in our setup.

Figure 1

Spectral function of the electronic excitations. The insets show the schematic structure of the spectrum. Shaded (blue) regions, continuum of the substrate superconductor; solid (red) lines, 1D states of the QW; and dashed (green) lines, edge states in the 2D SC induced by the coupling to the QW. The states in the QW are located (a) outside and (b) within the superconducting gap. (c) is the case where the coupling is strong enough that the bonding and antibonding states act as a potential barrier and edge states in the 2D superconductor emerge. (d) shows that no zero energy states are formed in an $s$-wave superconductor under the same conditions as (c).

Figure 2

(a) Density of states with different values of $\mu$. This plot shows the topological phase transition at $\mu ={\mu }^{*}$. There are no states inside the gap when $\mu <{\mu }^{*}$, while there are fourfold degenerate zero energy states when $\mu >{\mu }^{*}$. (b) Probability distribution of the zero energy states. The size of the system is $160\times 24$ for the substrate and $140\times 1$ for the wire. Solid red circles show the probability distribution in the wire, which has peaks at both ends.

Figure 3

Schematic figure of Majorana states at the ends of the wire (red circles). As discussed in the main text, the QW can be regarded as two copies of the Kitaev model (solid lines). There are two ways to combine two of them into ordinary fermions: nonlocal and local ones.