Flow of local quantum fluids: Conservation laws and vertex corrections from many-body linear-response theory with local self-energy

In non-diffusive conduction regimes of strongly correlated quantum electron systems, electromagnetic perturbations simultaneously probe the electronic dynamics in time and space: the exchanged energy ω excites retarded, i.e., frequency-dependent, many-body interactions, while the probing spatial modulation renders the response spatially nonlocal, i.e., dependent on the external wave vector q⃗. This work derives the nonlocal electrodynamic response of such dynamical quantum fluids assuming local but frequency-dependent self-energies and particle-hole irreducible two-particle vertex functions, and preserving charge/mass conservation. The latter is ensured by Bethe-Salpeter equations for renormalized interaction vertices, entering the Kubo formalism for two-particle correlation functions (e.g., for density, currents, momentum, stress). Within such a framework, exact symmetry criteria for the absence of vertex corrections are inferred. In particular, vertex corrections vanish at $q=0$ for single-particle dispersions that are even and bare interaction vertices that are odd with respect to specific momentum-space point group transformations, including inversion for vector vertices, and mirror reflections or two- or higher-fold rotations for tensor vertices. If the dispersion is isotropic, vertex corrections vanish from the transverse part (⃗) of vectorial vertices (such as for electric/particle current) at any finite ω and q⃗. These cancellations extend to the full current-current correlation function for quadratic isotropic dispersion. Further symmetry breaking, multiband effects, and the additional imposition of momentum conservation, are discussed, with application to the Hall viscosity of Landau levels. Explicit expressions for generic nonlocal correlation functions are derived for Fermi liquids and non-Fermi liquids.