Direct manifestation of band topology in the winding number of the Wannier-Stark ladder

Topological quantum phases of matter have been a topic of intense interest in contemporary condensed matter physics. Extensive efforts are devoted to investigate various exotic properties of topological matters including topological insulators, topological superconductors, and topological semimetals. For topological insulators, the dissipationless transport via gapless helical edge or surface states is supposed to play a defining role, which unfortunately has proved difficult to realize in experiments due to inevitable backscattering induced in the sample boundary. Motivated by the fundamental connection between topological invariants and the Zak phase, here, we show that the non-trivial band topologies of both two and three-dimensional topological insulators, characterized by the Chern numbers and the $\mathbb{Z}_2$ invariants, respectively, are directly manifested in the winding numbers of the Wannier-Stark ladder (WSL) emerging under an electric field. We use the Floquet Green's function formalism to show that the winding number of the WSL is robust against interband interference as well as non-magnetic impurity scattering.