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 locating 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 lattice symmetry broken. 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.