High order band topology and fractional charges of vortex bound states

In a one-dimensional system described by the SSH model, a half-integer electric charge can be localized at the domain wall between topological and trivial systems. A similar phenomenon can also occur in certain two dimensional systems in which fractional electric charge can be localized at a vortex core. We show that the origin of the zero-mode bound to a vortex comes from the higher order topological physics. Considering several concrete examples, we show that vortex bound states can generally be understood as a spatial charge pump between higher order topological phase and trivial phase. In particular, when PT symmetry exists, we show that the pumping of the second Stiefel-Whitney invariant w_2 corresponds to the number of zero modes localized at the vortex core.