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Journal of Physics A Mathematical and Theoretical

Exact solution of a quantum spin chain with competing bulk and boundary terms

Yusong Cao, Jian Wang, Yi Qiao, Junpeng Cao, Wen‐Li Yang

Published Jun 15, 2021• 0 citations
Publisher

Abstract

Abstract We obtain the exact solution of an anisotropic quantum spin chain including nearest neighbor (NN), next nearest neighbor (NNN), chiral three-spin interactions and non-diagonal boundary terms. Because the NNN couplings are involved, we find that the off-diagonal boundary reflections can not only induce the anisotropic Dzyloshinsky–Moriya interactions at boundaries but also enhance the anisotropy of first and last bonds. These properties are absent if only the NN spin-exchanging interactions are considered. Due to that the U (1) symmetry is broken, we analytically solve the system by using the off-diagonal Bethe ansatz. The inhomogeneous T − Q relation, energy spectrum and associated Bethe ansatz equations are given explicitly. This paper provides an universal scheme to construct new integrable models with certain interesting interactions.

DOI
10.1088/1751-8121/ac0b8b

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Abstract We obtain the exact solution of an anisotropic quantum spin chain including nearest neighbor (NN), next nearest neighbor (NNN), chiral three-spin interactions and non-diagonal boundary terms. Because the NNN couplings are involved, we find that the off-diagonal boundary reflections can not only induce the anisotropic Dzyloshinsky–Moriya interactions at boundaries but also enhance the anisotropy of first and last bonds. These properties are absent if only the NN spin-exchanging interactions are considered. Due to that the U (1) symmetry is broken, we analytically solve the system by using the off-diagonal Bethe ansatz. The inhomogeneous T − Q relation, energy spectrum and associated Bethe ansatz equations are given explicitly. This paper provides an universal scheme to construct new integrable models with certain interesting interactions.
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.
Abstract We propose a method to construct new quantum integrable models. As an example, we construct an integrable anisotropic quantum spin chain which includes the nearest-neighbor, next-nearest-neighbor and chiral three-spin couplings. It is shown that the boundary fields can enhance the anisotropy of the first and last bonds, and can induce the Dzyloshinsky–Moriya interactions along the z -direction at the boundaries. By using the algebraic Bethe ansatz, we obtain the exact solution of the system. The energy spectrum of the system and the associated Bethe ansatz equations are given explicitly. The method provided in this paper is universal and can be applied to constructing other exactly solvable models with certain interesting interactions.
The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbor, next-nearest-neighbor and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the U(1)-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous T−Q relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant J=−1 although the Bethe Ansatz equations are the inhomogeneous ones.
A novel Bethe ansatz scheme is proposed to investigate the exact physical properties of an integrable anisotropic quantum spin chain with competing interactions among the nearest, next nearest neighbor and chiral three spin couplings, where the boundary condition is the twisted one. The eigenvalue of the transfer matrix is characterized by its zero roots instead of the traditional Bethe roots. Based on the exact solution, the conserved momentum and charge operators of this U(1)-symmetry broken system are obtained. The ground state energy and density of rapidities are calculated. It is found that there exist three kinds of elementary excitations and the corresponding dispersion relations are obtained, which gives a different picture from that with periodic boundary condition.
We study the exact physical quantities of a competing spin chain which contains many interesting and meaningful couplings including the nearest neighbor, next nearest neighbor, chiral three spins, Dzyloshinsky-Moriya interactions and unparallel boundary magnetic fields in the thermodynamic limit. We obtain the density of zero roots, surface energies and elementary excitations in different regimes of model parameters. Due to the competition of various interactions, the surface energy and excited spectrum show many different pictures from those of the Heisenberg spin chain.