The Yangian relations of Heisenberg spin chain model

Guijiao Du,Chengcheng Zhou,Kang Xue

doi:10.1038/s41598-021-94050-6

Abstract

In this paper, we investigate the Yangian relations of Heisenberg spin chain systems. Firstly, we consider the closed XXZ spin chain model, through the Heisenberg spin XXZ model, we found the Hamiltonians for one kind system of three adjacent partial particles interaction systems. The model’s constitution rules of energy levels and energy states which expand from the few-particle system to multi-particle system have good regularity. In this system, we found Yangian’s law and illustrate it through graphs. Secondly, we further consider the closed XXZ spin chain’s generalization of other three neighboring particles interaction systems from few-particle system to multi-particle system. Finally, we also discussed the laws of the three adjacent particles system of some models, they are the XXZ model with twist boundary condition, the open XXZ spin chain model and the XXZ model containing the next neighbor. In addition, not only XXZ model, XXX model, XY model and Ising model, but the relevant laws of spin-1 systems of these models were also discussed, they have similar rules to the XXZ model. Through calculation and research, the eigensystems of these models all have good Yangian and constitution laws.

Highlights

Through the analysis of the Hamiltonian of the system we studied, it is found that the interaction of three adjacent particles in the multi-particle system has good laws
“Yangian of closed spin chain of N particles” section, we considered the periodic boundary, we will study N particles closed XXZ spin chain with twist boundary condition (TBC), the Hamiltonian have the following forms: HN − I ⊗ HN−1 = h1,2 + hN,1 − hN,[2] or HN − HN−1 ⊗ I = hN,1 + hN−1,N − hN−1,1, it is found that the eigenstates law of the interaction model of three neighboring particles in N particles system extended from threeparticle systems are the same as closed XXZ spin chain, except for the specific representation of the eigenstates are different. the degeneracies of the model are 2N−2, the Yangian has the same forms as Fig. 1
Through the eigensystem of these three particles, we can get the eigensystem of the system composed of the first, second and the last particle in the muli-particle system

Summary

OPEN The Yangian relations of Heisenberg spin chain model

We investigate the Yangian relations of Heisenberg spin chain systems. We consider the closed XXZ spin chain model, through the Heisenberg spin XXZ model, we found the Hamiltonians for one kind system of three adjacent partial particles interaction systems. The Yangian symmetry is the realization of integrability, and the extension of Lie algebra in Physics. This paper is organized as follows: in “Yangian of closed XXZ spin chain of N particles” section, we consider three particles interaction system in the closed XXZ spin chain model. The conditions of Yangian and Lie algebra of XXZ spin chain will be illustrated in Fig. 1 and corresponding formula representations will be given. The XXX spin chain, XY spin chain model and Ising model’s Yangian rules be studied in “Studies on the laws of XXX model, XXZ model with other conditions and other models” section.

Yangian of closed XXZ spin chain of N particles

Summary

Additional information