Navier-Stokes flows meet topological electronics*

Electrons in metals often flow ballistically like billiard balls colliding with impurities or phonons. However, in very clean samples and at intermediate temperatures, electrons are more likely to scatter with other electrons and can thus conserve their energy and momentum for long times. In this regime, electrons flow like water governed by the celebrated Navier-Stokes equation. There are numerous transport peculiarities observed in this regime such as negative non-local resistance, charge and heat transport decoupling and conductance higher than for ballistic transport. However, so far quantum effects seem rather absent in the collective dynamics.
Topological materials are excellent platforms for studying non-trivial hydrodynamic flows. These materials possess an internal structure encoded in their electronic wave functions (Berry curvature) which is able to alter the electron dynamics. Chief among them is the anomalous Hall motion where electrons move perpendicular to the electric field even without an external magnetic field. In this work, we derive novel Navier-Stokes equations governing the electron flow in the presence of Berry curvature and discuss possible transport consequences of hydrodynamics meeting topology.