The two-dimensional Hubbard model on the square lattice is studied in the presence of lattice distortions in the adiabatic approximation. The self energy is computed within perturbation theory up to second order, which provides a way for studying the quasiparticle dispersion. We compute numerically the second order contribution to the self-energy using a standard Fast Fourier Transform Algorithm for finite system sizes. The stability of the lattice distortions is investigated and a schematic phase diagram is drawn. The Fermi surface is analyzed for densities near to half filling, the presence of lattice distortion change some spectral properties of the model and gives an anisotropic interacting FS. The spectral function is calculated along several lines in momentum space and the renormalized quasiparticle dispersion is obtained. The behavior of the density of states is shown for different values of the intrasite repulsion U in the different phases.
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