Mott-Hubbard phase transition, gapped electron liquid in insulator stat

Within the framework of a mean-field approach the Mott-Hubbard phase transition is considered in the Hubbard and Falicov-Kimball models for half-filled occupation. It is shown that a static Z_2-field forms an insulator state on the lattice with a double cell, its strength is determined by the Hubbard interaction. An uniform configuration of the Z_2-field corresponds to a gapless spin liquid state, the configuration, at which the lattice with a double cell is formed, corresponds to a gapped fermion liquid, fermions move in this field. Due to the presence of a static field in an insulator state, a formation of a gapped electron liquid is similar to the gapless Majorana spin liquid in Kitaev's model [1]. A gap in the spectrum is calculated depending on the magnitude of the Hubbard interaction for the chain, square and cubic lattices. The proposed approach allows us to describe the Mott-Hubbard phase transition and the insulator state within the same formalism for an arbitrary dimension for different models.