For almost forty years our comprehension of the metallic behavior in many materials has been founded on the Landau physical intuition about interacting Fermi systems. The idea of Landau, that interacting fermions could be regarded as free particles with renormalized parameters, is at the basis of the standard picture of the solids in terms of single independent electrons delocalized throughout the systems. The extraordinary success of this scenario is demonstrated by the impressive number of predictions and results, on which has root a large part of the actual technology.
The Fermi-liquid concept has been also extended to systems showing a strong electron-electron interaction, i.e. strongly correlated electrons systems. Under suitable conditions the low temperature metallic properties of these systems can be interpreted in terms of renormalized quasiparticles.

Nevertheless, recent experiments on some strongly correlated materials have shown remarkable deviations from the Fermi liquid predictions concerning different physical observables, such as specific heat C, resistivity ρ or susceptibility χ. The theoretical understanding of the breakdown of the Fermi liquid paradigm observed heavy fermion systems or in high Tc superconductors is one of the open challenges in the correlated electrons physics. Many ideas have been put forward to explain the observed non-Fermi liquid behavior, without finding an absolute consensus so far. Among them we can distinguish three main directions: overscreening in Kondo models, Kondo disorder and quantum criticality.

A common feature can be recognized among some of these ideas, namely the existence of a physical mechanism pushing to zero the coherence temperature below which the Fermi liquid forms. In one case (Kondo disorder) this mechanism is associated to the presence of a certain degree of disorder. However, a more widely accepted mechanism for the formation of a non-Fermi liquid state is the proximity to a quantum phase transition (QPT) or to a quantum critical point (QCP). Within this scenario the breakdown of the Fermi liquid properties occurs in the neighborhood of T = 0 transition between a magnetically ordered phase (e.g. antiferromagnetic) and a paramagnetic one. In this regime the strong coupling between the fluctuations of the order parameter and the electrons may prevent the formation of a Fermi liquid phase with long-lived quasiparticles.

Among the different approach to non-Fermi liquid problem based on quantum criticality we can mention the Hertz-Millis theory, where the paramagnons of the ordered phase “dress” the conduction electrons to produce the NFL behavior. Another approach is the local quantum critical theory, in which the competition between local Kondo screening and long wavelength magnetic fluctuations drives the system into a critical regime where non-Fermi liquid properties can arise. This approach is based on a suitable extension of the dynamical mean-field theory. However, local quantum criticality, even providing useful insights to the non-Fermi liquid problem, recquires some simplifying approximations to solve the mean-field equations, spoiling DMFT approach of many of its benefits. Dynamical Mean Field Theory is a powerful theoretical tool to investigate strongly correlated electrons systems. Among other things, this method has permitted to obtain a satisfactory description of the Mott metal-insulator transition in simplified models, such as Hubbard. Lately, the application to realistic calculations, thru an ab-initio plus
DMFT algorithm, has greatly increased our knowledge about real materials.

At the heart of the DMFT approach are the simplifications on the lattice quantum many-body problem arising in the limit of infinite dimension. These simplifications permit to map the lattice problem onto an effective single impurity problem, that has to be solved in a
self-consistent way. In this respect DMFT can be considered as the quantum generalization of the classic mean-field theory, introduced so far to deal with spin models.

In this thesis we shall show that a new approach to the heavy fermions physics can be based on the DMFT solution of one of the canonical model of this area, namely the periodic Anderson
model. In particular we demonstrate that, contrary to conventional expectations, a non-Fermi liquid state is readily obtained from this model within the DMFT framework. In agreement with the quantum criticality scenario, this novel NFL state is located in the neighborhood of a quantum phase transition, but unlike the standard quantum criticality scenario sketched before, the relevant quantum transition here is a Mott transition. Thus, the present study sheds a different light onto the NFL problem, showing that the coupling to long wavelength magnetic fluctuations (absent in DMFT) is not a prerequisite for the realization of a NFL scenario. Local temporal magnetic fluctuations alone can provide sufficient scattering to produce an incoherent metallic state. The presence of such large local magnetic fluctuations in our model has origin in
the competition between magnetic interactions, namely the super-exchange antiferromagnetic interaction between the correlated electrons and the ferromagnetic interaction indirectly driven
by the delocalization of the doped charges. Thus, we are able to obtain a DMFT description of the non-Fermi liquid phase in heavy
fermion systems which is based on the proximity to a Mott point, i.e. Mottness scenario. Our study shows that the PAM, solved within DMFT, may be considered as a “bare bones” or min-
imal approach able to capture the physical scenario for the formation of a NFL state and that is in qualitative agreement with some observed phenomenology in heavy fermion systems.