Semiclassical Modeling of Isotropic Non-Heisenberg Magnets for Spin and Linear Quadrupole Excitation Dynamics

Equations describing one-dimensional non-Heisenberg model are studied by use of generalized coherent states in real parameterization, and then dissipative spin wave equation for dipole and quadrupole branches is obtained if there is a small linear excitation from the ground state. Finally, it is shown that for such exchange-isotropy Hamiltonians, optical branch of spin wave is nondissipative. 1. Introduction Many condensed matter systems are fully described by use of effective continuum field models. Topologically nontrivial field configurations have an important role in modeling systems with reduced spatial dimensionality [1]. Magnetic systems are usually modeled with the help of the Heisenberg exchange interaction [2–4]. However, for spin , the general isotropic exchange goes beyond the purely Heisenberg interaction bilinear in spin operators and includes higher order terms of the type with up to [5]. Due to the spin states, the complex parameters are necessary to describe each of them, and this corresponds with the degrees of freedom. Two degrees of freedom are omitted, one because of normalization condition and the other for arbitrary phase decrease, hence 4S parameters are required to completely modeled the remainder degrees of freedom of spin states [6]. Particularly, in case with the isotropic nearest neighbor exchange on a lattice is derived by use of the Hamiltonian Here are the spin operators acting at a site , and are, respectively, the bilinear (Heisenberg) and biquadratic exchange integrals. The model (1) has been discussed recently in connection with bosonic gases in optical lattices [7] and in the context of the deconfined quantum criticality [8, 9]. Hamiltonian (1) is a special form presented in [10] and because of importance of quadrupole excitation in ferromagnetic Materials, it is considered here. This paper does not consider the antiferromagnetic and nematic states. Considering the effects of both dipole and quadrupole branches gives a nonlinear approximation. If higher order multipole effects are considered, the approximation is more accurate but at the same time, deriving the equations is too complicated. In this paper, only the effect of quadrupole branch for Hamiltonians described by (1) is considered. Study of isotropic and anisotropic spin Hamiltonian with non-Heisenberg terms is complicated due to quadrupole excitation dynamics [5, 11, 12]. Antiferromagnetic property of this excitation in states near the ground proves the existence of it, and Dzyaloshinskii calculated the effect of this excitation [13]. Also, numerical