Magnetic Solitons for Non Heisenberg Anisotropic Hamiltonians in Linear Quadrupole Excitations

We discuss system with non-isotropic non-Heisenberg Hamiltonian with nearest neighbor exchange within a mean field approximation process. We drive equations describing non-Heisenberg non-isotropic model using coherent states in real parameters and then obtain dispersion equations of spin wave of dipole and quadrupole branches for a small linear excitation from the ground state. In final, soliton solution for quadrupole branches for these linear equations obtained.


I. INTRODUCTION
During the past decade study of nonlinear behavior of magnetic crystals has been attracted large attention, specially it accompany with the progress in some other fields such as development of theory of nonlinear differential equation, achieving new laboratory results and also potential applications in other branches of science and technology [1,2].
Particles with spin S ≥ 1 are more interesting among the other nano particles [3,4]. This is because of existing of complexity in their behavior due to their multipole dynamic spin excitations. In such systems, the number of necessary parameters for complete description of macroscopic properties increases up to 4S, that S stands for magnitude of system spin.
Also it worthwhile, the process of achieving classical spin equations and dynamic multipoles is based on coherent states that are obtained in SU (2S + 1) group [5].
We consider unitary anisotropic Hamiltonian as form of: Which,Ŝ x i ,Ŝ y i ,Ŝ z i are spin operators in lattice i, and δ is anisotropy coefficient. This is Hamiltonian of one dimensional ferromagnetic spin chain observed in compositions like CSN iF 3 [6].
In this paper the goal is to obtain classical equation for stated Hamiltonian and finding the answer of spin wave for small linear excitations upper than the ground state. Coherent states issued nearest approximation to classical state i.e. pseudo classical, because they minimize

II. COHERENT STATES IN SU(3) GROUP
Coherent states are special quantum states that their dynamic is very similar to behavior of their classical system. The kind of coherent state that is used in a problem depends on symmetry of existent operators. With considering existent symmetry in operators of Hamiltonian (1), coherent states in SU(3) group is used for accurate description and considering all multipole excitations. In this group, ground state considered as (1, 0, 0) T and its single-site coherent state is written as: In above equation, Where x t = ∂ ∂t and H is classical energy of system obtained by averaging Hamiltonian (1) on coherent states (2).
Two other terms appear when acquiring Lagrangian of spin system. The first is Kinetic term that has Berry phase characteristics issued from quantum interference of Instanton paths and has important role in quantum phenomenons such as spin tunneling and the second is boundary term that depends on boundary values of path. Both of term have no role in classical dynamic of spin excitations and so are not considered here.
The first equation is third order differential equation. So change of dipole moment in Hamiltonian (1) is not of the form of soliton. Solution of this equation has the following forms: If rewrite nonlinear Klein-Gordon equation (9) as: Where Numerical solution of (11) is plotted in figure (1). In this computation we consider α = 10 5 and β = 10 10 . Analytical solution of above nonlinear Klein-Gordon equation is the following form: Where C is constant.

IV. CONCLUSION
In this paper, we study semi-classic theory for spin systems with spin S = 1 that contain anisotropic exchange terms. it is shown that for anisotropic ferromagnet, value of average quadruple torque is not constant (g t = 0) and its dynamic contains rotational term around classical spin vector (γ t = 0) and another dynamics that relates to change of length of quadruple torque. There are no such excitations in regular magnets and their dynamics is achieved by use of average value of Heisenberg spin Hamiltonian. Also it is shown that soliton solutions are of the kind of non topologic Hilomorphic solitons for quadruple excitations.