Three-dimensional flux phases of the t-J model in a layer structure

We investigate flux phases of the three-dimensional (3D) t-J model for a layer structure. For some range of parameters, the system is stabilized by a uniform fictitious flux, which is commensurate to the electron number density. This state corresponds in 2D lattices to the commensurate flux phase, although in the 3D system we have the flux orientation as another variational parameter to lower the energy. Exchange of two carriers gives an extra phase factor to the wave function; this is not allowed in a continuous space but is in a lattice structure. The system exhibits 2D superconductivity confined in a plane perpendicular to the commensurate flux