Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement

A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby $s$-wave superconductor may be in a topological nontrivial state, in which it has a zero-energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero-energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase, even in the ideal clean single channel limit.

Figure 1

Andreev end-state energy $\varepsilon$, normalized to the bulk excitation gap ${\Delta }_p$, as a function of adiabaticity parameter $\sigma$. The red circles are obtained from a numerical calculation for which the wire is terminated by a smooth potential $V\left(x\right)$ of the form (17) with $a=5\mu$. Other parameters used in this numerical calculation are ${\varepsilon }_{\mathrm{so}}=0.1\mu$, $\Delta =0.05\mu$, $B=0.275\mu$. The solid blue curve corresponds to Eq. (13) of the main text.

Figure 2

Andreev end-state energy $\varepsilon$ as a function of the Zeeman energy $B$ (main panel) and induced superconducting gap parameter $\Delta$ (inset). The parameters of the numerical calculation are ${\varepsilon }_{\mathrm{so}}=0.1\mu$, $\Delta =0.04\mu$ (main figure), $B=0.5\mu$ (inset). The confining potential has the form (17) with $a=5\mu$ and $\sigma =1.273(h/p_{\mathrm{F}})$.

Figure 3

Andreev end-state energy $\varepsilon$ vs the induced superconducting gap parameter $\Delta$ with (squares) and without (circles) an abrupt termination of the order parameter $\Delta$ at position $x_{\mathrm{N}}$. The inset shows the functional form of the superconducting order $\Delta \left(x\right)=\Delta \theta (x-x_{\mathrm{N}})$. In both cases the wire is terminated by a smooth potential $V\left(x\right)$ of the form (17) with parameters with $a=5\mu$ and $\sigma \approx 3.183h/p_{\mathrm{F}}$. Other parameters used in this numerical calculation are ${\varepsilon }_{\mathrm{so}}=0.1\mu$ and $B=0.275\mu$.

Figure 4

Andreev end-state energies $\varepsilon$ for a two-dimensional wire with two transverse channels, normalized by the bulk excitation gap, as a function of the adiabaticity parameter $\sigma$. The Hamiltonian is given by the two-dimensional extension of Eq. (1), which has the spin-orbit coupling term $\alpha p_x{\sigma }_y{\tau }_z-\alpha p_y{\sigma }_x$. Parameters in the numerical calculation are $B=0.1667\mu$, and wire width $W=1.225h/p_{\mathrm{F}}=0.054\xi$. The spin-orbit energies of the two transverse bands (calculated for $B=0$) are ${\varepsilon }_{\mathrm{so},1}=0.074\mu$, ${\varepsilon }_{\mathrm{so},2}=0.042\mu$.