Density-matrix renormalization-group study of the spin-½ $\mathrm{XXZ}$ antiferromagnet on the Bethe lattice

Ground-state properties of a spin-½ $\mathrm{XXZ}$ antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point $J_{\mathrm{xy}}=J_z$ and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in $\mathrm{XY}$ plain) for $J_{\mathrm{xy}}<J_z$ ($J_{\mathrm{xy}}>J_z$). We also compare our results with those for other systems, say, the $\mathrm{XYZ}$ model on the square lattice to discuss a possible type of symmetry breaking.