We perform a systematic study of incoherent transport in the high temperature crossover region of the half-filled one-band Hubbard model. We demonstrate that the family of resistivity curves displays characteristic quantum critical scaling of the form ρ(T, δU ) = ρc(T )f (T /To(δU )), with To(δU ) ∼ |δU | zν , and ρc(T ) ∼ T . The corresponding β-function displays a "strong coupling" form β ∼ ln(ρc/ρ), reflecting the peculiar mirror symmetry of the scaling curves. This behavior, which is surprisingly similar to some experimental findings, indicates that Mott quantum criticality may be acting as the fundamental mechanism behind the unusual transport phenomena in many systems near the metal-insulator transition.

The experimentally established phase diagram of the half-filled Hubbard model features the existence of three distinct finite-temperature regimes, separated by extended crossover regions. A number of crossover lines can be defined to span those regions, which we explore in quantitative detail within the framework of dynamical mean-field theory. Most significantly, the high temperature crossover between the bad metal and Mott-insulator regimes displays a number of phenomena marking the gradual development of the Mott insulating state. We discuss the quantum critical scaling behavior found in this regime, and propose methods to facilitate its possible experimental observation. We also introduce the concept of quantum Widom lines and present a detailed discussion that highlights its physical meaning when used in the context of quantum phase transitions.

We develop a systematic typical medium dynamical cluster approximation that provides a proper description of the Anderson localization transition in three dimensions (3D). Our method successfully captures the localization phenomenon both in the low and large disorder regimes, and allows us to study the localization in different momenta cells, which renders the discovery that the Anderson localization transition occurs in a cell-selective fashion. As a function of cluster size, our method systematically recovers the reentrance behavior of the mobility edge and obtains the correct critical disorder strength for Anderson localization in 3D.