Examine the phase diagram of two different mixed-dimensional $t-J_z-J_{\perp}$ models on a square lattice. Here the hopping amplitude $t$ is nonzero only along the $x$ direction. In the first boson model, the spin exchange amplitudes $J_{\perp}$ are negative and isotropic along the $x$ and $y$ directions of the lattice, and $J_z$ are isotropic and positive. Low-energy physics is characterized by spin-charge separation. Holes hop as free fermions in an easy-planar ferromagnetic background. In the second model, $J_{\perp}$ is restricted to the $x$ axis, but $J_z$ remains isotropic and positive. This model does not rely on particle statistics and exhibits stripe patterns of antiferromagnetic N{\’e}el ordering at low temperatures and high hole concentrations, similar to mixed-dimensional $t-J_z$ and $tJ$ models. indicate. At lower hole concentrations, we find very strong first-order transitions and hysteresis loops up to a very high hole doping of 14(1)%.
