Majorana states in helical Shiba chains and ladders

Motivated by recent proposals to realize Majorana bound states in chains and arrays of magnetic atoms deposited on top of a superconductor, we study the topological properties of various chain structures, ladders, and two-dimensional arrangements exhibiting magnetic helices. We show that magnetic domain walls where the chirality of a magnetic helix is inverted support two protected Majorana states giving rise to a tunneling conductance peak twice the height of a single Majorana state. The topological properties of coupled chains exhibit nontrivial behavior as a function of the number of chains beyond the even-odd dichotomy expected from the simple ${\mathbb{Z}}_2$ nature of coupled Majorana states. In addition, it is possible that a ladder of two or more coupled chains exhibit Majorana edge states even when decoupled chains are trivial. We formulate a general criterion for the number of Majorana edge states in multichain ladders and discuss some experimental consequences of our findings.

Figure 1

We study the properties of Shiba chains (corresponding to a single row) and ladders fabricated by depositing magnetic atoms on top of a superconductor by STM. The magnetic moments give rise to electronic impurity states that are coupled to each other.

Figure 2

(a) Helical magnetic order in the chain may give rise to topologically nontrivial phase and Majorana states localized at both ends. (b) Domain wall separating two regions of opposite chirality supports two Majorana states that are degenerate for a planar texture.

Figure 3

Topological phase diagram for a planar texture as a function of the pitch angle $\theta$ and Zeeman energy for $t_x=\Delta$, $\mu =4\Delta$. The energy gap vanishes near $\theta =0$ and $\theta =2\pi$; therefore the robust nontrivial states are located in the interior region away from the boundaries.

Figure 4

(a) Spectrum of a finite chain with 80 lattice sites with a domain wall in the middle corresponding to parameters $t_x=\Delta$, $\mu =4\Delta$. At critical Zeeman energy the system undergoes a topological phase transition and exhibits four zero-energy Majorana bound states. Panel (b) shows the domain-wall profile and indicates the locations of the four Majorana states corresponding to the zero-energy states in (a).

Figure 5

(a) Number of Majorana end states in a ladder of two coupled chains $N_z=2$ as a function of tilt angle and Zeeman field corresponding to parameters $t_x=\Delta$, $t_z=\Delta$, $t_d=0$, $\mu =4\Delta$, and $\theta =\pi /2$. The white area describes a trivial parameter region without Majorana end states. (b) Same as (a) but for three coupled chains.

Figure 6

(a) Colors indicate the number of MBS at each end (topological transverse channels) in a ladder of two coupled chains $N_z=2$ as a function vertical hopping. The plot parameters are $t_x=\Delta$, $t_d=0$, $\mu =4\Delta$, and $\theta =\pi /2$. (b) Same as (a) but for $N_z=3$.

Figure 7

Number of topological channels (MBS pairs) in the ladder as a function of transverse hopping amplitude in the case where decoupled chains are in the trivial phase. The plot parameters are $t_x=\Delta$, $t_d=0$, $\mu =5\Delta$, $B_0=2.1\Delta$, and $\theta =\pi /2$.