A Topological Sum-Rule Causes the Total Suppression of the Hall Response

We present a topological sum-rule for the transverse polarization valid for out-of-equilibrium quantum states in two-dimensional lattices. Whenever the state equally spreads over all bands at a fixed energy, the topological sum-rule implies the identical suppression of the polarization, regardless of the magnetic field strength. As a remarkable consequence, the Hall response robustly vanishes even in absence of particle-hole symmetry and with arbitrary strong magnetic fields. We show that these out-of-equilibrium states are commonly realized in standard (Landauer) quantum transport settings and we rely on DMRG to show that our results equally apply to strongly interacting regimes.