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Zeeman and Davydov splitting of Frenkel excitons in the antiferromagnet CuB$_2$O$_4$

The optical spectra of antiferromagnetic copper metaborate CuB$_2$O$_4$ are characterized by an exceptionally rich structure of narrow absorption lines due to electronic transitions within the magnetic Cu$^{2+}$ ions, but their unambiguous identification and behavior in magnetic field remain far from being fully understood. We studied the polarized magneto-absorption spectra of this tetragonal antiferromagnet with a high spectral resolution in the range of $1.4055-1.4065$ eV in magnetic fields up to 9.5 T and temperatures from 1.6 up to $T_N = 20$ K. We observed a set of eight absorption lines at $T=1.6$ K in magnetic fields exceeding 1.4 T which we identified as arising from Frenkel excitons related to the ground and the first excited state of Cu$^{2+}$ ions. The number of these excitons is defined by the presence of the four Cu$^{2+}$ ions with the doubly-degenerate spin state $S = 1/2$ at the 4$b$ positions in the crystallographic unit cell. The energies of these excitons are determined the exchange interaction of 0.5 meV of Cu$^{2+}$ ions in the excited state with surrounding ions and by the Davydov splitting of 0.12 meV. In large magnetic field the observed Zeeman splitting is controlled by the anisotropic $g$-factors of both the ground and excited states. We developed a theoretical model of Frenkel excitons in magnetic field that accounts for specific features of the spin structure and exchange interactions in CuB$_2$O$_4$. The model was used for fitting the experimental data and evaluation of Frenkel exciton parameters, such as the Davydov splitting, the molecular exchange energy, and the $g$-factors of the ground and excited states of the Cu$^{2+}$ ions.