Topological superconductivity and anti-Shiba states in disordered chains of magnetic adatoms

Regular arrays of magnetic atoms on a superconductor provide a promising platform for topological superconductivity. In this work, we study the effects of disorder in these systems, focusing on vacancies realized by missing magnetic atoms. We develop approaches that allow treatment of ferromagnetic dense chains as well as long-range hopping ferromagnetic and helical Shiba chains at arbitrary subgap energies. Vacancies in magnetic chains play an analogous role to magnetic impurities in a clean $s$-wave superconductor. A single vacancy in a topological chain gives rise to a low-lying “anti-Shiba” state below the band edge of a regular magnetic chain. Proliferation of the anti-Shiba band formed by a finite density of hybridized vacancy states leads to deterioration of the topological phase, which exhibits unusual fragility in a particular parameter region in dilute chains. We also consider local fluctuation in the Shiba coupling and discuss how vacancy states could contribute to experimental verification of topological superconductivity.

Figure 1

Schematic representation of the systems studied in this work, illustrating (a) the dense chain with nearest-neighbor hopping, (b) the ferromagnetic Shiba chain, and (c) the helical Shiba chains. The grayed-out sites represent vacancies giving rise to low-lying anti-Shiba states.

Figure 2

Diagram of the ${\mathbb{Z}}_2$ invariant of a 50-site dense chain (a) in the pure limit and (b) with five vacancies averaged over 200 configurations. Both figures were obtained through applying the Pfaffian inavriant in real space as described in the text. Blue and yellow correspond to the topological and trivial phase, respectively; the red line is the analytical $k$-space solution for the phase boundaries obtained from Eq. (4). Remaining parameters are $\mu =4,{\alpha }_R=1$, and $\mathrm{\Delta }=1$.

Figure 3

(a) Normalized DOS of a dense chain with PBC, for 1000 sites with a single vacancy. The red line corresponds to the vacancy energy obtained through numerical calculation of the $T$-matrix. (b) Positive subgap bound state energies of a chain with two vacancy sites as a function of the distance between the vacancies. The spectrum has been calculated in a chain with 1000 sites. (c) The DOS of a single vacancy disorder realization of a periodic chain with 2000 sites and 100 vacancies. (d) The DOS for a disorder-averaged ($2\times {10}^6$ configurations) periodic chain with 80 sites with four vacancies. In all four figures, the parameters are $t=3,\mu =5,B=2,{\alpha }_R=1$, and $\mathrm{\Delta }=1$.

Figure 4

Distribution of the lowest-lying positive energy for 50 000 realizations of a periodic dense chain with 100 sites of which ten are vacancies. Parameters are $t=2.8,\mu =5,B=2,{\alpha }_R=1$, and $\mathrm{\Delta }=1$. For comparison, the gap energy for a pure system is $\approx 0.67\mathrm{\Delta }$.

Figure 5

(a) ${\mathbb{Z}}_2$ phase diagram of the finite ferromagnetic Shiba chain with Rashba SOC, obtained by evaluating the Pfaffian invariant. Blue and yellow correspond to $Q=-1$ and $Q=1$ phases, respectively. The red curves are exact phase boundaries of an infinite system obtained in Appendix pp1. Parameters used are $\varsigma =0.01\text{and}{\xi }_0=50a$ with 50 magnetic sites for the Pfaffian diagram. (b) $\mathbb{Z}$-valued winding number diagram for an infinite chain with the same parameters. The $\left|N\right|=2$ phase, supporting two Majorana end states in open chains, is not visible in the ${\mathbb{Z}}_2$ diagrams. (c) Energy gap diagram for the same parameters.

Figure 6

Effect of vacancies on the ferromagnetic Shiba chain. (a) Ratio of the single-vacancy bound state energy $E_{\mathrm{vac}}$ to the smallest positive energy $E_{\mathrm{gap}}$ of a clean system. Remaining parameters are $\varsigma =0.01\text{and}{\xi }_0=50a$. Note also the diagonal lines where the impurity energy is fine-tuned to near zero. The maximum values have been capped at 1 to avoid divergences at gap closings. (b) Relative gap plotted along the black line in (a). The vacancy-based gap closings are clearly visible, being wider than those caused by topological phase changes. (c) Pfaffian invariant $Q$ for a 50-site system with five vacancies, averaged over 400 configurations. The parameters are otherwise the same as in the previous figures. Additional diagonal gapless lines are seen next to the original single-vacancy lines. (d) Same, but with 10 vacancies.

Figure 7

The effect of disorder in the Shiba coupling $\alpha$ on the topology of the ferromagnetic Shiba chain. An average over (a) 100 configurations for a uniform disorder of 5% in $\alpha$; (b) 200 configurations for 20% disorder. Other parameters are $\varsigma =0.01\text{and}{\xi }_0=50a$. The length of the PBC chain is 50 sites.

Figure 8

(a) Disorder-averaged energy gap $\overline{E}$ for a 100-site ferromagnetic Shiba chain with 10 vacancies, averaged over 1000 configurations. The tube containing the average value represents one standard deviation. Other parameters are $\alpha =0.9,\varsigma =0.01,\text{and}{\xi }_0=50a$. (b) and (c) Distribution of the lowest positive energy for a 50-site periodic ferromagnetic Shiba chain with 5 and 10 vacancies, respectively. Distribution is for 10 000 configurations. Other parameters are $k_{F}a=20.5,\alpha =1,\varsigma =0.01,\text{and}{\xi }_0=50a$. The gap for a clean system is $\approx 0.11\mathrm{\Delta }$.

Figure 9

(a) ${\mathbb{Z}}_2$ phase diagram of a helical Shiba chain of 100 sites with $k_{H}a=\pi /8,\theta =\pi /2,{\xi }_0=50a$. Blue and yellow correspond to values $Q=-1$ and $Q=1$ of the Pfaffian invariant. The red curves represent exact phase boundaries of infinite system, obtained in Appendix pp1. (b) Energy gap diagram of an infinite helical Shiba chain with the same parameters. The diagonal line visible in the middle here, but not in the ${\mathbb{Z}}_2$ diagram, corresponds to the phase transition between phases $N=1$ and $N=-1$ of $\mathbb{Z}$-valued winding number invariants.

Figure 10

(a) Single-vacancy bound state energy divided by the energy gap of a clean system. Similarly to the ferromagnetic case, a line with zero bound-state energy is observed. The figure is calculated for a chain with 48 sites and $k_{H}a=\pi /8,\theta =\pi /2,\text{and}{\xi }_0=50a$. (b) Topological ${\mathbb{Z}}_2$ invariant for a helical Shiba chain with 48 sites of which 6 are vacancies, averaged over 400 configurations. Otherwise, the parameters are the same as in (a). The red curves in both figures represent the exact topological phase boundaries for an infinite system.

Figure 11

Topological ${\mathbb{Z}}_2$ phase diagrams for a helical Shiba chain with 48 sites and (a) 5% disorder in $\alpha$, averaged over 300 configurations. (b) Disorder in the planar angles of the magnetic moments: ${\phi }_i\to {\phi }_i\pm \delta {\phi }_i$, where $\delta {\phi }_i\in [-\pi /8,\pi /8]$, averaged over 100 configurations. Parameters used are $k_{H}a=\pi /8,\theta =\pi /2,\text{and}{\xi }_0=50a$.