A system of self-consistent integral equations for the superconducting gap is formulated and solved taking account of the instability of the normal phase of bilayer cuprates against charge-density waves. The critical parameters are calculated as a function of the wave vector, temperature, and doping index. It is found that the region in which superconductivity coexists with d-type charge-density waves depends strongly on the doping index. The effective energy-gap parameter, determined as the interval between the peaks of the density of states, can have a local minimum at temperatures T<T _c .

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

The microscopic mechanism of the interaction of exchange-coupled Fe and V spins in a $\text{Fe}{\text{V}}_2{\text{O}}_4$ ferrimagnet with the electric field has been described. The derived effective energy operator makes it possible to calculate the electric polarization vector using the information about the magnetic structure. The calculated electric polarization is consistent with the available experimental data.