Flux Phase as a Possible Ordered State in t-J Model

In high-Tc cuprate superconductors, violation of time-reversal symmetry (${\cal T}$) has been observed experimentally. In order to explain this phenomenon, we consider a flux phase in the t-J model. The flux phase has a free energy higher than that of a superconducting (SC) state, but it may occur near a [110] surface of a $d_{x^2-y^2}$-wave superconductor where the SC order is strongly suppressed. In this case the system would break ${\cal T}$ locally near the surface. In this short note we estimate the bare transition temperature of the flux phase, $T_{FL}$, assuming the absence of the SC state. It is found that $T_{FL}$ is finite for rather large doping rate $\delta$ ($\delta \lesssim 0.15$) if the shape of Fermi surface of the system is favorable to this state.