The famous bulk-edge correspondence is one of the striking properties that has strongly motivated
enormous interest for topological insulators in the past decades. However, in a rather surprising way,
recent experiments imaging the non-equilibrium current pattern in disordered Chern insulators have shown that
most of the Hall current happens to flow inside the bulk of the spectrum, and not on narrow edge channels,
contrary to the most common expectation. I will present a recent attempt (in a collaboration with D. Kovrizhin and
R. Moessner) to account for these striking observations, which relies on two key ingredients: a smooth confining
electrostatic potential and quenched disorder. Our main result is the existence of wide meandering conduction channels,
that carry the quantized Hall current. Preliminary numerical results suggest, possibly for the first time, an intriguing
coexistence of extended energy eigenstates with sharply localized states, at all energies. This presentation will be in two parts.
I will begin to recall the key relevant concepts that we have used, such as the anomalous velocity in a Chern band, the spectral
flow of the Chern-band-projected Hamiltonian in the presence of an electric field, the role of incommensurability of the
electric field direction with respect to the underlying crystalline lattice, and of quenched disorder. After the break, the second
part will discuss experimental results, then our attempt to understand them, and I will end by sharing some questions on the
possible coexistence of extended and localized states at all energies.
