Yangian In The Berry, The Phase Applications

C.N.Yang[1] and R.J.Baxter[2] separately established Quantum Yang-Baxter Equation(For short, QYBE)[3][4][5] in 1960s. Since then the investigations on quantum integrable models [6][7][8][9] have been greatly promoted. Worthy of mention especially is that the Yangian and quantum algebra was established by V.G.Drinfeld[10][11][12]in 1985 that offered a cogent mathematic method for the studies about the symmetry of quantum integrable models in physics. After several decades, people have made great progress both in different kinds of realizations of Yangian and in the investigations on quantum integrable models, giving new physical understanding and theoretical results. For example, the Yangian symmetry and RTT integrability of hydrogen atom[13] had been reported at length[14][15].For the simplest interactive system of two particles with spin 1/2,the operator of Lie algebra can only realize the transition among the triplets, however, in order to realize the transition between the triplets and the singlet, the operators of Yangian must be involved, that is ,Yangian goes beyond Lie algebra in Quantum Mechanics. In this paper, we mainly apply Yangian to the Berry's Phase[16] which is significative in physics, and find the Berry's phase of the two spin coupling system and the transiting operators between its topological states, so we realize the transition between the states with different weights.In this paper, we take the example of the system whose Hamiltonian concludes the generators of Yangian. On one side, such spin system provides us with the simplest model to investigate the properties of quantum system, especially when they are put into the time-dependent magnetic field and electronic field. On the other, such models have a wide range of applications to spin systems, condense matter physics and high energy physics. Through the calculation of the questions above, we may have a further understanding of the models in physics, meanwhile, we can better master the realization of Yangian algebra in physics. Yangian can connect the states with different weight factors,it is the extension of the operators in quantum mechanics.