Quantifying wave-function overlaps in inhomogeneous Majorana nanowires

A key property of Majorana zero modes is their protection against local perturbations. In the standard picture, this protection is guaranteed by a high degree of spatial nonlocality of the Majoranas, namely a suppressed wave-function overlap, in the topological phase. However, a careful characterization of resilience to local noise goes beyond mere spatial separation and must also take into account the projection of wave-function spin. By considering the susceptibility of a given zero mode to different local perturbations, we find the relevant forms of spin-resolved wave-function overlaps that measure its resilience. We quantify these overlaps and study their dependence with nanowire parameters in several classes of experimentally relevant configurations. These include nanowires with inhomogeneous depletion and induced pairing, barriers, and quantum dots. Smooth inhomogeneities have been shown to produce near-zero modes, so-called pseudo-Majoranas, below the critical Zeeman field in the bulk. Surprisingly, their resilience is found to be comparable or better than that of topological Majoranas in realistic systems. We further study how accurately their overlaps can be estimated using a purely local measurement on one end of the nanowire, accessible through conventional transport experiments. In uniform nanowires, this local estimator is remarkably accurate. In inhomogeneous cases, it is less accurate but can still provide reasonable estimates for potential inhomogeneities of the order of the superconducting gap. We further analyze the zero-mode wave-function structure, spin texture, and spectral features associated with each type of inhomogeneity. All our results highlight the strong connection between internal wave-function degrees of freedom, nonlocality, and protection in smoothly inhomogeneous nanowires.

Figure 1

Inhomogeneous nanowires. Sketch of an inhomogeneous nanowire, hosting Majorana zero modes of overlap ${\mathrm{\Omega }}_s$. The overlap may be estimated by a local quantity $\eta$ measured by a local probe. Five types of inhomogeneous profiles of the electrostatic potential $\varphi \left(x\right)$ and pairing $\mathrm{\Delta }\left(x\right)$ are considered: uniform, ${\mathrm{S}}^{\prime }\mathrm{S}$ (superconductor-superconductor), NS (normal-superconductor), barrier-S, and dot-S. The latter two are subtypes of the general NS case.

Figure 2

Uniform nanowires. Panels (a) and (b) show the spectrum (top panel) and the Majorana overlaps ${\mathrm{\Omega }}_{\max }$ (red shaded region), ${\mathrm{\Omega }}_s$ (solid red), ${\mathrm{\Omega }}_0$ (dotted red), $\delta N$ (solid green), local estimator $\eta$ (black), and splitting $E_0$ (blue) of the lowest lying state (bottom panel) as a function of Zeeman $B$ in uniform nanowires with $\mu =0.2$ meV, $\mathrm{\Delta }=0.3\mathrm{meV}$, and $\alpha =0.4\mathrm{eV}\AA$. The nanowire length is $L=1.2\mu \mathrm{m}$ (a) and $L=3\mu \mathrm{m}$ (b). The Majorana wave functions along the nanowire are shown to the right, at three values of $B$ (numbered circles), together with a sketch (above) of the corresponding band-topological phase of the nanowire. In panel (c), we show the position in the $(\eta ,{\mathrm{\Omega }}_s)$ plane of the near-zero modes with $E_0\le 10\mu \mathrm{eV}$ in an ensemble of nanowire configurations uniformly distributed within the following parameter ranges: $\mu \in (0,2.5)$ meV, $\mathrm{\Delta }\in [0,0.5]$ meV, $\alpha \in [0.1,1]\mathrm{eV}\AA$, and $L\in [0,2]\mu \mathrm{m}$. Panel (d) shows the corresponding probability density $P(\eta ,{\mathrm{\Omega }}_s)$. The Pearson correlation factor of the nonlocality estimator is $r=0.98$ [slightly less as we filter into bins of increasing Fermi energy; see the small red, green, and blue subpanels to the right of panels (c) and (d)].

