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International Journal of Modern Physics B

INTEGRABLE XYZ HAMILTONIAN WITH NEXT-NEAREST-NEIGHBOR INTERACTION

Lai Yun-zhong, Zhan-Ning Hu, Jun Liang, Fu‐Cho Pu

Published Mar 20, 1999• 0 citations
Publisher

Abstract

In this paper, we construct a Hamiltonian of the impurity model with next-nearest-neighbor interaction within the framework of the open boundary Heisenberg XYZ spin chain. This impurity model is an exactly solved one and it degenerates to the integrable XXZ impurity model under the triangular limit. It is the first approach to add the impurities and next-nearest-neighbor interaction to the integrable completely anisotropic Heisenberg spin chain. We find also that the impurity parameters in the bulk are real when the cross parameter is imaginary for the Hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interaction is introduced. The eigenvalue of the Hamiltonian and the Bethe ansatz equations for the trigonometric limit case are derived also.

DOI
10.1142/s0217979299000710

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In this paper, we construct a Hamiltonian of the impurity model with next-nearest-neighbor interaction within the framework of the open boundary Heisenberg XYZ spin chain. This impurity model is an exactly solved one and it degenerates to the integrable XXZ impurity model under the triangular limit. It is the first approach to add the impurities and next-nearest-neighbor interaction to the integrable completely anisotropic Heisenberg spin chain. We find also that the impurity parameters in the bulk are real when the cross parameter is imaginary for the Hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interaction is introduced. The eigenvalue of the Hamiltonian and the Bethe ansatz equations for the trigonometric limit case are derived also.
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.
We propose an integrable model of the spin-1/2 Heisenberg chain coupled to two impurity moments. With the open boundary conditions at the impurity sites, the model can be exactly solved for arbitrary impurity spin and arbitrary exchange constants between the bulk and the impurities. The absence of redundant terms in the Hamiltonian makes the model very reasonable. The Hamiltonian is diagonalized via algebraic Bethe ansatz. It is found that the impurity spins can only be screened (partially for S>1/2S>1/2S>1/2) for antiferromagnetic coupling between the impurity and the bulk. Otherwise the impurity spins cannot be screened. The residual entropy of the ground state and the Kondo temperature are also derived explicitly based on the Bethe ansatz and the local Fermi-liquid theory.
Abstract We rigorously prove that the S=1/2S=1/2S=1/2 S = 1 / 2 anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S=1/2S=1/2S=1/2 S = 1 / 2 XYZ chain with Z magnetic field, whose proof technique is employed in our result.
We construct an integrable quantum spin chain that includes the nearest-neighbor, next-nearest-neighbor, chiral three-spin couplings, Dzyloshinsky–Moriya interactions and unparallel boundary magnetic fields. Although the interactions in bulk materials are isotropic, the spins nearby the boundary fields are polarized, which induce the anisotropic exchanging interactions of the first and last bonds. The U(1) symmetry of the system is broken because of the off-diagonal boundary reflections. Using the off-diagonal Bethe ansatz, we obtain an exact solution to the system. The inhomogeneous T–Q relation and Bethe ansatz equations are given explicitly. We also calculate the ground state energy. The method given in this paper provides a general way to construct new integrable models with certain interesting interactions.