The Yangian relations of Heisenberg spin chain model

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.


Yangian of closed XXZ spin chain of N particles
In this section, we consider three particles interaction of closed XXZ spin chain model, the research contents are the rules of Lie algebra and Yangian of this model, eigensystem rules also be studied. The interaction between the first, second and the last particles are considered in this part, the interaction law of the other three particles will be shown in the next part. According to the study contents, we will recall the Lie algebra and Yangian. As we know, symmetry is an important concept in physics, since 1985, V. G. Drinfeld established Yangian theory, numerous studies have found that many physical models are Yangian symmetry. There are close relations between Lie algebra and Yangian, any simple Lie algebra A could associate with Hopf algebra Y(A) which is a deformation of the universal enveloping algebra for the polynomial current Lie algebra. In this paper, we use the Yangian operator J α , (α = ±, 3) to describe the transition between different energy levels and Lie algebra operatorI α (α = ±, 3) represent transfers between states at the same level. The details are illustrated in Fig. 1.
Many models have Yangian symmetry, such as Heisenerg models, Hubbard model. In studying the spin chain model, we find that Yangian of a multi-particle system has a good rule if only three particles are considered to interact. We will use spin-1 2 XXZ model with periodic boundary condition as an example to explain the rules. As we know, the Hamiltonian of XXZ spin chain is H N = N i,j=1 h ij = J N i,j=1 Here we consider the diamagnetic case ( J = 1 ). When we consider the interaction of XXZ closed spin chain between 1, 2 and N particles (the other three particles interaction law will be presented in the next part), the Lie algebra and Yangian of the system will show a perfect rule and the Hamiltonian of N particles system is as follows, When N=3, the eigensystem conditions of the system are: . The corresponding eigenstates of three particles system are as follows,  Figure 1. The Yangian and Lie algebra relations of three adjacent particle system in XXZ spin model with periodic boundary condition. This figure shows the transitions between states of the first, second and the last particles' interaction of N-particle closed XXZ spin chain models. Lie algebra operators I α represent transitions between states in the same energy level. Yangian operators J α represent transitions between states in different energy states. www.nature.com/scientificreports/ As the number of particles studied grows, The eigenvalues rule of N-particle system are (2) |φ 1 � = | ↓↓↓�, |φ 2 � = | ↑↑↑�, 2 N−2 +1 = 2 N−2 +2 = · · · · · · = 2 N−1 = 1 4 (2 + �), 2 N−1 +1 = 2 N−1 +2 = · · · · · · = 3 * 2 N−2 = − 1 4 (1 + � + 9 + 4� 2 − 4�), 3 * 2 N−2 +1 = 3 * 2 N−2 +2 = · · · · · · = 2 N = − 1 4 (1 + � − 9 + 4� 2 − 4�).  www.nature.com/scientificreports/ The degeneracies of the four eigenvalues are same ( 2 N−2 ). The eigenvalues of N-particle system have good regularity, and the eigenstates law of N-particle closed XXZ spin chain system can be obtained from the eigenstates of three-particle system. According to the eigenstates of three particles system, we can get the eigenstates of four particles system firstly, the details are as follows, . During the study, we found that the eigensystem of the model of the first, the second and the last of four particles system can be obtained from the eigensystem of three particles system. From the above research content, we can discover the law of structure that the eigenstates of the four-particle system can be obtained by adding ↑, ↓ at the penultimate position of three-particle engenstates. Five-particle system's eigenstates are also obtained from four-particle system by adding ↑, ↓ at penultimate position. The same holds true when we extend the scope of the study to all multi-particle system. In the course of studying this system, we can know that the system's energy levels have a high degree of degeneracy ( 2 N−2 ) and clear regularity.
In this system, the research object is the first, second and the last particle interaction of N-particle. From the previous introduction, we can see that there are two important applications of Yangian: describing symmetry and constructing transition operators. The relations between Lie algebra operators and Yangian operators are as follows: According to the system's eigensystem and particles interaction laws of this system, the Lie algebra I α , (α = ±, 3) and Yangian J α , (α = ±, 3) transition relations among eigenstates have the following forms, (a| ↑↓↑↓� + | ↓↑↑↓� + | ↓↓↑↑�)  www.nature.com/scientificreports/ The above formulas are the concrete representations of Lie algebra and Yangian transition operators, where | � represents the eigenstate in the system of N particles, and n is the number of particles in the system. The Yangian operators express the transition between different states at different energy levels and Lie algebra operators represents the transfer between different states in the same energy level. We will illustrate specific transitions between states in Fig. 1. Figure 1 shows the energy levels of three-particle interaction (the first, second and the last particles) of N-particle closed XXZ spin chain model, it clearly shows the four energy levels of this system and corresponding energy level transitions. The Hamiltonian of this system is H It is found that the same Hamiltonian can be constructed by another way of XXZ model's Hamiltonian. As we know, the Hamiltonian of Heisenberg XXZ closed chain model is H Nxxz = n i,j=1 1 2 (s + i s − j + s − i s + j ) + �s 3 i s 3 j , we can reconstruct the same Hamiltonian in the following way, H Nxxz − H (N−1)xxz . Different particle number models correspond to different action spaces, so we add the identity matrix. This ensures that the action space is the same and does not destroy the structure of Hamiltonian. Take four particles system for example: In the example above, we can get the same Hamiltonian when we put the identity matrix in front of the three-particle Hamiltonian, but if we put the identity matrix behind the three-particle Hamiltonian, we will construct a new system with different laws. Since the identity matrix constructs different system at different locations and correspond to different laws, the research content of this part will be presented in the next section.

