Flux-density wave and superconducting instability of the staggered-flux phase

We study the stability of the staggered-flux phase in the t-J model away from half-filling using a systematic large-N slave-boson approach. Found below a critical doping concentration δc=a(J/t)², is a flux-density-wave instability with an incommensurability wave number {~}δ3/4. The instability towards a modulated-flux state is due to low-lying phase fluctuations of the valence bonds. When the doping parameter exceeds δc, we find a fully gapped d-wave superconducting state due to residual quasiparticle interactions.