Vortex superlattice induced second-order topological mid-gap states in first-order topological insulators and superconductors

Topological defects such as vortex and dislocations, support zero-energy localized states as a reflection of the bulk topology, in first-order topological insulators and superconductors. Furthermore, emergent first-order topological mid-gap states have been discovered driven by the magnetic vortex superlattice. However, whether the higher-order topological mid-gap states would emerge from the first-order topological insulators and superconductors with the vortex superlattice remains elusive. In this work, we propose vortex superlattice could induce second-order topological mid-gap states with staggered lattice spacings for vortices in first-order topological insulators and superconductors. These higher-order topological mid-gap states originate from the staggered tunneling between vortex-induced bound states and the emergent π flux on vortex superlattices, as an intrinsic exhibition of the interplay between vortices and bulk topology for the first-order topological states. Our work uncovers higher-topological characteristics of topological-defect superlattice in first-order topological states, and develops a controllable environment for the creation and exploration of higher-order topological states.