Effective field theory of Berry Fermi liquid from the coadjoint orbit method

We construct an effective field theory for an interacting Fermi liquid with nonzero Berry curvature at zero temperature, called the Berry Fermi liquid. We start with the extended phase space formalism, incorporating physical time into the configuration space. This approach allows us to include the time dependence of the background gauge fields ``covariantly'' into the symplectic structure. Upon restricting to the physical hypersurface, the effective action that lives on the coadjoint orbit becomes the minus free energy on the extended phase space. We also derive the action perturbatively in external fields using the canonical variables. For applications, we compute both linear and nonlinear electrical responses using the Kubo formula and identify contributions from the electric and magnetic dipole moments, which stem from interactions breaking parity and time-reversal symmetry. The anomalous Hall effect is confirmed using the kinetic theory.