Two types of topological transitions in finite Majorana wires

Motivated by the recent advances in studying Majorana states in nanowires under conditions of the superconducting proximity effect, we address the correspondence between the common topological transition in an infinite system and a topological transition of another type that manifests itself in the positions of the poles of the scattering matrices. We establish a universal dependence of the pole positions in the vicinity of the common transition on the parameter controlling the transition, and discuss the manifestations of the pole transitions in the differential conductance.

Figure 1

(a) The setup: the Majorana wire of length $L$ at the top of a superconductor is connected to a normal-metal lead. The total scattering matrix at low energy incorporates that of the contact ($S_{\mathrm{c}}$) and the energy-dependent scattering matrix describing propagation in the wire, $S_{\mathrm{w}}$. (b) Sketch of the (continuous) spectrum in the limit of infinite; a Majorana level emerges upon the common topological transition. (c) The common topological transition becomes a crossover for finite $L$. The quantized energy levels (real parts of the pole energy positions) are sketched versus the control parameter $a$. The lowest level reaches 0 at the point of the pole topological transition. Dashed lines give the imaginary parts of the pole positions for the lowest level.

Figure 2

The real (a) and and imaginary (b) parts of the pole energy positions versus the control parameter $\tilde{a}$ at two values of the reflection amplitude $r$: $r=0.86$ (green dots, almost isolated) and $r=0.34$ (red crosses, almost transparent). The part of (a) within the rectangle is replotted in (c). The pole topological transition occurs at $a_0=-1.2$ for $r=0.86$ and $a_0=-3.3$ for $r=0.34$. (d) The dependence of the transition point $a_0$ on $r$.

Figure 3

The universal bias-dependent contribution to the differential Andreev conductance of the contact versus energy/voltage. Solid, dashed, dotted curves correspond to $r=0.9,0.5,0.1$ respectively (${\tilde{a}}_0\approx -0.5,-1.7,-3.5$). (a) $\tilde{a}=2$, long before the transition. (b) $\tilde{a}=0$, the bulk topological transition, no sharp features in the conductance curves. (c) $\tilde{a}=-2$ in the crossover interval. (d) $\tilde{a}=-4$ long after the transition. The Majorana at the far end of the wire is manifested as a narrow dip at zero bias.