Staggered flux state of electrons in a two-dimensionalt−Jmodel

The competition between the staggered flux state, or the d-density wave state, and the d-wave pairing state is analyzed in a two-dimensional $t-J$ model based on the U(1) slave boson mean-field theory. Not only staggered flux of spinons but also staggered flux of holons are considered. In this formalism, the hopping order parameter of physical electron is described by the product of hopping order parameters of spinons and holons. The staggered flux amplitude of electrons is the difference of staggered flux amplitude of spinons and that of holons. In the $\pi$-flux phase of spinons, staggered fluxes of spinons and holons cancel completely and the staggered flux order of electrons does not exist. However, in the staggered flux phase of spinons whose staggered flux amplitude is not $\pi ,$ fluxes does not cancel completely and the staggered flux amplitude of electron remains. Thus the phase transition between these two phases, the $\pi$-flux phase and the staggered flux phase of spinons, becomes a second-order transition in the physical electron picture. The order parameter which characterizes this transition is the staggered flux order parameter of electrons. A mean-field phase diagram is shown. It is proved analytically that there is no coexistence of staggered flux and d-wave pairing. The temperature dependence of Fermi surface and excitation gap at $(0,\pi )$ are shown. These behaviors are consistent with angle-resolved photoemission spectroscopy experiments.