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 flux 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 optimaldoping 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 flux 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 pseudo-gap 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.