  In this paper, we discuss the possible emergence of SYK non-Fermi liquid in
the incoherent critical metal phase generated by the Hubbard-type four-fermion
on-site random interactions which follow the Gaussian distribution. The SYK
behaviors here are similar to the $q=4$ SYK model except an additional factor
$N$. The phase transitions to the Fermi liquid regime and high temperature DMFT
insulting regime are possible through the turning of related parameters,
including the temperature, coherence scale, SYK coupling, and Coulomb boson
field. Taking the nonlocal Coulomb interaction (with pseudospin interaction) as
an example, we also discuss the existence of stable gapless bosonic excitations
in SYK compressible non-Fermi liquid phase away from the quantum critical point
(and thus prevent the condensation), which is due to the preserved SU(2) local
gauge symmetry in incoherent critical metal phase with strong SYK coupling and
at intermedia temperature. Note that here the compressibility plays the role of
effective coupling in the low-energy SYK bosonic action, and is a measurable
quantity in non-fermi liquid phase. We found that in this special case, the
global U(1) and U(1)$\otimes$U(1) symmetries are also preserved. Our results
are useful in detecting the effect of Hubbard parameter on the SYK behaviors in
non-Fermi liquid incoherent metal phase and how the on-site interacting model
enters the holographic picture at intermiedia and high temperature regimes, as
well as the relation between the phase transitions with the symmetry breaking.
