  We explore the magnetic properties of a two-dimensional Hubbard model on an
inhomogeneous square lattice, which provides a platform for tuning the
bandwidth of the flat band. In its limit, this inhomogeneous square lattice
turns into a Lieb lattice, and it exhibits abundant properties due to the flat
band structure at the Fermi level. By using the determinant quantum Monte Carlo
simulation, we calculate the spin susceptibility, double occupancy,
magnetization, spin structure factor, and effective pairing interaction of the
system. It is found that the antiferromagnetic correlation is suppressed by the
inhomogeneous strength and that the ferromagnetic correlation is enhanced. Both
the antiferromagnetic correlation and ferromagnetic correlation are enhanced as
the interaction increases. It is also found that the effective $d$-wave pairing
interaction is suppressed by the increasing inhomogeneity. In addition, we also
study the thermodynamic properties of the inhomogeneous square lattice, and the
calculation of specific heat provide good support for our point. Our intensive
numerical results provide a rich magnetic phase diagram over both the
inhomogeneity and interaction.
