  Filling-control metal-insulator transition on the two-dimensional Hubbard
model is investigated by using the correlator projection method, which takes
into account momentum dependence of the free energy beyond the dynamical
mean-field theory. The phase diagram of metals and Mott insulators is analyzed.
Lifshitz transitions occur simultaneously with metal-insulator transitions at
large Coulomb repulsion. On the other hand, they are separated each other for
lower Coulomb repulsion, where the phase sandwiched by the Lifshitz and
metal-insulator transitions appears to show violation of the Luttinger sum
rule. Through the metal-insulator transition, quasiparticles retain nonzero
renormalization factor and finite quasi-particle weight in the both sides of
the transition. This supports that the metal-insulator transition is caused not
by the vanishing renormalization factor but by the relative shift of the Fermi
level into the Mott gap away from the quasiparticle band, in sharp contrast
with the original dynamical mean-field theory. Charge compressibility diverges
at the critical end point of the first-order Lifshitz transition at finite
temperatures. The origin of the divergence is ascribed to singular momentum
dependence of the quasiparticle dispersion.
