Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain

We study a chain of magnetic moments exchange coupled to a conventional three-dimensional superconductor. In the normal state the chain orders into a collinear configuration, while in the superconducting phase we find that ferromagnetism is unstable to the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wave vector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band crosses the Fermi energy, wherein the spiral phase becomes stabilized by the spontaneous opening of a $p$-wave superconducting gap within the band. This leads to the possibility of electron-driven topological superconductivity with Majorana boundary modes using magnetic atoms on superconducting surfaces.

Figure 1

Magnetic atoms on the surface of a superconductor induce local spin-polarized, subgap YSR states (black curves). YSR wave-function overlap (shaded red) leads to hopping and $p$-wave pairing due to hybridization with singlet Cooper pairs in the host. This hybridization can drive the spin chain into a spiral state that, in turn, induces spontaneous topological superconductivity within the YSR chain.

Figure 2

(a) Exact ground-state phase diagram of Eq. (1) as a function of lattice spacing and YSR energy $\varepsilon$ [Eq. (5)] for integer $n$ ($n=15$ shown) and $k_F\xi =2000$. Outside the range $n+1/4\lesssim k_{F}a/\pi \lesssim n+3/4$, the spin chain is an antiferromagnet (AF) with $q=\pi /a$, while inside this range it is a spiral (S) whose wave vector is shown schematically in grayscale and plotted in (b) along the line $k_{F}a=15.5\pi$. The topologically nontrivial superconducting (${\mathrm{TS}}^{\pm }$) regions are bounded by the red curves, across which either a first-order transition to the AF state or a continuous transition into the trivial S state occurs. The latter transition occurs approximately along the dotted lines: ${\varepsilon }^{\pm }/\mathrm{\Delta }=-(k_{F}a\mathrm{mod}\mp \pi )/k_{F}a$. The system is gapless along the ferromagnetic line $\varepsilon /\mathrm{\Delta }=\left[\pi /2-\left(k_{F}a\mathrm{mod}\pi \right)\right]/k_{F}a$ separating the topologically distinct ${\mathrm{TS}}^{\pm }$ regions with topological invariants $\pm 1$. (b) Ground-state wave vector $q$ of the magnetic spiral plotted as a function of $\varepsilon$. The solid line corresponds to the exact solution of Eq. (1), the long-dashed line to the $T$-matrix approximation of Eq. (6), and the dotted line to the YSR approximation of Eq. (7). The inset shows the magnetic order near the topological superconducting (${\mathrm{TS}}^{\pm }$) phases bounded by the vertical dashed-dotted lines.

Figure 3

Spectrum $E_k$ of the YSR band, Eq. (7). For $q\ne 0$ a spectral gap opens and the system gains the condensation energy, corresponding to the gray area between the solid and dashed lines for momenta near $\pm k_{*}$. This occurs only if the pairing at these points is nonzero: ${\mathrm{\Delta }}_{k_{*}}\ne 0$. When the zeros of $h_k$ and ${\mathrm{\Delta }}_k$ (depicted by dots) coincide, the energy (7) is minimized at $q=0$, corresponding to a gapless electron system and a ferromagnetic spin chain. This phase occurs along lines in the phase diagram given by Eq. (8).