In this dissertation, Spin-Fermion (SF) models for diluted magnetic semiconductors and high temperature superconducting cuprates are constructed and studied with unbiased numerical techniques.

A microscopic model to describe magnetically doped III-V semiconductors is proposed. This model includes the appropriate lattice geometry, as well as, magnetic, spin-orbit, and Coulomb interactions and contains no free parameters. Its study using state-of-the-art numerical techniques provides results in excellent agreement with experimental data for Mn doped GaAs. For the first time, Curie-Weiss behavior of the magnetization is obtained numerically and the values of the Curie temperature are reproduced in a wide range of Mn doping and compensations. We observed that for x ≥ 3%, the holes are doped into the valence band and uniformly distributed in the material. This could support the "valence band" scenario regarding this material.

Phononic degrees of freedom, which are often neglected in studies of high T_c cuprates, are considered in a numerical study of a spin-fermion model. Both diagonal and off-diagonal electron-phonon interactions are considered. While diagonal terms tend to stabilize ordered structures such as stripes, the off-diagonal terms introduce disorder making this structures more dynamical. Our results indicate that phonons play a role in the stabilization of stripe-like states.