The Yangian Symmetry Of Three-spin-1/2 And Four-spin-1/2 Heisenberg XXX Models

The Yang-Baxter Equation were constructed in Quantum Physics by Chen-Ning Franklin Yang in 1967,and R. J. Baxter also constructed similar theory in 1972. This equation is the solid basics of the quantum integrable models, which promotes the investigation in this field, and there are some outstanding results in this domain. The Yangian and Quantum Group Theory, established by V. G. Drinfel’d in 1985, offer a mathematical method for the investigation about the quantum integrable models in physics.The Yangian theory, established by Drinfel’d in 1985, offers a mathematical method for the investigation about the Yangian symmetry of quantum integrable models in physics. Some researchers have found that many physical models possess the Yangian symmetry, such as one-dimensional Hubbard chain, Haldane-Shastry model, and so on. We denote Yangian algebra associated with the Lie algebra A as Y(A)-For instance, the Yangian algebra associated with the Lie algebra sl(2) is Y(sl(2)).The model system we studied in this paper is the one-dimensional Heisenberg XXX model which is described by a nearest neighbor Hamiltonian in the form of where Si represents spin-1/2 operator located on the i th site and J is the exchange coupling constant. The reason for this choice, but its simplicity, also for Heisenberg XXX model has always provided insights into many-body problem.It is well known that the Heisenberg XXX model possesses Lie algebra sl(2) symmetry. However, Lie algebra sl(2) cannot describe the total symmetry of this model. In order to solve this problem, researchers constructed the Yangian realizations, with the Yangian Y(sl(2)) theory utilizing in this system, do not commute with the Heisenberg XXX model. This motivated us to investigate the symmetry for Heisenberg XXX model. To this end, this paper investigates Yangian symmetry for the three-spin-1/2 and four-spin-1/2 Heisenberg XXX models. Applying Yangian Y(sl(2)) theory to three-spin-1/2 and four-spin-1/2 systems, we construct Yangian Y(sl(2)) realizations for these systems. This paper constructs the Yangian realization with a novel method. And the Yangian realizations constructed in this paper commute with the corresponding Hamiltonian systems.Consequently, utilizing the Yangian operators, we also investigate the transitions between different eigenstates in the degenerated subspaces. Finally, this paper shows that these Yangian realizations can be used to describe three-spin-1/2 and four-spin-1/2 Heisenberg XXX models.