Taming the shape of the topological states

Topological systems with their peculiar robust edge state have drawn lots of attention in recent years. Edge states in these systems are exponentially localized at the interfaces between two structures with different topological characteristics. The energy eigenvalue associated with the edge state lies in the middle of the gap and is protected by symmetries associated with the structure. Consequently, it is not trivial to construct a lattice that has no obvious symmetry, and yet it has a mid-gap state that is robust against disorder. Furthermore, for practical application, it is crucial to have protected states that are not necessarily localized at the edge, or in some cases, it is mandatory to have it as an extended state. Here we propose a general approach that allows us to engineer the shape of the mid-gap state by employing a mathematical transformation in lattice unit cells. To show the power of our method we make examples in 1D and 2D lattices allowing the observation of a mid-gap state with flipped edge state to the opposite edge and transformed edge state to an extended state. Our work paves the way toward new applications of topological effects such as imaging and far-field topological sensing.