Antiferromagnet with two coupled antiferromagnetic sublattices in a magnetic field

We discuss the magnon spectrum of an antiferromagnet (AF) in a magnetic field $h$ consisting of two interpenetrating AF sublattices coupled by the exchange interaction at T=0. One-ion easy-plane anisotropy is also taken into account. We calculate using the 1/S expansion the gap in the spectrum which is a manifestation of the order-by-disorder effect in this system and the optical magnon mode splitting. Both of these phenomena originate from the inter-sublattice interaction. We calculate also the gap value at $h\approx h_c$ in the leading order of the small parameter $(h_c-h)/h_c$ using the magnon Bose-Einstein condensation treatment, where $h_c$ is the saturation field. By comparing results obtained within these two approaches we conclude that the 1/S expansion gives a qualitatively correct result at $h\sim h_c$ even at large one-ion anisotropy but it overestimates the gap value. The application is discussed of these results to the actively studied AF of the considered type $\rm NiCl_2$-$\rm 4SC(NH_2)_2$ (DTN).