Scientific Reports
The eigenvalues of this system are identical to the eigenvalues of "Yangian of closed XXZ spin chain of N particles" section, Eq. (2). Through the above eigenstates Eq. (7), we can find a very similar place with previous formulas of Eqs. (3) and (4). They have the same coefficients and similar structures, but on closer inspection, the eigenstates are different. Similarly, we can generalize the multi-particle eigensystem from the first three-particle eigensystem (Eq. 7), when we get four particles eigenstates from three-particle eigenstates, we need to put operators ↑, ↓ in the second positive, same thing with five particles. By contrast, we can still derive the Hamiltonian for this system using the Heisenberg closed XXZ spin chain model: 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. Not only do we know how the first, the second and the last particles interact, but we also know how the first, the last and the penultimate particle interact. In a multi-particle system, the eigenstates of the interactions of the other three adjacent particles are illustrated by the following Fig. 2. Figure 2 shows the interaction models of any three adjacent particles in the N-body particles system. The heavy solid line indicates that the two particles' interaction terms are added ( h ij ), and the dotted line indicates that the two particles' interaction terms are subtracted ( −h ij ). In Fig. 2, we use double arrows to represent systems with Hamiltonian of H N = h 1,2 + h N,1 − h N,2 . This system consists of 1, 2 and N particles. By adding operators ↑, ↓ to the penultimate position of the eigenstates of the first three-particle model, the eigenstates ( H N = h 1,2 + h N,1 − h N,2 ) of the multi-particle system can be obtained. The dotted line arrows represent the system of 1, N and N − 1 particles ( H N = h N,1 + h N−1,N − h N−1,1 ), for N particle systems, we can obtain the corresponding eigenstate by adding operators ↑, ↓ in the second positive and we mark this case with a dotted line. In Fig. 2, there are two other generalizations, one is to add operators ↑, ↓ at the front of the eigenstates, consisted of the N, N − 1 and N − 2 particles, it is represented by solid line arrow, and the other is to add an operator at the last position of the eigenstates, represented by dotted arrows, the three adjacent particles studied are 1, 2, 3 ; 2, 3, 4; 3, 4, 5; ...... particles. All of them have the same eigenvalue case as Eq. (2), the eigenstates are different from each other, but the law is the same.
In "Introduction" section, we study the 1, 2 and N particles interaction terms model in closed Heisenberg XXZ spin chain model of N particle system, the rules of eigensystem of this model are studied. The degenerate case of the system is 2 N−2 , the corresponding Yangian are studied, it is specifically expressed as Eq. (6) and Fig. 1. In this section, on the basis of "Introduction" section, we examine in detail the eigensystems laws of any three adjacent particle interactions, the composition of the Hamiltonian and its relationship to the closed XXZ spin chain model are also be researched. According to the studied laws, we can get the eigensystem of the model composed of any three adjacent particle system in N particles closed XXZ spin chain model system. In the next section, we will study the corresponding laws of other spin chain models in the Heisenberg model and extending the scope of the study to high-spin systems.

