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Journal of Physics A Mathematical and General

Algebraic Bethe ansatz for a spin-½ quantum linear chain with competing interactions

M. A. Rego-Monteiro

Published Nov 9, 2000• 0 citations
Publisher

Abstract

We propose a generalization of the formula that gives an integrable quantum 1D spin Hamiltonian with nearest-neighbour interactions as a logarithmic derivative of a vertex model transfer matrix in order to include in this scheme more realistic integrable models. We compute exactly this generalized formula using the R matrix of the XXX model, obtaining the Majumdar-Ghosh Hamiltonian plus a charge-like interaction term. We diagonalize this Hamiltonian using the quantum inverse scattering method and present the Bethe ansatz equations of the model.

DOI
10.1088/0305-4470/33/46/101

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We propose a generalization of the formula that gives an integrable quantum 1D spin Hamiltonian with nearest-neighbour interactions as a logarithmic derivative of a vertex model transfer matrix in order to include in this scheme more realistic integrable models. We compute exactly this generalized formula using the R matrix of the XXX model, obtaining the Majumdar-Ghosh Hamiltonian plus a charge-like interaction term. We diagonalize this Hamiltonian using the quantum inverse scattering method and present the Bethe ansatz equations of the model.
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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.
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We present an 'algebraic treatment' of the analytical Bethe ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic gl ( )-spin chain possessing on each site an arbitrary gl ( )-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite-dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. The local Hamiltonian associated with a given integrable spin chain needs to be computed case by case. A general formula is still lacking at that point. As examples, we recover the usual Hamiltonian and Bethe equations for closed and open spin chains; we treat the alternating spin chains and the closed spin chain with an impurity. We also compute the Hamiltonian and Bethe equations for the open spin chain model used in the context of large- N QCD.
Quantum spin chains arise naturally from perturbative large- N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbour type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of . We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.