Berry-Landau Fermi-liquid theory: transport in presence of quantum geometry

Landau Fermi-liquid theory characterizes interacting metals through quasiparticles and their residual interactions. It is a challenge to incorporate non-trivial quantum geometry -- as encoded by Berry phases and the quantum metric -- as a fundamental ingredient. We formulate a Berry-Landau Fermi-liquid theory for spinless fermions within an isolated band crossing the Fermi surface and derive the Landau functional through the Nozières--Luttinger construction. The instantaneous response to a perturbation of the Bloch waves generates both an anomalous Berry-connection potential in the action and an interaction-induced quantum-geometric contribution to the quasiparticle current. The conserved physical charge current is then obtained via the electromagnetic Ward identity as a combinati of this quantum-geometric current and the bare drift current. Therefore, the intrinsic anomalous Hall conductivity is fixed by the Berry-curvature integral of the dressed quasiparticle band while the Drude weight contains both conventional and quantum-geometric contributions. In the presence of Galilean symmetry, the Drude weight is protected against interaction renormalization. In the flat/narrow-band limit, transport is dominantly quantum-geometric and can be thermally enhanced. These results establish quasiparticle occupations, Landau interactions, and the quantum geometry carried by quasiparticles as the fundamental low-energy ingredients of a Berry-Landau Fermi liquid.