Second-order spin-wave results for the quantum XXZ and XY models with anisotropy

A second-order spin-wave analysis is given for the quantum Heisenberg antiferromagnet with anisotropy on the one-dimensional linear chain, two-dimensional honeycomb lattice, and three-dimensional simple-cubic and body-centered-cubic lattices. A similar analysis is given for the quantum XY model with anisotropy on the one-dimensional linear chain, two-dimensional triangular and honeycomb lattices, and three-dimensional simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The singular behavior at the isotropic point is discussed in detail. We find that the quantum spin reduction of the isotropic Heisenberg antiferromagnet on a two-dimensional honeycomb lattice is much stronger than that on a square lattice.