Antiferromagnetic resonance in a spin-gap magnet with strong single-ion anisotropy

Quasi-one-dimensional magnet NiCl$_2\,{\cdot}\,$4SC(NH$_2$)$_2$ denoted as DTN remains disordered in zero magnetic field down to $T = 0:$ the $S_z = 0$ ground state is separated from $S_z =\pm1$ excitations by a gap caused by strong single-ion easy-plane anisotropy acting on the Ni$^{2+}$ ions. When a magnetic field is applied along the principal axis of anisotropy, the gap closes in a field above $B_{c1} = 2.18$ T and the field-induced antiferromagnetic order arises. There are two excitation branches in this field-induced phase, one of which should be the Goldstone mode. Recent studies of the excitation spectrum in the field-induced ordered phase of the DTN magnet (T. Soldatov et al., Phys. Rev. B 101, 104410 (2020)) have revealed that the Goldstone mode acquires a gap in the excitation spectrum of the field-induced phase at a small deviation of the applied magnetic field from the tetragonal axis of the crystal. In this work, a simple description of both magnetic resonance branches in the ordered phase of a quasi-one-dimensional quantum $S = 1$ magnet with strong single-ion anisotropy is proposed. This approach is based on a combination of an effective strong coupling model for an anisotropic spin chain and the classical antiferromagnetic resonance theory. This description reproduces the experimental results semi-quantitatively without additional parameters.