Studies on the laws of XXX model, XXZ model with other conditions and other models
In the previous two parts, we studied the relevant laws of the closed XXZ spin chain model, in this part, we extended the research scope to the open and closed XXX spin chain model, XY spin chain and Ising model. We will also consider the next nearest neighbor particle interaction, the twist boundary conditions and spin-1 system laws of these models.
(a| ↑↑↑↓� + | ↑↑↓↑� + | ↓↑↑↑�) (a| ↑↓↑↓� + | ↑↓↓↑� + | ↓↓↑↑�) (c| ↑↓↑↓� + | ↑↓↓↑� + | ↓↓↑↑�) (d| ↑↑↓↓� + d| ↓↑↑↓� + | ↓↑↓↑�) |ψ 16  www.nature.com/scientificreports/ The periodic boundary conditions which required S N+1 ≡ S 1 (N is particle's number)are often considered in the study of closed Heisenberg spin chain studies, then the twist boundary condition was introduced. In 1990, Bill Sutherland and B. Sriram Shastry considered a one-dimensional quantum many-body system on a ring of length L with M particles 29 , each with a charge of q. When the ring is crossed by a magnetic flux of strength, according to the Aharonov-Bohm effect which is discussed by Beyers and Yang on a quantum many-body system 30 , the twist boundary condition Ψ N (x 1 , ..., x j + L, ...x N ) = e iΦ Ψ N (x 1 , ..., x j , ...x N ) is generated. Here we will also study the Heisenberg model with twist boundary conditions. When we consider the two ends of the chain are attached with a twist of angle ∅ around the z-axis, the boundary condition is s + n+1 = s + 1 e i∅ , s − n+1 = s − 1 e −i∅ , s z n+1 = s z 1 . Thus the Hamiltonian of XXZ model is . According to this boundary term, H xxzt has neither translational symmetry nor isotropy, the translational invariance can be restored if we twist all neighboring bonds in the chain by an angle φ = ∅ N . The Hamiltonian can be rewrite the forms 31 . In "Yangian of closed XXZ spin chain of N particles" section, we considered the periodic boundary, in this section, we will study N particles closed XXZ spin chain with twist boundary condition (TBC), the Hamiltonian have the following forms: , 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 2 N−2 , the Yangian has the same forms as Fig. 1.
The open XXZ spin chain system is also considered in this paper. It constructed Hamiltonian in the same way as the "Yangian of closed XXZ spin chain of N particles" section, the Hamiltonian of the model is www.nature.com/scientificreports/

Summary
In this paper, we consider the interaction model of any three adjacent particles of Heisenberg spin chain systems. Firstly, we consider the first, second and the last particles model of the closed XXZ spin chain, the system has four sets of eigenvalues and their degeneracies are 2 N−2 . 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. The Hamiltonian of the same three adjacent particle systems can be constructed by a closed XXZ spin chain model in the form of H N = H Nxxz − I ⊗ H (N−1)xxz , the Yangian rule of the model is shown in Fig. 1. Secondly, we study the eigensystem and Yangian laws of other similar three particles systems of the closed XXZ spin chain model. In a multi-particle system, the constitutive laws of the interaction model between any close three particles and the generalization laws of their eigenstates have also been found. The generalizations of the model are illustrated in detail in Fig. 2. Last, we also studied the open-chain, next-nearest neighbor and twist boundary conditions of XXZ spin chain model. Corresponding research results have also been obtained for the XXZ spin chain system in the above case. The XXX, XY, Ising model and the spin-1 chain system are also be discussed. According to the research, we found that the eigensystems of these systems are generalized to N particle systems in the same way as the XXZ model.