Extended dynamical mean-field theory and $\mathrm{GW}$ method

We develop the extended dynamical mean-field theory (E-DMFT) with a view towards realistic applications. (1) We introduce an intuitive derivation of the E-DMFT formalism. By identifying the Hartree contributions before the E-DMFT treatment, it allows us to handle systems in symmetry-breaking phases within a simple formalism. (2) We make an implementation of E-DMFT through a real Hubbard–Stratonovich transformation to decouple the nonlocal two-particle interactions. We apply it to a three-dimensional $U-V$ model, with U the on-site and V the nearest-neighbor interactions, and investigate the behavior of the various Green’s functions, especially the density susceptibility, as the density instability is approached. We obtain the phase diagram at a finite temperature. (3) We present a formalism incorporating E-DMFT with cellular DMFT. (4) We suggest an improvement of the E-DMFT approach by combining it with a generalized GW method. The method combines the local self-energy from E-DMFT and the nonlocal ones from the perturbative calculation of $\mathrm{GW}.\mathrm{}$ We apply the method to a one-dimensional $U-V$ model with two sublattices carrying different chemical potentials. By comparing with those from density matrix renormalization group calculations, we show that the results are shifted in the correct direction due to the $\mathrm{GW}$ contributions. (5) In order to handle the generic Coulomb repulsion within E-DMFT, we describe a method to tailor E-DMFT so that the proper momentum dependence can be kept in general response functions.