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Strong coupling Hartree-Fock approximation in the dynamical mean-field theory

In the limit of infinite spatial dimensions a thermodynamically consistent theory, which is valid for arbitrary value of the Coulombic interaction ($U<\infty$), is built for the Hubbard model when the total auxiliary single-site problem exactly splits into four subspaces with different ``vacuum states&#39;&#39;. Some analytical results are given for the Hartree-Fock approximation when the 4-pole structure for Green&#39;s function is obtained: two poles describe contribution from the Fermi liquid component, which is ferromagnetic and dominant for small electron and hole concentrations (``overdoped case&#39;&#39; of high-$T_c$&#39;s), whereas other two describe contribution from the non-Fermi liquid, which is antiferromagnetic and dominant close to half filling (``underdoped case&#39;&#39;).