Nature of Mott Insulator in 2D Hubbard Model by Eliashberg Theory

We introduce an Eliashberg theory in the particle-hole channel for the Hubbard model over the square lattice that exploits the perfect nesting shown by the Fermi surface at half filling[1]. It results in Eliashberg equations for the wavefunction renormalization 1/Z and for the gap Δ_SDW in quasi-particle excitations at half filling. They are solved within the approximation that the corners of the diamond Fermi surface control the physics at half filling. We find that the wavefunction renormalization 1/Z vanishes as (t/U)² at strong on-site repulsion U compared to the hopping matrix element t. By comparison with mean field theory, the SDW gap is accordingly reduced down in size by the wavefunction renormalization to Δ_SDW ∼ t²/U. In other words, the SDW gap is of order the antiferromagnetic exchange coupling constant J. These results will be checked against direct numerical solutions of the former Eliashberg equations.
[1] J.P. Rodriguez and R. Melendrez, J. Phys. Commun. 2, 105011 (2018).