Renormalized mean-field theory of flux phases

A renormalized Hamiltonian approach is applied to construct commensurate flux phases for the t-J model. The assumption of nonuniform phases and modulus for the bond hopping amplitudes is necessary and (sufficient) condition of self-consistency. The supercell of the modulation is closely related to the filling. The self-consistent coupled equations are solved numerically on finite square clusters of sizes as large as 20x20. The differences in kinetic energy and in spin-spin correlation with respect to the t=0 uniform commensurate flux state show ∼-(t/J)1/2 and ∼t/J behaviors, respectively. When t/J≥1, time-reversal symmetry is only ''weakly broken'' and the Fermi level lies in a pseudogap with a depressed density of state