Visualizing the connection between edge states and the mobility edge in adiabatic and nonadiabatic topological charge transport

The ability to pump quantized amounts of charge is one of the hallmarks of topological materials. An archetypical example is Laughlin's gauge argument for transporting an integer number of electrons between the edges of a quantum Hall cylinder upon insertion of a magnetic flux quantum. This is mathematically equivalent to the equally famous suggestion of Thouless that an integer number of electrons is pumped between two ends of a one-dimensional quantum wire upon sliding a charge-density wave over a single wavelength. We use the correspondence between these descriptions to visualize the detailed dynamics of the electron flow during a single pumping cycle, which is difficult to do directly in the quantum Hall setup because of the gauge freedom inherent in its description. We find a close correspondence between topological edge states and the mobility edges in charge-density wave, quantum Hall, and other topological systems. We illustrate this connection by describing an alternative, nonadiabatic mode of topological transport that displaces precisely the opposite amount of charge compared to the adiabatic pump. We discuss possible experimental realizations in the context of ultracold atoms and photonic waveguide experiments.