© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1507.03242 · Journal of Physics A Mathematical and Theoretical

Slavnov and Gaudin–Korepin formulas for models without U (1) symmetry: the XXX chain on the segment

S. Belliard, Rodrigo A. Pimenta

Published Mar 18, 2016• 18 citations

Abstract

Abstract We consider the isotropic spin Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin–Korepin formula, i.e., the square of the norm, is also obtained.

DOI: 10.1088/1751-8113/49/17/17lt01

Related papers

Slavnov and Gaudin-Korepin Formulas for Models without U(1) Symmetry: the Twisted XXX Chain

Universit&, S. Belliard, #233 +2 more

We consider the XXX spin-1 2 Heisenberg chain on the circle with an arbitrary twist.We characterize its spectral problem using the modified algebraic Bethe anstaz and study the scalar product between the Bethe vector and its dual.We obtain modified Slavnov and Gaudin-Korepin formulas for the model.Thus we provide a first example of such formulas for quantum integrable models without U(1) symmetry characterized by an inhomogenous Baxter T-Q equation.

arXiv:1506.06550•Symmetry Integrability and Geometry Methods and Applications•Dec 4, 2015•34 citations

Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

N. Cirilo António, N. Manojlović, Igor Salom

We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.

arXiv:1405.7398•Nuclear Physics B•Oct 15, 2014•20 citations

Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model

N. Manojlović, Igor Salom

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.

arXiv:1705.02235•Nuclear Physics B•Aug 3, 2017•12 citations

Scalar product of twisted XXX modified Bethe vectors

S. Belliard, N A Slavnov, B Vallet

Abstract We consider closed XXX spin chains with broken total spin symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained formulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin–Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.

arXiv:1805.11323•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2018•18 citations

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

Universit&, S. Belliard, #233 +2 more

We consider XXX spin-1/2 Heisenberg chain with non-diagonal boundary conditions.We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors.In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix.

arXiv:1906.06897•Symmetry Integrability and Geometry Methods and Applications•Sep 2, 2019•13 citations

Why scalar products in the algebraic Bethe ansatz have determinant representation

S. Belliard, N. A. Slavnov

Abstract We show that the scalar products of on-shell and off-shell Bethe vectors in the algebralic Bethe ansatz solvable models satisfy a system of linear equations. We find solutions to this system for a wide class of integrable models. We also apply our method to the XXX spin chain with broken U(l) symmetry.

arXiv:1908.00032•Journal of High Energy Physics•Oct 1, 2019•39 citations

Abstract We consider the isotropic spin Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin–Korepin formula, i.e., the square of the norm, is also obtained.

We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.

Abstract We consider closed XXX spin chains with broken total spin symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained formulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin–Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.