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The aim of this Ph.D. thesis was to investigate superconducting properties inthe presence of Zeeman magnetic field in systems with local fermion pairing onthe lattice. The study also concerned the evolution from the weak coupling(BCS-like) limit to the strong coupling limit of tightly bound local pairs(BEC) with increasing attraction, both in the ground state and at finitetemperatures, within the spin-polarized extended Hubbard model. The analysiswas also extended to the case of spin dependent hopping integrals (massimbalance), with special attention paid to the BCS-BEC crossover physics in theground state. The methods used included: the mean field approximation(BCS-Stoner type) and the estimation of the phase coherence temperature withinthe Kosterlitz-Thouless (KT) scenario in two dimensions. The BCS-BEC crossoverwas also analyzed in three dimensions, at finite temperatures, within the spinpolarized Attractive Hubbard Model (AHM), going beyond the mean fieldapproximation. In this case, the critical temperatures of the superconductingtransition were determined within the self-consistent T-matrix method. Thestrong coupling expansion was applied to map the spin polarized AHM onto themodel of hard-core bosons on the lattice.

The influence of magnetic field on the superconducting properties and the BCS-BEC crossover in systems with local fermion pairing