The 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 $\hbar \omega$ 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 $\vec{q}$. This work determines the exact nonlocal electrodynamic response of such dynamical quantum fluids under the assumptions of local, frequency-dependent interactions and 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 framework, it is shown that vertex corrections generally vanish at $q=0$ for single-particle dispersions with inversion symmetry and for bare interaction vertices that are odd with respect to specific point group transformations in momentum space, including inversion for vector vertices, and mirror reflections or two- or higher-fold rotations for tensor vertices. In addition, for quadratic dispersion vertex corrections identically vanish from the current-current correlation function, at any momentum $\vec{q}$ and frequency $\omega$. The robustness of these criteria against further symmetry breaking, multiband effects, and additionally imposing momentum conservation, is discussed, with application to the Hall viscosity of Landau levels. Explicit expressions for generic nonlocal correlation functions are derived for Fermi liquids (with well-defined quasiparticle peaks) and non-Fermi liquids (devoid of quasiparticles), for arbitrary local self-energies.