A theoretical study of the effects of disorder on the Mn-Mn exchange interactions for Ga1-xMn xAs diluted magnetic semiconductors is presented. The disorder is intrinsically considered in the calculations, which are performed using an ab initio total energy density-functional approach, for a 128 atoms supercell, and by considering a variety of configurations with 2, 3 and 4 Mn atoms. Results are obtained for the effective J$^{Mn-Mn}_n$, from first (n = 1) all the way up to sixth (n = 6) neighbors via a Heisenberg Hamiltonian used to map the magnetic excitations from ab initio total energy calculations. One then obtains a clear dependence in the magnitudes of the J$^{Mn-Mn}_n$ with the Mn concentration x. Moreover, we show that, in the case of fixed Mn concentration, configurational disorder and/or clustering effects lead to large dispersions in the Mn-Mn exchange interactions. Also, calculations for the ground-state total energies for several configurations suggest that a proper consideration of disorder is needed when one is interested in treating temperature and annealing effects.

We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved ‘minus-sign problem', and two ways to stabilize the algorithm at low temperatures.