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We analyze the evolution from the weak coupling (BCS-like limit) to the strong coupling limit of tightly bound local pairs (LP's) in the 2D asymmetric attractive Hubbard model, in the presence of the Zeeman magnetic field ($h$). The broken symmetry Hartree approximation is used. We also apply the Kosterlitz-Thouless (KT) scenario to determine the phase coherence temperatures. We obtain that for the spin dependent hopping integrals ($t^{\uparrow}\neq t^{\downarrow}$) the homogeneous polarized superfluid (SC$_M$) phase in the ground state for the strong attraction and lower filling can be stabilized. We find a topological quantum phase transition (Lifshitz type) from the unpolarized superfluid phase (SC$_0$) to SC$_M$ and tricritical point in the ($h-\mu$) and spin polarization ($P$) vs. attraction ($U<0$) ground state phase diagrams. The finite temperatures phase diagrams for $t^{\uparrow}\neq t^{\downarrow}$ are constructed.

On the BCS-BEC Crossover in the 2D Asymmetric Attractive Hubbard Model