Lattice-Symmetry-Assisted Second-Order Topological Superconductors and Majorana Patterns

We propose a realization of the lattice-symmetry-assisted second-order topological superconductors with corner Majorana zero modes (MZM) based on two-dimensional topological insulators (2DTI). The lattice symmetry can naturally lead to the anisotropic coupling of edge states along different directions to the in-plane magnetic field and conventional $s$-wave pairings, thus leading to a single MZM located at the corners for various lattice patterns. In particular, we focus on the 2DTI with $D_{3d}$ lattice symmetry and found different types of gap opening for the edge states along the armchair and zigzag edges in a broad range of parameters. As a consequence, a single MZM exists at the corner between the zigzag and armchair edges, and is robust against weakly broken lattice symmetry. We propose to realize such corner MZMs in a variety of polygon patterns, such as triangles and quadrilaterals. We further show their potentials in building the Majorana network through constructing the Majorana $Y$ junction under an in-plane magnetic field.

Figure 1

(a) The Majorana isosceles triangle in six orientations. (b) The Majorana right triangles in six orientations. Each Majorana triangle is equivalent with a Majorana nanowire along the six orientations. (c) $Y$ junction made from three Majorana isosceles triangles.

Figure 2

(a) The electronic band structures along armchair (left panel) and zigzag (right panel) edges. The solid and dashed curves represent the edge state dispersions with and without in-plane magnetic field. (b) The relative energies of Dirac points $\mathrm{\Delta }{E}_{\mathrm{DP}}=E_{\mathrm{DP}}(\theta )-E_{\mathrm{DP}}(\pi /2)$ (the black dashed curve) and the edge state gap (the blue curve) as a function of the edge cut directions with the magnetic field along the $x$ direction (black arrow). (c) The gap at the zigzag (blue curves) and armchair edges (red curves) as a function of the in-plane magnetic field directions with different amplitudes. (d) The wave function plot with its eigenvalue closest to zero. The black arrow indicates the magnetic field direction.

Figure 3

(a) The gaps for the zigzag and armchair edges as a function of the superconducting gap ${\mathrm{\Delta }}_{\mathrm{sc}}$ with the in-plane magnetic field, $M/{\lambda }_{\mathrm{so}}=0.1$, along the $x$ direction. (b) and (c) The color plot of the gap at the armchair and zigzag edges, respectively, under the in-plane magnetic field and superconducting amplitudes. (d) The edge state gap as a function of the in-plane magnetic field direction with $M/{\lambda }_{\mathrm{so}}=0.1$ and ${\mathrm{\Delta }}_{\mathrm{sc}}/{\lambda }_{\mathrm{so}}=0.04$. For this plot, we choose the armchair and zigzag edges along $x$ and $y$ directions, respectively.

Figure 4

(a) and (b) The density plot of MZMs in Majorana isosceles and right triangles. (c) Majorana triangles with the edges have a 5° deviation from the armchair and zigzag edges, respectively. (d) $Y$ junction made from three Majorana isosceles triangles. The black arrow indicate the in-plane magnetic field direction. The insets plot the energies of the eigenstates.