How bad can a metal get?

A striking feature of strongly correlated electron materials is the existence of bad metallic behaviour. The resistivity can monotonically increase with temperature to values which are much larger than the Mott-Ioffe-Regel limit, a value corresponding (in a simple Drude model) to a value where the mean-free path is smaller than a lattice constant.
Key questions are

• Does the resistivity ever "saturate" at high temperatures?
• If so, to what value is the maximum possible resistivity?
• Could this be related to the large spectral weight spread out over a broad spectral range in the optical conductivity? (see the next figure below).

There is a nice summary of the issues in section VIIH of this RMP by Basov and Timusk.

Gunnarsson and collaborators have come up with a simple argument to address these questions [see this nice RMP colloquium].

At high enough temperatures there is no Drude peak in the optical conductivity. However, the latter must still satisfy the f-sum rule, which relates the total "low energy" spectral weight to the average kinetic energy, E_K. The spectral weight may be spread over the non-interacting band width W.
These observations can be summarised in the figure and equation below which provides an estimate for the dc conductivity  sigma(0) on the scale e^2/(h d) where d is the interlayer spacing.

A recent PRB by Bergeron, Hankevych, Kyung, and Tremblay calculates the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. The curves below is for a doping p=0.2 , close to optimal,  and a temperature of T=0.2t.

Qualitatively, it seems consistent with the predictions of Gunnarsson et al.