Bosonization with constraints: Application to the one-dimensional t-J model

The bosonization scheme for constrained fermions in one dimension (1D) is constructed. It is shown that the constrained bosonization is similar to the unconstrained one: the only difference is a phase shift simple. This bosonization scheme is applied to the 1D t-J model. For small J/t the model is equivalent to the Luttinger liquid for charge and spin. When J/t≥π sin(πδ)/cos(2πδ) (with doping δ) the holon liquid is unstable and one has only a spin liquid. This corresponds to phase separation