Relation between d-density wave of electron and staggered flux of spinon
  A $d_{x^2-y^2}$-density wave (ddw) order of electron in two-dimensional t-J model is analyzed in saddle point level using the U(1) slave boson formalism. We considered not only the staggered flux (s-flux) order of spinon but also the s-flux order of holon. This analysis provides the relation between the s-flux order of spinon and the ddw order of electron. We discovered a new phase in the phase diagram. In this phase, there is a s-flux order of spinon, but no ddw order of electron.Our results are that 1) a region of electron ddw exists, 2) there is no coexistence of ddw and $d_{x^2-y^2}$-wave pairing (singlet-RVB) in all region of phase diagram, and that 3) the ground state is a purely $d_{x^2-y^2}$ wave superconducting state. 
Koichi Hamada and Daijiro Yoshioka