The Application Of Yangian To Solve The Geometric Phase In Comoposite Systems

The Quantum Yang-Baxter Equation is one successful theory which deals with nolinearquestion. The theory of Yangian is one important development of the theory of the QuantumYang-Baxter Equation. Yangian is most used to describe the symmetry of some physicalsystems and so on.The concept of geometric phase first introduced by Pancharatnam in his study ofinterference between light waves in distinct states of polarization and rediscovered by Berryfor quantal systems undergoing cyclic adiabatic evolution has been refined and applied duringthe past years. Aharenov and anandam removed the need of adiabatic external parameters andpointed out that the geometric phase could be considered the anholonomy associated with thecurvature of the projective Hilbert space. Samuel extended the geometric phase to noncyclicand nonunitary evolutions. Geometric phase is an important conception in quantummechanics.In this paper, we mainly study the impact of Yangian on geometric phase. Firstly we studythe geometric phase of the system with Hamiltonian H=1/2α(σ|→)₁ ·B| →( t ) + J ( (σ₁⁺)( σ₂⁺) + h·c ),andthrough the operator Q we give a general form of the Berry phase of the followinghamiltonian forms:The differences among them are studied. Although the form of the interaction aredifferent, we could obtain the same forms of Berry phase. Compared the forms of Berry phasewith one particle in rotary magnetic field, we make an conclusion that both of them have thesame forms .