Interplay between superconductivity and flux phase in the t-J model

We study the phase diagram of the t-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 Bur or bond-order state starts near optimal doping at T=0 and rises quickly with decreasing doping. The transition temperature T-c 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 T-c in the underdoped region. Treating for simplicity the Bur phase as commensurate the superconducting and flux phases coexist in the underdoped region below T-c, whereas a pure flux phase exists above T-c with a pseudogap of d-wave symmetry An 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.