Theory of the Kondo lattice:: competition between Kondo effect and magnetic order

We present here a review of some theoretical features of the Kondo lattice problem, which can describe anomalous experimental properties of strongly correlated electron systems with mainly cerium, ytterbium or uranium. In such Kondo systems, there exist both a Kondo effect on each site and strong intersite magnetic interactions. We present firstly the general case of the Kondo lattice and the mean-field approximation. We then discuss the case of the "underscreened" Kondo lattice with a strong coexistence between the Kondo effect and a ferromagnetic order, as for example observed in uranium compounds such as UTe. We also discuss the Kondo-spin glass competition with eventually an additional magnetically ordered phase (ferromagnetic or antiferromagnetic one) and we can account for phase diagrams of some cerium and uranium disordered alloys, such as CeNi1-xCux. Finally, the introduction of a transverse magnetic field in the Kondo-spin glass problem yields a quantum critical point and allows a better description of the phase diagrams of disordered alloys.