We have seen in the previous chapter how the dynamics of a two-dimensional quantum magnetic system on a square lattice can be described in the framework of the CP1/nonlinear sigma model. It is quite appealing, not only from the standpoint of basic principles but also from the point of view of modelling real materials, to investigate the behavior of electrons in the presence of this magnetic background. From this perspective, it would be interesting to study, among other issues: what would be the effective electronic interactions generated by the magnetic background, how these would depend on the phase transitions undergone by the underlying magnetic system, according to the values of different control parameters; what would be the role of skyrmion topological defects on the physical properties of the associated electrons; and how would electron or hole doping affect this interplay.

On the other hand, from the experimental point of view, several advanced materials have been obtained recently, the phase diagram of which present a very rich set of phases displaying different types of order. These are typically superconducting, magnetic or charge orders. Among these materials, we find heavy fermions such as CeCoIn5, high-Tc cuprates such as La2−x SrxCuO4 and iron pnictides, such as Sr1−x Kx Fe2As2. The richness of phases observed in such materials suggests there could be an underlying interaction responsible for the observed output, depending on the values of internal as well as external control parameters such as coupling constants and temperature, respectively. The spin-fermion model [174] describes this kind of system. Here we develop a quantum field theory approach the spinfermion system and investigate the possible effective interactions that are induced among the electrons by the (AF) magnetically ordered substrate.

Itinerant Electrons and Ordered Localized Spins

The Hamiltonian

We envisage a system containing both localized and itinerant electrons, the former belonging to atomic orbitals fixed to the sites of a square lattice. These generate localized magnetic dipole moments, which interact with nearest neighbors according to the SO(3), AF Heisenberg model. The itinerant electrons, conversely, have their kinematics determined by a tight-binding Hamiltonian, containing a hopping between nearest neighbors.