© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Scientific Reports

The Yangian relations of Heisenberg spin chain model

Guijiao Du, Kang Xue, Chengcheng Zhou

Published Jul 16, 2021• 1 citation

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.

DOI: 10.1038/s41598-021-94050-6

Related papers

Remarks towards the spectrum of the Heisenberg spin chain type models

Č. Burdík, Jan Fuksa, A. P. Isaev +2 more

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two-component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model (“small polaron model”) is considered. Finally, we discuss some questions relating to the general open Hecke chain models.

arXiv:1412.3999•Physics of Particles and Nuclei•May 1, 2015•2 citations

Study of the ground state of the one-dimensional Heisenberg spin-1/2 chain

K.R. de Ruiter

We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.

•Jan 1, 2018•0 citations

Integrable Multi-impurity Open XXZ Chain with Next-Nearest-Neighbor Interaction

Lai Yun-zhong, Ai-zhen Zhang, Zhan-Ning Hu +2 more

The Hamiltonian of one-dimensional integrable multi-impurity XXZ spin chain with open boundary cases is constructed by using the method of adding parameters to the vertical matrices. This is the first approach to add multi-impurity and next-nearest-neighbor interaction to the integrable anisotropic Heisenberg spin chain. It is found that all of the impurity parameters in the bulk must be pure imaginary numbers if the cross parameter is real for getting the hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interactions are introduced. The Hamiltonian can be also applied to describe the many body systems with two kinds of particles if all of the impurity parameters are equal.

Chinese Physics Letters•Jun 1, 1999•0 citations

The Yangian symmetry for three-spin-1/2 and four-spin-1/2 Heisenberg XXX models

Ying Liu, Kang Xue, Gangcheng Wang +2 more

In this paper, we investigate 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. We show that these Yangian realizations can be used to describe three-spin-1/2 and four-spin-1/2 Heisenberg XXX models. The Yangian operators as shift operators are also studied in this paper.

International Journal of Quantum Information•Feb 1, 2014•0 citations

Integrable quantum spin chains

Anastasia Doikou

Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...

•Jan 1, 1999•0 citations

Phase Diagram ofS=1 XXZ Chain with Next-Nearest-Neighbor Interaction

Takahiro Murashima, Keigo Hijii, K. Nomura +1 more

The one dimensional S=1 XXZ model with next-nearest-neighbor interaction $α$ and Ising-type anisotropy $Δ$ is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Néel phase, and then we confirm the universality class. Moreover, we map this model onto the non-linear $σ$ model and analyze the phase diagram in the $α$ $\ll$ -1 and $Δ$ $\sim$ 1 region by using the renormalization group method.

arXiv:cond-mat/0405385•Journal of the Physical Society of Japan•May 15, 2005•9 citations

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.

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two-component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model (“small polaron model”) is considered. Finally, we discuss some questions relating to the general open Hecke chain models.

We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.

The Hamiltonian of one-dimensional integrable multi-impurity XXZ spin chain with open boundary cases is constructed by using the method of adding parameters to the vertical matrices. This is the first approach to add multi-impurity and next-nearest-neighbor interaction to the integrable anisotropic Heisenberg spin chain. It is found that all of the impurity parameters in the bulk must be pure imaginary numbers if the cross parameter is real for getting the hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interactions are introduced. The Hamiltonian can be also applied to describe the many body systems with two kinds of particles if all of the impurity parameters are equal.

Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...

The one dimensional S=1 XXZ model with next-nearest-neighbor interaction

α

and Ising-type anisotropy

Δ

is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Néel phase, and then we confirm the universality class. Moreover, we map this model onto the non-linear

σ

model and analyze the phase diagram in the

α

\ll

-1 and

Δ

\sim

1 region by using the renormalization group method.