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The singlet-triplet magnetism and induced spin fluctuations in the high-$T_c$ copper oxides

High-$T_c$ cuprates like $La_{2-x}Sr_{x}CuO_{4}$ and $YBa_{2-x}Cu_{3}O_{6+x}$ are considered as a system of the electron and hole polar pseudo-Jahn-Teller $CuO_{4}$ centers $[{CuO}_{4}^{7-}]_{JT}$ and $[CuO_{4}^{5-}]_{JT}$, respectively, or a system of the local bosons moving in a lattice of the hole centers. Ground manifold of the polar centers includes three terms $^{1}A_{1g}$ (Zhang-Rice singlet), $^{1}E_{u}$, $^{3}E_{u}$ with different spin multiplicity, orbital degeneracy and parity that provides an unconventional multi-mode behaviour of the cuprates. The spin subsystem of the copper oxides within the polar Jahn-Teller $CuO_{4}$ centers model is a two-component spin liquid and corresponds to a singlet-triplet magnet with possible noncollinear spin configurations. In the framework of a modified mean field approximation some kinds of spin ordering are discussed including as a trivial singlet or triplet states as a pure quantum singlet-triplet mixed state. A local boson movement is accompanied by a modulation of the spin density on the site resulting in the so called induced spin fluctuations. Some unconventional features of the induced spin fluctuations are considered including an appearance of the induced longitudinal ferrimagnetism with an appropriate contribution to the spin susceptibility, a possibility to observe and examine the charge fluctuations with the help of the traditional magnetic methods such as the magnetic inelastic neutron scattering and the spin lattice relaxation experiments. The suggested model, in comparison with the nearly antiferromagnetic Fermi-liquid model, represents new approach to the description of the spin system of the high-$T_c$ cuprates.