Strong Localization of Majorana End States in Chains of Magnetic Adatoms

A recent experiment [Nadj-Perge et al, Science 346, 602 (2014)] gives possible evidence for Majorana bound states in chains of magnetic adatoms placed on a superconductor. While many features of the observed end states are naturally interpreted in terms of Majorana states, their strong localization remained puzzling. We consider a linear chain of Anderson impurities on a superconductor as a minimal model and treat it largely analytically within mean-field theory. We explore the phase diagram, the subgap excitation spectrum, and the Majorana wave functions. Owing to a strong velocity renormalization, the latter are localized on a scale which is parametrically small compared to the coherence length of the host superconductor.

Figure 1

Representative phase diagrams for the adatom chain as a function of the Shiba state energy $E_s$ of an individual impurity and the hopping amplitude $w$ between $d$ levels. The colors indicate the topological index (grey: topologically trivial; red or green: topological phase with index $\pm 1$). We chose $E_{d,\downarrow }=100\mathrm{\Delta }$, $k_{F}a=4.3\pi$, $k_{h}a=0.26\pi$, and ${\xi }_0/a=\infty$. The panels correspond to (a) symmetric adatom $d$ bands ($E_{d,\uparrow }=-100\mathrm{\Delta }$) and (b) asymmetric adatom $d$ bands ($E_{d,\uparrow }=-300\mathrm{\Delta }$). Here, $E_{d,\sigma }={\epsilon }_d-\sigma U⟨m⟩/2$.

Figure 2

Excitation spectra $E_k$ for $ka/\pi \in [0,1]$ and (a) $\mathrm{\Gamma }=64\mathrm{\Delta }$, (b) $\mathrm{\Gamma }=16\mathrm{\Delta }$, and (c) $\mathrm{\Gamma }=0.16\mathrm{\Delta }$. We choose $k_{F}a=4.3\pi$, $k_{h}a=0.26\pi$, $E_{d,\downarrow }=100\mathrm{\Delta }$, $E_{d,\uparrow }=-19900\mathrm{\Delta }$, $w=90\mathrm{\Delta }$, and ${\xi }_0/a=\infty$. The dashed lines are subgap dispersions of the impurity chain without coupling to the SC. The blue curves are exact dispersions. The red curves are calculated using Eq. (12) for $\mathrm{\Gamma }\gg \mathrm{\Delta }$ and Eq. (11) for $\mathrm{\Gamma }\ll \mathrm{\Delta }$. Notice that the horizontal axis in (c) is restricted to a very narrow range and that the deviation between the red and blue curves is indeed small. Panels (d), (e), and (f) show Majorana wave functions ${|{\psi }^M(i)|}^2$ (blue lines) obtained for a finite chain of length $L=1500a$. Only the first 120 sites $i$ are shown. (d) and (e) are plotted on a logarithmic scale and the red dashed lines are fits using Eq. (2) for the Majorana localization length. (f) is plotted on a linear scale. (Inset) Decay over the first 600 sites.

Figure 3

Local density of states of particle excitations, computed in the center (blue lines) and at the end (red lines), for a chain of length $L=300a$ and hybridizations (a) $\mathrm{\Gamma }=64\mathrm{\Delta }$ and (b) $\mathrm{\Gamma }=16\mathrm{\Delta }$. Other parameters are as in Fig. 2.