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. 
E. Cappelluti and R. Zeyher