  High-$T_c$ superconductivity (SC) was recently found in the bilayer material
La$_3$Ni$_2$O$_7$ (La327) under high pressures. We study the bilayer
two-orbital Hubbard model derived from the band structure of the La327. The
model is solved by cluster dynamical mean-field theory (CDMFT) with natural
orbitals renormalization group (NORG) as impurity solver at zero temperature,
considering only normal states. With hole doping, we have observed sequentially
the Mott insulator (Mott), pseudogap (PG), non-Fermi liquid (NFL), and Fermi
liquid (FL) phases, with quantum correlations decreasing. The ground state of
the La327 is in the NFL phase with Hund spin correlation, which transmits the
Ni-$3d_{z^2}$ ($z$) orbital inter-layer AFM correlation to the
Ni-$3d_{x^2-y^2}$ orbitals. When the $\sigma$-bonding state of the $z$ orbitals
($z+$) is no longer fully filled, the inter-layer antiferromagnetic (AFM)
correlations weaken rapidly. At low pressures, the fully filled $z+$ band
supports a strong inter-layer AFM correlations, potentially favoring
short-range spin density wave (SDW) and suppressing SC. Hole doping at low
pressures may achieve a similar effect to high pressures, under which the $z+$
band intersects with the Fermi level, and consequently the spin correlations
weaken remarkably, potentially suppressing the possible short-range SDW and
favoring SC.
