Dynamical Mean Field Theory (DMFT) does not take into account spacial correlations? It is often said that the Dynamical Mean Field Theory (DMFT) does not take into account spacial correlations. What does this mean in layman terms?
Does that mean that we assume:
$$
\langle n_i n_j \rangle - \langle n_i \rangle \langle n_j \rangle = 0
$$
where $i \neq j$ and $n_i = a_i^\dagger a_i$? Or does it mean something else?
 A: DMFT assumes that the self-energy $\Sigma_{nn'}(\omega, k)$, which is the correction to the noninteracting Green's function $G^{-1} = G_0^{-1} - \Sigma$, is local, meaning $\Sigma_{nn'}(\omega, k) = \Sigma_{nn'}(\omega)$.
This is an exact result for the Hubbard model in the limit of infinite coordination number (nearest neighbors). Hence, DMFT is more useful for 3D (especially of high coordination number lattices), which perhaps is why it's one of the most common post-DFT methods out there.
DMFT solves the Hubbard model by decoupling the lattice into sites with strong correlation, each interacting with a mutual bath; the bath's parameters are determined self-consistently. There are a few derivations out there for DMFT. It can be thought of as a dynamical extension to Density Functional Theory, in the sense that your auxiliary potential is now time-dependent, allowing for a description of excited/equilibrium states. It's not quite a 1-1 analogy, but close enough if you don't plan on being a DMFT practitioner.
