Non-local effects in the fermion Dynamical mean field framework. Application to the 2D Falicov-Kimball model

We propose a new, controlled approximation scheme that explicitly includes the effects of non-local correlations on the $D=\infty$ solution. In contrast to usual $D=\infty$, the selfenergy is selfconsistently coupled to two-particle correlation functions. The formalism is general, and is applied to the two-dimensional Falicov-Kimball model. Our approach possesses all the strengths of the large-D solution, and allows one to undertake a systematic study of the effects of inclusion of k-dependent effects on the $D=\infty$ picture. Results for the density of states $\rho(\omega)$, and the single particle spectral density for the 2D Falicov-Kimball model always yield positive definite $\rho(\omega)$, and the spectral function shows striking new features inaccessible in $D=\infty$. Our results are in good agreement with the exact results known on the 2D Falikov-Kimball model.