Flux phase and its competition with superconductivity in the t-J model

The phase diagram of the t-J model is investigated within the X-operator formalism using the Baym-Kadanoff theory and a 1/N expansion. In this way, no auxiliary fields are introduced. For finite Coulomb repulsion, the system shows a strong competition between d-wave superconductivity and d-wave flux phase, leading to a strong suppression of superconductivity in the flux state. The underdoped region is characterized by flux order in the normal state, with a d-wave gap in the excitation spectrum, and by coexistence of both superconductivity and flux phases below the superconducting transition temperature T-c. The balance between the two phases is determined by the short-range Coulomb repulsion, while the long-range part of Coulomb interaction prevents phase separation and leads to incommensurate charge-density-wave state far away from the superconducting region.