Antiferromagnetic nonreciprocity of light emission in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>CuB</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">O</mml:mi> <mml:mn>4</mm

Nonreciprocity of light emission, when the radiation intensity differs for two opposite propagation directions, is a rare phenomenon in solids because it requires a violation of the crystal symmetry with respect to time-reversal. Such a violation via time-reversal symmetry breaking can occur either due to an applied magnetic field or due to a magnetic ordering. We perform a detailed theoretical and experimental study of the photoluminescence (PL) nonreciprocity in the noncentrosymmetric tetragonal antiferromagnet ${\mathrm{CuB}}_{2}{\mathrm{O}}_{4}$, where this effect reaches 80% below the N\'eel phase transition temperature of ${T}_{N}=20\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. The effect is observed for three sets of extremely narrow exciton and exciton-magnon PL lines, associated with Frenkel excitons on the ${\mathrm{Cu}}^{2+}$ ions in the magnetic $4b$ subsystem. A strong manifestation of the nonreciprocity of emission is found in certain geometries for the commensurate antiferromagnetic phase, as well as in other phases with incommensurate spin ordering. In accordance with the magnetic symmetry of ${\mathrm{CuB}}_{2}{\mathrm{O}}_{4}$, the nonreciprocity of emission is observed for light propagation along certain directions within the easy (001) plane. A rigorous quantum-mechanical analysis of the wave functions of the initial and final states of the ${\mathrm{Cu}}^{2+}$ ions responsible for the PL is performed for various experimental geometries of the crystallographic axes and the applied magnetic field. The analysis confirms that the nonreciprocity of emission from Frenkel excitons in ${\mathrm{CuB}}_{2}{\mathrm{O}}_{4}$ is due to the interference of magnetic-dipole and electric-dipole transitions of antiferromagnetically ordered $4b$ spins of the ${\mathrm{Cu}}^{2+}$ ions, in good agreement with the experimental data.