Temperature and filling dependence of the superconducting $pi$-phase in the Penson-Kolb-Hubbard model

We investigate in the Hartree approximation the temperature and filling dependence of the superconducting π phase for the Penson-Kolb-Hubbard model. Due to the presence of the pair-hopping term, the phase survives for repulsive values of the on-site Coulomb interaction, exhibiting an interesting filling and temperature dependence. The structure of the self-consistent equations peculiar to the π phase of the model allows us to explicitly solve them for the chemical potential. The phase diagrams are shown and discussed in dimension 2 and 3. We also show that, when a next-nearest-neighbors hopping term is included, the critical temperature of the superconducting region increases, and the corresponding range of filling values is shifted away from half-filling. Comparisons with known exact results are also discussed.