We study the phase diagram of the $t\ensuremath{-}J$ model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at
T=0 and rises quickly with decreasing doping. The transition temperature
Tc for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of
Tc in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below
Tc, whereas a pure flux phase exists above
Tc with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.
We study the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model of a doped Mott insulator in connection to high-
Tc superconductors. The nearest neighbor Coulomb interaction
(V) is treated quantum mechanically on equal footing as the antiferromagnetic exchange interaction
(J). Motivated by the
SU(2) symmetry at half-filling, we construct a large-
N theory which allows a systematic study of the interplay between staggered flux order and superconductivity upon doping. We solve the model in the large-
N limit and obtain the ground state properties and the phase diagram as a function of doping. We discuss the competition and the coexistence of the staggered flux and the
d-wave superconductivity in the underdoped regime and the disappearance of superconductivity in the overdoped regime.