Conductivity of interacting spinless fermion systems via the high dimensional approach

Spinless fermions with repulsion are treated non-perturbatively by classifying the diagrams of the generating functional $\Phi$ in powers of the inverse lattice dimension $1/d$. The equations derived from the first two orders are evaluated on the one- and on the two-particle level. The order parameter of the AB-charge density wave (AB-CDW) occurring at larger interaction is calculated in $d=3$. The Bethe-Salpeter equation is evaluated for the conductivity $\sigma(\om)$ which is found to have two peaks within the energy gap $2\Delta$ in the AB-CDW: a remnant of the Drude peak and an excitonic resonance. Unexpectedly, $\sigma_{\rm\scriptscriptstyle DC}$ does not vanish for $T\to 0$