Backflow in a Fermi Liquid

We calculate the backflow current around a fixed impurity in a Fermi liquid. The leading contribution at long distances is radial and proportional to 1/r². It is caused by the current induced density modulation first discussed by Landauer [IBM J. Res. Dev. 1, 223 (1957)]. The familiar 1/r³ dipolar backflow obtained in linear response is only the next-to-leading term, whose strength is calculated here to all orders in the scattering. In the charged case the condition of perfect screening gives rise to a novel sum rule for the phase shifts. Similar to the behavior in a classical viscous liquid, the friction force is due only to the leading contribution in the backflow while the dipolar term does not contribute.