  The XXZ model on a square lattice in the presence of a transverse magnetic
field is studied within the spin wave theory to investigate the resulting
canted antiferromagnet. The small and large field regimes are probed separately
both for easy-axis and easy-plane scenarios which reveal an unentangled
factorized ground state at an intermediate value of the field. Goldstone modes
are obtained for the field-free $XY$ antiferromagnet as well as for the
isotropic antiferromagnet with field up to its saturation value. Moreover, for
an easy-plane anisotropy, we find that there exists a non-zero field, where
magnon degeneracy appears as a result of restoration of an U(1) sublattice
symmetry and that, across that field, there occurs a magnon band crossing. For
completeness, we then obtain the system phase diagram for $S=1/2$ via large
scale quantum Monte Carlo simulations using the stochastic series expansion
technique. Our numerical method is based on a quantization of spin along the
direction of the applied magnetic field and does not suffer from a
sign-problem, unlike comparable algorithms based on a spin quantization along
the axis of anisotropy. With this formalism, we are also able to obtain powder
averages of the transverse and longitudinal magnetizations, which may be useful
for understanding experimental measurements on polycrystalline samples.
