Phase Separation in Mixtures of Hard Core Bosons

A theoretical investigation of boson versions of the $t-J$ and $t-J_z$ models on the square lattice is carried out. In the $t-J_z$ model, phase separation between a hole-rich and a hole-free phase occurs, at sufficiently low hole doping, for arbitrarily small values of $J_z$. The boson $t-J$ model, instead, features a uniform ground state at any doping for $J/t\le 1.5$. No evidence of a striped ground state is found. Relevance of this study to the corresponding fermion models is discussed. Fermi statistics is found to enhance the tendency toward phase separation; in particular, phase separation is predicted, at low doping, in the fermion $t-J_z$ model, at all values of $J_z$.