Fig. 3 | Communications Physics

Fig. 3

From: Non-hermitian topology as a unifying framework for the Andreev versus Majorana states controversy

Fig. 3

Trivial and non-trivial zero modes in inhomogeneous nanowires. The microscopic property that governs the absence (presence) of an exceptional point (EP) bifurcation for \(B\ < \ {B}_{{\rm{c}}}\) is wave function locality (non-locality), such as in a quantum dot state (local case, a); and with smoothly confined Andreev bound states (ABS) for \(B\ < \ {B}_{{\rm{c}}}\) (non-local case, b). The corresponding profiles \(\mu (x)\) and \(\Delta (x)\) are shown in the top inset panels. Panels (cd) show the complex spectrum versus Zeeman field for (c) the quantum dot state (trivial parity crossing); and (d) the smoothly confined ABS for \(B\ < \ {B}_{{\rm{c}}}\). The decay rates (imaginary energy) of the two lowest states are shown in dashed red. Exceptional points (circles) appear as decay rate bifurcations accompanied by real energies (solid red) stabilised at zero. The trivial parity crossing in (c) does not show an EP bifurcation for \(B\ < \ {B}_{{\rm{c}}}\) since the non-Hermitian topological criterion \(\nu \equiv ({\gamma }_{0}-| {E}_{0}| )/{\Gamma }_{0}\ > \ 0\) is not fulfilled at low Zeeman fields (e). The EP bifurcation occurs far from this trivial parity crossing (near \(B={B}_{{\rm{c}}}\), as expected for a long uniform wire). On the contrary, a smooth inhomogeneity gives rise to stable zero modes after an EP at \(B\ < \ {B}_{{\rm{c}}}\) (d), with non-Hermitian topological criterion \(\nu \ > \ 0\) (f). The decay asymmetry \(\gamma /\Gamma\) changes accordingly with sharp increases from \(\gamma /\Gamma =0\) to \(\gamma /\Gamma \to 1\) at \(B={B}_{{\rm{c}}}\) (g) and \(B\ < \ {B}_{{\rm{c}}}\) (h). Detailed parameters can be found in the Supplementary Table 1

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