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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 (SC) and antiferromagnetism (AF). Stability of the phases is studied with respect to temperature. The coexistence region exists for the sufficiently large Coulomb repulsion ($U>U_{cr}$), and in the vicinity of the half-filled band (hole doping $δ< δ_{cr}$). The critical hole doping is relatively small ($δ_{cr} \approx 0.006$ for $J/|t| = 1/3$) and linear with respect to $J$. The decrease of $U_{cr}$ is proportional to $J$, except the limit of small $J$ ($J/|t|< 0.03)$, where $U_{cr}$ 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 $T \rightarrow 0$, SC phase is stable even for large hole-doping (such as $δ= 0.5$). Additional paramagnetic (PM) phase appears for large $δ$ or small $U$ at non-zero temperature. When temperature increases, both SC and AF+SC phase regions are reduced.