Superconductivity, antiferromagnetism and phase separation in the   two-dimensional Hubbard model: A dual-fermion approach
  The dual-fermion approach offers a way to perform diagrammatic expansion around the dynamical mean-field theory. Using this formalism, the influence of antiferromagnetic fluctuations on the self-energy is taken into account through ladder-type diagrams in the particle-hole channel. The resulting phase diagram for the (quasi-)two-dimensional Hubbard model exhibits antiferromagnetism and d-wave superconductivity. Furthermore, a uniform charge instability, i.e., phase separation, is obtained in the low doping regime around the Mott insulator. We also examine spin/charge density wave fluctuations including d-wave symmetry. The model exhibits a tendency towards an unconventional charge density-wave, but no divergence of the susceptibility is found. 
Junya Otsuki, Hartmut Hafermann, Alexander I. Lichtenstein
cond-mat.str-el cond-mat.supr-con