Majorana fermions in a topological superconducting wire out of equilibrium: Exact microscopic transport analysis of a $p$-wave open chain coupled to normal leads

Topological superconductors are prime candidates for the implementation of topological-quantum-computation ideas because they can support non-Abelian excitations such as Majorana fermions. We go beyond the low-energy effective-model descriptions of Majorana bound states (MBSs) to derive nonequilibrium transport properties of wire geometries of these systems in the presence of arbitrarily large applied voltages. Our approach involves quantum Langevin equations and nonequilibrium Green's functions. By virtue of a full microscopic calculation we are able to model the tunnel coupling between the superconducting wire and the metallic leads realistically, study the role of high-energy nontopological excitations, predict how the behavior compares for an increasing number of odd versus even number of sites, and study the evolution across the topological quantum phase transition (QPT). We find that the normalized spectral weight in the MBSs can be remarkably large and goes to zero continuously at the topological QPT. Our results have concrete implications for the experimental search and study of MBSs.

Figure 1

Schematic depiction of a Kitaev chain in terms of MFs (solid circles) for $\nu =0$. Two generic cases with an odd ($N=3$) and even ($N=4$) number of sites are shown side by side. The solid and dashed lines correspond to $\left|\Delta \right|\pm \gamma$ matrix elements, respectively. The two dotted lines at each end represent equal-strength contacts with the respective lead. Since for $\nu =0$ there is no matrix element between the two MFs within each site, one can see the existence of two independent Majorana (conduction) channels with $N$ MFs each. Notice those two channels are always mirror equivalent for odd $N$ and contain an exact zero mode each. For $\left|\Delta \right|=\gamma$ and any odd or even $N$, the dashed lines are absent and there are two zero-energy MBSs localized one at each end and decoupled from the rest of the chain.

Figure 2

$dI/dV$ (in units of $e^2/h$) vs $eV$ for ${\gamma }_p=1$, ${\gamma }_p^{\prime }=\gamma =0.1$ ($p=L,R$), and $\left|\Delta \right|=0.3$. Everywhere $\nu =0$ except for $\nu =0.1,0.25$ in (g) and (h), respectively. Wire lengths are as follows: (a) $N=2$, (b) $N=6$, (c) $N=16$, (d) $N=100$, and (e) $N=3$, (f)–(h) $N=101$.

Figure 3

$I$-$V$ characteristics for a symmetrically biased wire. $N=15$, ${\gamma }_p=10{\gamma }_p^{\prime }=1$ ($p=L,R$), and $\gamma =0.3$. In (a) $\Delta$ changes as $0,0.1,...,0.6$ while $\nu =0$, and in (b) $\nu =0.65$.

Figure 4

Normalized spectral weight of the MBSs for ${\gamma }_p=10{\gamma }_p^{\prime }=1$ ($p=L,R$), and (a) $\Delta =0.3$ or (b) $\gamma =0.3$.