Heisenberg–Dirac–van Vleck vector model for a 1D antiferromagnetic chain of localized spins $S$ = 1

Magnetic properties of anisotropic crystals with localized spins $S$ = 1 are investigated; for these crystals, the Hamiltonian is derived in the Heisenberg-Dirac-van Vleck form, which includes biquadratic contributions apart from bilinear terms. The ground-state energy of the antiferromagnetic chain of spins $S$ = 1 is calculated in the model of nearest neighbors, and the interaction constant is renormalized using the renorm group method in the case of coarsening of the system. The temperature criterion for the formation of long-range order in the system is obtained. The excitations of this chain in the linear approximation have a dispersion relation differing from that for antiferromagnets with spin $S$ = 1/2 and are separated by an energy gap from the ground state. Allowance for nonlinear contribution leads to the formation of a solitary wave in the form of a dark-bright soliton.