Nonlinear equations describing one-dimensional non-Heisenberg ferromagnetic model are studied by the use of generalized coherent states in a real parameterization. Also, dissipative spin wave equation for dipole and quadruple branches is obtained if there is a small linear excitation from the ground state.

In the past decades, magnets with spin value

Indeed, only dipole branch effect has been used for describing magnets with spin value

Using the effects of both dipole and quadrupole branches results in a nonlinear approximation. The use of higher-order multipole effects yields more accurate approximations which demand more complicated equations. In this paper, only the effect of quadruple branch for Hamiltonians described by (

In classical physics term, the number of parameters required for a full macroscopic description of the magnet behavior is equal to

Here,

In order to calculate the effect of quadrupole excitation, first, the classical equivalent of Hamiltonian (

Obtaining coherent states for spin

Calculating the average values of spin operator.

Obtaining classical spin Hamiltonian equation using previously calculated values.

Computing Lagrangian equation by the use of Feynman path integral over coherent states and then computing classical equations of motion.

For finding nonlinear equations of magnet behavior, it is necessary to substitute resulted Hamiltonian in classical equations of motion. Solutions of these nonlinear equations result in soliton description of magnet that is not needed here.

Calculating ground states of magnet and then linearizing the nonlinear equations around the ground states for small excitation.

At the end, calculating spin wave equation and dispersion equation.

In what follows, the mathematical descriptions of the above steps are presented.

In quantum mechanics, coherent states are special kind of quantum states that their dynamics are very similar to their corresponding classical system. These states are obtained by act of Weil-Heisenberg group operator on vacuum state. Vacuum state of SU(3) group is

Coefficients

Two angles,

In order to obtain the classical equivalent of Hamiltonian (

The average spin values in SU(3) group are defined as [

The above classical Hamiltonian is substituted in equation of motion that was obtained from the Lagrangian, and the result is classical equations of motion:

In this paper, only the linearized form of (

In this paper, only dispersion of spin wave in neighborhood of the ground states is studied. For this purpose, small linear excitations from the ground states, as shown in (

In this situation, the linearized classical equations of motion are

Consider functions

Substitute of these equations in (

These equations are dispersion equations of spin wave near the ground states in SU(3) group.

In this paper, describing equations of one-dimensional anisotropic non-Heisenberg Hamiltonians are obtained using real-parameter coherent states. It was indicated that both dipole and quadruple excitations have different dispersion if there is small linear excitation from the ground state.

In addition, it was indicated, that for anisotropic ferromagnets, the magnitude of average quadruple moment is not constant and its dynamics consists of two parts. One part is rotational dynamics around the classical spin vector (

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