4-pole analysis of the two-dimensional Hubbard model
  The electronic states of the two-dimensional Hubbard model are investigated by means of a 4-pole approximation within the Composite Operator Method. In addition to the conventional Hubbard operators, we consider other two operators, which come from the hierarchy of the equations of motion. These operators carry information regarding surrounding spin and charge configurations. By means of this operatorial basis, we can study the physics related to the energy scale J=4t^2/U. Results present, in addition to the main two bands, a quasi-particle peak at the Fermi level, shadow bands and band flatness at (pi, 0) point. 
Satoru Odashima, Adolfo Avella, Ferdinando Mancini