Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions

We report the emergence of zero-energy states in the trivial phase of a short nanowire junction with a strong spin-orbit coupling and magnetic field, formed by strong coupling between the nanowire and two superconductors. The zero-energy states appear in the junction when the superconductors induce a large energy shift in the nanowire, such that the junction naturally forms a quantum dot, a process that is highly tunable by the superconductor width. Most importantly, we demonstrate that the zero-energy states produce a $\pi$ shift in the phase-biased supercurrent, which can be used as a simple tool for their unambiguous detection, ruling out any Majorana-like interpretation.

Figure 1

(a) 1D NW (cyan) coupled to the middle of two 2D SCs (red) by $\mathrm{\Gamma }$. A short central region of the NW is left uncoupled, giving a short S-N-S junction with a $\varphi$ superconducting phase difference. (b) Effective chemical potential profile deep into the $S$ parts of the NW as a function of SC width $L_y$. Markers are representative points at which three cases are studied: ideal (triangle), PB (cross), and QD (dot).

Figure 2

Zeeman field-dependent spectrum at $\varphi =0$ for ideal ($L_y=41a$) (a), PB ($L_y=11a$) (b), and QD ($L_y=21a$) (c) cases. Vertical dashed green lines in (a)–(d) mark the topological phase transition, while the red arrow in (c) marks the start of zero-energy levels. (d) Local spin projection at the junction $S_{x=L_{\mathrm{NW}}/2}^{(x)}$ in the lowest level $E_0$ for cases (a)–(c) as a function of $B/B_c$. Cyan (magenta) marks spin up (down) while marker size denotes magnitude. (e),(f) Color plot of $E_0$ as a function of $\mathrm{\Gamma }$ and $B$ for $L_y=41a$ (e) and $L_y=21a$ (f) cases. The green line marks the topological phase transition, the red line marks the start of the supercurrent $\pi$ shift, and the dotted white line marks $\mathrm{\Gamma }=0.7$. The filled circles in (f) denote colored markings in (c).

Figure 3

(a)–(d) Phase-dependent low-energy spectrum in the QD case ($L_y=21a$), obtained at the color-marked $B$ values in Fig. 2. Inset in (a): scaled free energy $\tilde{F}=(F-F_{\min })/(F_{\max }-F_{\min })$ for (a)–(d), with $F_{\min (\max )}$ the minimum (maximum) of $F$ in each case. Probability density of the lowest state, $|{\mathrm{\Psi }}_0{|}^2\times {10}^3$, in (a)–(d) at $\varphi =0$ (e) and $\varphi =\pi$ (f).

Figure 4

(a) Color plot of the supercurrent for the QD case ($L_y=21a$) as a function of $\varphi$ and $B$. Topological phase transition (green dashed line), beginning of ABS crossings in phase-dependent energy spectrum (magenta), and zero-energy crossing at $\varphi =0$, i.e., the red arrow in Fig. 2 (white). Total supercurrent (b), with contributions from $E_0$ (c) and $E_1$ (d) energy levels at the color-marked $B$ values Fig. 2, repeated in (a).