Analytical and semianalytical tools to determine the topological character of Shiba chains

We introduce three analytical and semianalytical tools that allow one to determine the topological character of impurity Shiba chains. The analytical methods are based on calculating the effective Green's function of an infinite embedded chain using the $T$-matrix formalism and describing the chain as a line impurity. We thus provide a solution to the longstanding size-effects problem affecting the only general alternative method, the numerical tight-binding analysis. As an example we consider a chain of magnetic impurities deposited on an $s$-wave superconducting substrate with Rashba spin-orbit and we calculate its topological phase diagram as a function of the magnetic impurity strength and the chemical potential. We find a perfect agreement between all our techniques and a numerical analysis.

Figure 1

Schematics of the systems considered. Panel (a) shows a chain of magnetic impurity atoms (green) on the surface of the superconductor (dark blue). For approach I (see main text), the system is infinite in both directions. For approach III, the surface is finite in the $x$ direction and infinite in the $y$ direction. In panel (b) the red atom is an additional probe point-impurity, as described in approach II in the main text. All numerical tight-binding solutions are performed for systems finite in both directions.

Figure 2

(a) The chiral invariant, $\nu$, plotted as a function of the chemical potential and the magnetic impurity strength, as obtained from the effective wire Green's function. (b) The zero-energy density of states, $\rho$, in the presence of a probe point-impurity; shown in the inset is the total density of states in the absence of the probe. (c), (d) The Majorana polarization $C$ and the energy corresponding to the lowest-energy state calculated using a tight-binding model for a chain of 60 sites introduced into a two-dimensional system of size $21\times 80$. We take $\lambda =0.2t$ and $\mathrm{\Delta }=0.4t$.

Figure 3

The quasi-one-dimensional invariant, ${\nu }_{\mathrm{Q}1\mathrm{D}}$, for (a) $N_x=51$ and (b) $N_x=11$.

Figure 4

The quasi-one-dimensional invariant for $\lambda =\mathrm{\Delta }=0.005t$ for $N_x=1001$. Panels (b) and (c) demonstrate the parts of the phase diagram close to $\mu /t=0$ and $\mu /t=4$ at a higher resolution.

Figure 5

A schematic of the magnetic impurity sites (green atoms) above the superconducting surface (dark blue atoms).

Figure 6

A comparison of the lowest-energy state energy ${\epsilon }_1$ and its Majorana polarization $C$ for the impurities defined as sites above the two-dimensional lattice. Here we take $\lambda =0.2t$ and $\mathrm{\Delta }=0.4t$. Either $t^{\prime }$ or $U$ are kept constant and the other term is varied, as labeled in the figures.