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 ${\mathrm{\delta }}_{\mathit{c}}$=a(J/t${)}^2$, is a flux-density-wave instability with an incommensurability wave number ∼${\mathrm{\delta }}^{3/4}$. The instability towards a modulated-flux state is due to low-lying phase fluctuations of the valence bonds. When the doping parameter exceeds ${\mathrm{\delta }}_{\mathit{c}}$, we find a fully gapped d-wave superconducting state due to residual quasiparticle interactions.