© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:nlin/0204066 · Journal of Physics A Mathematical and General

On the algebraic Bethe ansatz for the XXX spin chain: creation operators beyond the equator

A. E. Kovalsky, G. P. Pronko

Published Jul 30, 2002• 0 citations

Abstract

Considering the XXX spin-1/2 chain in the framework of the algebraic Bethe ansatz, we make the following short comment: the product of the creation operators corresponding to the recently found solution of the Bethe equations ‘on the wrong side of the equator’ [1] is just zero (not only its action on the pseudovacuum).

DOI: 10.1088/0305-4470/35/32/102

Related papers

Bethe equations `on the wrong side of the equator'

G. P. Pronko, Yu. G. Stroganov

We analyse Baxter's famous T-Q equations for the XXX ( XXZ ) spin chain and show that, apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe ansatz equations, there also exists a second solution which should correspond to the Bethe ansatz beyond . This second solution of Baxter's equation plays an essential role and together with the first one gives rise to all fusion relations.

arXiv:hep-th/9808153•Journal of Physics A Mathematical and General•Jan 1, 1999•89 citations

Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz

S. Belliard

We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al.The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain.We prove this result for the chains with small length.We obtain also an off-shell equation (i.e.satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

arXiv:1309.6165•Symmetry Integrability and Geometry Methods and Applications•Nov 22, 2013•97 citations

The Bethe ansatz

Fedor Levkovich-Maslyuk

Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.

arXiv:1606.02950•Journal of Physics A Mathematical and Theoretical•Jul 20, 2016•70 citations

Algebraic and Functional Bethe Ansatz for the Integrable XXX Spin Chain

Patrick Kaftan

Properties of the XXX spin chain will be obtained through quantum inverse scattering methods. Starting with the one dimensional homogeneous model, the algebraic Bethe ansatz will be applied to obtain the Hamiltonian and corresponding spectrum of the system. The model will be expanded into the inhomogeneous spin chain for the application of the functional Bethe ansatz. Comparing the results of both methods we obtain the full set of mathematical properties of the spin chain.

•Jan 10, 2020•0 citations

Families of solutions of the nested Bethe ansatz for theA2spin chain

G P Pronko, Yu G Stroganov

The full set of polynomial solutions of the nested Bethe ansatz is constructed for the case of the A 2 rational spin chain. The structure and properties of these associated solutions are more various than the case of the usual XXX ( A 1 ) spin chain but their role is similar.

arXiv:hep-th/9902085•Journal of Physics A Mathematical and General•Nov 9, 2000•32 citations

Modified algebraic Bethe ansatz for XXZ chain on the segment – III – Proof

Jean Avan, S. Belliard, Nicolas Grosjean +1 more

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-12 chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix.

arXiv:1506.02147•Nuclear Physics B•Aug 11, 2015•77 citations

We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al.The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain.We prove this result for the chains with small length.We obtain also an off-shell equation (i.e.satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.