Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling

There exists a variety of proposals to transform a conventional $s$-wave superconductor into a topological superconductor, supporting Majorana fermion midgap states. A necessary ingredient of these proposals is strong spin-orbit coupling. Here we propose an alternative system consisting of a one-dimensional chain of magnetic nanoparticles on a superconducting substrate. No spin-orbit coupling in the superconductor is needed. We calculate the topological quantum number of a chain of finite length, including the competing effects of disorder in the orientation of the magnetic moments and in the hopping energies, to identify the transition into the topologically nontrivial state (with Majorana fermions at the end points of the chain).

Figure 1

Proposed setup to obtain Majorana fermions at the end points of a chain of magnetic nanoparticles (typically Fe or Ni) on an $s$-wave superconducting substrate (typically Pb). Arrows indicate the orientation of magnetic moments of the nanoparticles.

Figure 2

Determinant of the reflection matrix as a function of $\mu$, for fixed $B_0=2t_0$ and $\Delta =0.9t_0$, ensemble averaged over 400 chains of nanoparticles of length $N$. Data are for ${\delta }_b=\infty$, ${\delta }_t=0$, corresponding to random and uncorrelated orientations of the magnetic moments of the nanoparticles, without randomness in the hopping energies.

Figure 3

Color scale plot of the determinant of the reflection matrix as a function of $\mu$ and ${\Delta }_0$, for fixed $B_0=2t_0$, calculated for a single disorder realization in a chain of $N=6000$ nanoparticles. The topologically nontrivial phase has $\mathrm{Det}r=-1$ (black region). The four panels are for different values of the disorder ${\delta }_t$ in the hopping energies and ${\delta }_b$ in the orientation of the magnetic moments [with ${\delta }_b=\infty$ corresponding to random and uncorrelated orientations; see Eq. (2.7)]. The dashed red curve in the top panel gives the phase boundary following from the self-consistent Born approximation.

Figure 4

Same as Fig. 3, but now as a function of ${\delta }_b$ and ${\delta }_t$ for fixed ${\Delta }_0,\mu$.

Figure 5

Excitation spectrum (black solid curves) of two magnetic particles with an angle $\theta ={70}^{\circ }$ between their magnetic moments, calculated from the Hamiltonian (2.1) at fixed $\mu =B_0=2{\Delta }_0$ as a function of the hopping energy $t_0$. The level crossings at the Fermi energy do not split because the ground states at the two sides of the crossing differ in fermion parity $\mathcal{P}\in \{0,1\}$ [red dashed curve, calculated from the Pfaffian of the antisymmetrized Hamiltonian, $\mathcal{P}=\frac{1}{2}-\frac{1}{2}\mathrm{sgn}\mathrm{Pf}\left({\sigma }_y{\tau }_{y}H\right)$].