Study of gossamer superconductivity and antiferromagnetism in the extended $t$-$J$-$U$ model

We have analytically studied the d-wave superconductivity (dSC) and antiferromagnetism for a two dimensional extended $t$-$J$-$U$ model, by use of a renormalized mean field theory with the Gutzwiller approximation. The purpose of introducing the $U$ term is to partially impose the no double occupancy constraint. The phase diagrams as functions of doping $\delta$ and $U$ are studied. Using the standard value of $t/J=3.0$ and in the large $U$ limit, we show that the antiferromagnetic order emerges and coexists with the dSC in the underdoped region below the doping $\delta\sim0.1$. The dSC order parameter increases from zero as the doping increases and reaches a maximum near the optimal doping $\delta\sim0.15$. In the small $U$ limit, only the dSC order survives while the AF order disappears. The effect of the long-range hopping terms $t^{'}$ and $t^{''}$ on the dSC and antiferromagnetism is also discussed.