Nonlinear Excitation In Two Ferromagnetic Chains With Anti-ferromagnetic Interaction And Its Square Root Algebra

The system discussed in this paper is composed by two one-dimensional chains. It is ferromagnetic interaction between internal nearest atoms of each chain,but the interaction of nearest atoms between two chains is anti-ferromagnetic. In this paper, we research the phenomenon about the ferromagnetic and the anti-ferromagnetic in condensed matter physics. People have made a lot of achievements about the ferromagnetic and anti-ferromagnetic chains. Such as magnetic spin waves and solitary waves in one-dimensional ferromagnetic and anti-ferromagnetic chains.Employing the double-sublattice mode and the coherent state ansatz, the properties of nonlinear excitation in one-dimensional two ferromagnetic chains with anti-ferromagnetic interaction was studied by Xu Chang-tan and Liu Fu-yi. The result showed that this system can excite non-topological soliton or non-topological reversed soliton or topological soliton or topological reversed soliton. Considering the effects of external potential, we got a mode which contains external potential.People have exploited two-mode model to investigate the influence of quantum fluctuation on the self-trapping of the Bose-Einstein Condensates,and got a lot of meaningful results. There is no reports that somebody have investigated the ferromagnetic and anti-ferromagnetic phenomenon with two-mode forms. We use the method of nonlinear quantum mechanics to discuss the excitation in one-dimensional two ferromagnetic chains with anti-ferromagnetic interaction.Because of the nonlinear item of model is relatively complicated, It is very difficult to get its double energy level which is not seen before this report. from the schrodinger differential equation, using two modulus approximations, we deduced two modulus matrix form of nonlinear excitation in one-dimensional two ferromagnetic chains with anti-ferromagnetic interaction and its corresponding Hamiltonian, Which is the foundation of discussing the tunneling, geometric phase, and problems as such. We obtain the matrix form of the two-level nonlinear excitation. We get the Hamilton of the system and its second quantized form, and further obtain the matrix form of pure quantum Hamilton in Fock states. Under the action of Fock states, we found that pure quantum Hamiltonian can no longer be described by Lie algebra because of the effects of the complex conjugate.Lie group and lie algebra which is closely related with the symmetry plays an important role in many physical areas. In quantum mechanics, the most common lie algebra is su(2) algebra and so(3) algebra which is used to study the angular momentum of the atomic theory. In recent years, people'more interest was caused by general deformation of lie algebra, it takes the quantum group as a special example, more than such a nonlinear algebraic structure which can be seen as"nonlinear expansion of lie algebra", many results have been reached. Square-root algebra is such an promotion of this nonlinear algebra.As this system contains a non-hermite Gaussian item and complex conjugate items, so we describe the model of the system by square root algebra. There was no report that nonlinear square-root algebra was used to describe a one dimensional anti-ferromagnetic interaction between two ferromagnetic chains. We described the model of system by square root algebra. This paper realized the Hamiltonian and angular momentum of the system by bosons. Both the Hamiltonian and angular momentum of the system described by square contains the generated units of square root algebra. We obtain a model of angular momentum which is changing over time. These equations are no longer closed.