Self-Organized Topological State with Majorana Fermions

Most physical systems known to date tend to resist entering the topological phase and must be fine-tuned to reach that phase. Here, we introduce a system in which a key dynamical parameter adjusts itself in response to the changing external conditions so that the ground state naturally favors the topological phase. The system consists of a quantum wire formed of individual magnetic atoms placed on the surface of an ordinary $s$-wave superconductor. It realizes the Kitaev paradigm of topological superconductivity when the wave vector characterizing the emergent spin helix dynamically self-tunes to support the topological phase. We call this phenomenon a self-organized topological state.

Figure 1

Chain of magnetic atoms on a superconducting substrate. (a) Schematic depiction of the system with the red spheres representing the adatoms and blue arrows showing their magnetic moments arranged in a spiral. (b) Two spin-degenerate branches of the normal-state spectrum of the system in the absence of magnetic moments modeled by the nearest-neighbor tight-binding model Eq. (1). (c) With the magnetic moments the two branches shift in momentum by $\pm G$ and the gap $JS$ opens at $q=0$, $\pi$. Dashed lines show the shifted spectral branches indicated in panel (b) with no gap for comparison.

Figure 2

The pitch of the spiral and the topological phase diagram. Panel (a) shows the spiral wave vector $G$ that minimizes the system ground-state energy $E_g(G)$ as a function of $\mu$, the latter in units of $t$. The parameters are $S=5/2$, $J=0.1t$, and $\Delta =0$ (red line), $\Delta =0.1t$ (blue line). The dashed line represents $G=2k_F$ whereas the green band shows the region in which $G$ must lie for the system to be topological for a given $\mu$. Panel (b) shows the topological phase diagram in the $\mu \mathrm{\text{−}}J$ plane, both in units of $t$, for $\Delta =0.1t$. To distinguish the two phases, we have calculated the Majorana number $\mathcal{M}$ as defined in Ref. 13. Topological phase (TSC) is indicated when $\mathcal{M}=-1$ whereas $\mathcal{M}=+1$ indicates the topologically trivial phase (N).

Figure 3

Spectral function of the adatom chain. $A_{\mathrm{eff}}(\omega ,q)$ defined in Eq. (11) is represented as a density plot for several representative values of the chemical potential $\mu$, with the appropriate self-consistently determined spiral pitch $G$ shown in Fig. 2a. In panel (a), $\mu$ is below the bottom edge of the chain band and the system is in the trivial phase. When $\mu$ reaches the band edge, a topological phase transition occurs through a gap closing shown in panel (b). Further increase of $\mu$ puts the system into the topological phase illustrated in (c) and (d) until the gap closes again near half filling (e) placing the system back into the trivial phase (f). A similar sequence of phases occurs for positive values of $\mu$ for which the spectral function can be obtained by simply flipping the sign of the frequency $\omega$. The dark band around the chemical potential reflects the bulk SC gap. In all panels, frequency $\omega$ is in units of $t$ whereas $JS=0.4t$, ${\Delta }_0=0.6t$, ${\rho }_0{r}^2=0.05t$, and $\delta =0.002t$ is used to give a finite width to the spectral peaks.