Topological superconductivity in a multichannel Yu-Shiba-Rusinov chain

Chains of magnetic atoms placed on the surface of an $s$-wave superconductor with large spin-orbit coupling provide a promising platform for the realization of topological superconducting states characterized by the presence of Majorana zero-energy modes. In this work, we study the properties of one-dimensional chains of Yu-Shiba-Rusinov states induced by magnetic impurities using a realistic model for the magnetic atoms that includes the presence of multiple scattering channels. These channels are mixed by spin-orbit coupling and, via the hybridization of the Yu-Shiba-Rusinov states at different sites of the chain, result in a multiband structure for the chain. We obtain the topological phase diagram for such a band structure and show that the inclusion of higher bands can greatly enlarge the phase space for the realization of topological states.

Figure 1

(a) Schematic setup of the proposed structure supporting multichannel YSR chain. Atoms with high magnetic moments form a ferromagnetic chain on the surface of thin film superconductor [42, 52]. Each magnetic atom creates multiple YSR states with distinct angular momentum quantum number $l$. Here, we consider $l=-1$, 0, and 1 states which form three bands: $s$ band from the $l=0$ state and $p$ bands from the $l=-1,+1$ states, in a quasi one-dimensional system. In the topological phase, there is an odd number of Majorana zero-energy modes at the opposite ends of the chain. (b) and (c) are schematic illustrations of the two limits: deep $s$-band limit and deep $p$-band limit, respectively. The dark (light) shaded area inside the superconducting gap $\mathrm{\Delta }$ schematically demonstrates the energy regime of the $s$ ($p$) YSR band in the two limits, in which the bandwidth $W$ (the vertical width of the shaded area) of each YSR band is small: $W\ll \mathrm{\Delta }$. In the deep $s$-band limit (b), the $s$-band is near the midgap region and is well separated from the $p$ bands; on the contrary, in the deep $p$-band limit (c), the $p$ bands are near the midgap region and well separated from the $s$ band.

Figure 2

Topological phase diagram for the deep $s$ band as a function of physical parameters. The dark and light colors represent topologically nontrivial and trivial phases, respectively. (a) Topological phase diagram for the magnetization in $\widehat{\mathbf{z}}$ direction, as a function of $J_0$ and $k_{F}a$ for $\alpha =0.3, J_1=0.4$, and ${\xi }_0=2a$. (b) Topological phase diagram for the magnetization in $\widehat{\mathbf{z}}$ direction as a function of $\alpha$ and $k_{F}a$ for $J_0=1.025, J_1=0.4$, and ${\xi }_0=2a$. (c) The phase boundary for the magnetization in $\widehat{\mathbf{z}}$ direction (blue dashed line) and for the magnetization in $\widehat{\mathbf{x}}$ direction (red solid line) indicates that by changing the magnetization one can drive the topological phase transition. (d) Calculated quasiparticle excitation gap for the parameter regime in the phase diagram (a) with the phase boundary indicated by the white line.

Figure 3

Quasiparticle excitation gap $E_g$ along different line cuts on the phase diagrams: (a) along the line cut A in Fig. 2 as a function of $J_0$ at $k_{F}a=37.2\pi$. (b) Along the line cut B in Fig. 2 as a function of $k_{F}a$ at $J_0=1.015$. (c) Along the line cut C in Fig. 2 as a function of $\alpha$ at $k_{F}a=36.8\pi$. Here, the shaded area in (a)–(c) indicates the topologically nontrivial phase as shown in Figs. 2 and 2. To see the location of the gap closing points in (c), we plot the hopping energy spectrum $h\left(k\right)$ in (d), the effective pairing energy spectrum $\mathrm{Im}\tilde{\mathrm{\Delta }}\left(k\right)$ in (e), and the whole energy spectrum of the particle band in (f), as a function of $k$ and $\alpha$.

Figure 4

The dependence of the normal-state energy spectrum (i.e., ${\tilde{\mathrm{\Delta }}}_{ij}=0$) on momentum $k$ for (a) $\alpha J_0=0, J_1=1.0125, k_{F}a=37.5\pi , {\xi }_0=2a$; and (b) $\alpha =0.3, J_0=0.4, J_1=1.0125, k_{F}a=37.5\pi , {\xi }_0=2a$. The normal-state spectrum consists of heavy-fermion and light-fermion bands which are hybridized by the SOC. The zoom-in figure of (b) near the Fermi level is shown in the inset.

Figure 5

(a) Topological phase diagram in the $(k_{F}a,J_1)$ plane in the deep $p$-band limit, for the case in which $\mathbf{S}\parallel \widehat{\mathbf{z}}, J_0=0.4, \alpha =0.3$, and ${\xi }_0=2a$. (b) Enlargement of the topological phase diagram shown in (a) around the region surrounded by the dashed line rectangle. (c) Calculated quasiparticle excitation gap for the parameter regime in the phase diagram (b) with the phase boundary indicated by a white line. (d) The quasiparticle excitation gap on the line cut A in (b) and (c) near the re-entrance region at $k_{F}a=36.4\pi$. (e) The quasiparticle excitation gap on the line cut B in (b) and (c) near the re-entrance region at $k_{F}a=37.6\pi$. Here, the shaded area in (d) and (e) indicates the topologically nontrivial phase as shown in (b). The integer numbers $-1$, 0, 1 shown in (d) and (e) are the winding numbers calculated for each gapped phase.

Figure 6

Comparison of the topological phase diagram: (a) the $s$ band system for $\alpha =0.3, J_1=0.4$, and ${\xi }_0=2a$; and (b) the $p$ band system for $\alpha =0.3, J_0=0.4$, and ${\xi }_0=2a$.

Figure 7

The dependence of the functions $A_0\left(k_{F,\lambda }a+i{\zeta }_{\lambda }^{-1}a\right), A_2\left(k_{F,\lambda }a+i{\zeta }_{\lambda }^{-1}a\right), B_1\left(k_{F,\lambda }a+i{\zeta }_{\lambda }^{-1}a\right)$ and $C_1\left(k_{F,\lambda }a+i{\zeta }_{\lambda }^{-1}a\right)$ on $k_{F,\lambda }a$. Here, we used ${\zeta }_{\lambda }=10a$.