  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.
