t−t′−J−U Model in Mean-Field Approximation: Coexistence of Superconductivity and Antiferromagnetism

We discuss the t−J−U model in the mean-field approximation. The role of spin-exchange coupling J and the second nearest hopping t ′ are examined in the context of the coexistence of superconductivity and antiferro-magnetism. Stability of the phases is studied with respect to temperature. The coexistence region exists for the sufficiently large Coulomb repulsion (U> Ucr), and in the vicinity of the half-filled band (hole doping δ < δcr). The critical hole doping is relatively small (δcr ≈ 0.006 for J/|t | = 1/3) and linear with respect to J. The decrease of Ucr is proportional to J, except the limit of small J (J/|t | < 0.03), where Ucr grows rapidly with decreasing J. The effect of the second nearest hopping is limited the phase diagram does not change in a qualitative manner when the t ′ value is changed. In the limit of T → 0, SC phase is stable even for large hole-doping (such as δ = 0.5). Additional paramagnetic phase appears for large δ or small U at non-zero temperature. When temperature in-creases, both SC and AF+SC phase regions are reduced