Figure 3

Smooth ${\mathrm{S}}^{\prime }\mathrm{S}$ nanowires. [(a), (b)] Equivalent of Fig. 2 in a nanowire with two superonducting halves, of lengths $L_{S^{\prime }}=1\mu \mathrm{m}$ and $L_S=2\mu \mathrm{m}$, with uniform pairing $\mathrm{\Delta }=0.5$ meV but different ${\varphi }_{S^{\prime }}=-0.2\mathrm{meV}$ and ${\varphi }_S=-0.7\mathrm{meV}$ (a) or ${\varphi }_S=-2.0\mathrm{meV}$ (b) at either side. In terms of Fermi energy differences $\mathrm{\Delta }\mu ={max}_x\left[\varphi \left(x\right)\right]-{min}_x\left[\varphi \left(x\right)\right]$, this corresponds to $\mathrm{\Delta }\mu =0.5$ meV (a) and $\mathrm{\Delta }\mu =1.8$ meV (b). The interface has smoothness length $\zeta =0.5\mu \mathrm{m}$. The rest of parameters as in Fig. 2. Panels (c) and (d) are computed by independently sampling all parameters, $B\in [0,2.7]$ meV, $\mathrm{\Delta }\in [0,0.5]$ meV, ${\varphi }_{S^{\prime },S}\in [-2.5,2.5]$ meV, $L_{S^{\prime },S}\in [0.2,1]\mu \mathrm{m}$, $\zeta \in [0,0.5]\mu \mathrm{m}$, and $\alpha \in [0.1,0.7]\mathrm{eV}\AA$. The binning windows are defined in terms of $\mathrm{\Delta }\mu$.

Figure 4

Smooth NS nanowires. Equivalent to Figs. 2 and 3, with identical model and sampling parameters as in the latter, except for a zero pairing ${\mathrm{\Delta }}_N=0$ on the left side and finite ${\mathrm{\Delta }}_S=0.5$ meV on the right side of the smooth junction. Note the similar wave functions of the smooth junction Majoranas as compared to the ${\mathrm{S}}^{\prime }\mathrm{S}$ case of Fig. 3.

Figure 5

Barrier-superconductor nanowires. Specific barrier-S configuration of the NS model, similar to the one discussed in Ref. [35], with positive ${\varphi }_N=2\mathrm{meV}$ and short $L_N=\zeta =0.1\mu \mathrm{m}$, which forms a smooth, Zeeman-polarized insulating barrier around $x=0$. Other parameters: ${\varphi }_S=-1.8$ meV, ${\mathrm{\Delta }}_S=0.3$ meV, $L_S=2\mu \mathrm{m}$, and $\alpha =0.4\mathrm{eV}\AA$.

Figure 6

Dot-superconductor nanowires. Specific dot-S configuration of the NS model, relevant to a number of experiments [7, 49], with negative ${\varphi }_N=-14\mathrm{meV}$ and short $L_N=0.15\mu \mathrm{m}$, which confines states in a quantum-dot region around $x=0$. Other parameters: ${\varphi }_S=-0.8$ meV, ${\mathrm{\Delta }}_S=0.5$ meV, $\zeta =0$, $L_S=2\mu \mathrm{m}$, and $\alpha =0.2\mathrm{eV}\AA$.

Figure 7

Majorana spin in ${\mathrm{S}}^{\prime }\mathrm{S}$ junctions. (a) Wave function $|{\mathbf{u}}^L|$ and $|{\mathbf{u}}^R|$ of the lowest eigenstate in an ${\mathrm{TS}}^{\prime }\mathrm{S}$ junction of increasing smoothness $\zeta =(0.0,0.1,1.0)\mu \mathrm{m}$. (b) The corresponding spin density $\langle {\sigma }_x\rangle$ along the Zeeman field. The shading under the spin density curves encodes the Majorana canting angle $\theta$ relative to the Zeeman field along $x$. Parameters: ${\varphi }_{S^{\prime }}=0$, ${\varphi }_S=-1$ meV, $L_{S^{\prime }}=L_S=1.8\mu \mathrm{m}$, ${\mathrm{\Delta }}_{S^{\prime }}={\mathrm{\Delta }}_S=0.4$ meV, $\alpha =0.4\mathrm{eV}\AA$.

Figure 8

Majorana spin in NS junctions. Same as Fig. 7 for an NS nanowire (${\mathrm{\Delta }}_N=0$).

Figure 9

Majorana spin in barrier-S junctions. Same as Fig. 7 for a barrier-S nanowire (${\mathrm{\Delta }}_N=0$, ${\varphi }_N=1$ meV).