Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Polonica A

Combinatorics of Lax Objects in Bethe Ansatz

Ryszard Stagraczyński

Published Aug 1, 2015• 0 citations
Publisher

Abstract

Algebraic Bethe Ansatz, also known as quantum inverse scattering method, is a consistent tool based on the YangBaxter equation which allows to construct Bethe Ansatz exact solutions.One of the most important objects in algebraic Bethe Ansatz is a monodromy matrix M , which is dened as an appropriate product of so-called Lax operators L (local transition operators).Monodromy matrix as well as each of Lax operators acts in the tensor product of the quantum space H with an auxiliary space C 2 .Thus M , when written in the standard basis of auxiliary space, consists of four elements A, B, C, D, which are the operators acting in quantum space H, where B and C are step operators and the remaining generate all constants of motion.In this work a consistent method of construction of the Bethe Ansatz eigenstates in terms of objects a, b, c, d i.e. matrix elements of the Lax operators in the auxiliary space is proposed.

DOI
10.12693/aphyspola.128.215

Related papers

Combinatorics of Lax Objects in Bethe Ansatz

Ryszard Stagraczyński

Algebraic Bethe Ansatz, also known as quantum inverse scattering method, is a consistent tool based on the YangBaxter equation which allows to construct Bethe Ansatz exact solutions.One of the most important objects in algebraic Bethe Ansatz is a monodromy matrix M , which is dened as an appropriate product of so-called Lax operators L (local transition operators).Monodromy matrix as well as each of Lax operators acts in the tensor product of the quantum space H with an auxiliary space C 2 .Thus M , when written in the standard basis of auxiliary space, consists of four elements A, B, C, D, which are the operators acting in quantum space H, where B and C are step operators and the remaining generate all constants of motion.In this work a consistent method of construction of the Bethe Ansatz eigenstates in terms of objects a, b, c, d i.e. matrix elements of the Lax operators in the auxiliary space is proposed.

Acta Physica Polonica A•Aug 1, 2015•0 citations

Algebraic Bethe Ansatz forN= 4

Dorota Jakubczyk, Tadeusz Lulek, Paweł Jakubczyk +1 more

In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2 . The form of this matrix in the basis of orbits of the translation group C 4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.

Journal of Physics Conference Series•Mar 1, 2010•0 citations

Bethe Ansatz, quantum circuits, and the F-basis

Roberto Ruiz, Alejandro Sopena, Esperanza López +2 more

The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as “algebraic Bethe circuits”, have been developed to prepare Bethe states for the spin- 1/2 1 / 2 XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the “F-basis” known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin- 1/2 1 / 2 XXZ model.

arXiv:2411.02519•SciPost Physics•Jun 12, 2025•8 citations

GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators

Stanislav Zdislavovich Pakuliak, E. Ragoucy, Université de Savoie, France +1 more

We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix.We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices.Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates.We obtain determinant representations for these matrix elements.Thus, we solve the inverse scattering problem in a weak sense.

arXiv:1502.01966•Symmetry Integrability and Geometry Methods and Applications•Jul 31, 2015•41 citations

The Algebraic Bethe Ansatz

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.

Cambridge University Press eBooks•Aug 19, 1993•1 citation

Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(A2(2))

N. Manojlović, Zoltán Nagy

We implement the Bethe ansatz method for the elliptic quantum group Eτ,η(A2(2)). The Bethe creation operators are constructed as polynomials of the Lax matrix elements expressed through a recurrence relation. We also give the eigenvalues of the family of commuting transfer matrices defined in the tensor product of fundamental representations.

arXiv:0704.3032•Journal of Mathematical Physics•Dec 1, 2007•2 citations
Algebraic Bethe Ansatz, also known as quantum inverse scattering method, is a consistent tool based on the YangBaxter equation which allows to construct Bethe Ansatz exact solutions.One of the most important objects in algebraic Bethe Ansatz is a monodromy matrix M , which is dened as an appropriate product of so-called Lax operators L (local transition operators).Monodromy matrix as well as each of Lax operators acts in the tensor product of the quantum space H with an auxiliary space C 2 .Thus M , when written in the standard basis of auxiliary space, consists of four elements A, B, C, D, which are the operators acting in quantum space H, where B and C are step operators and the remaining generate all constants of motion.In this work a consistent method of construction of the Bethe Ansatz eigenstates in terms of objects a, b, c, d i.e. matrix elements of the Lax operators in the auxiliary space is proposed.
Algebraic Bethe Ansatz, also known as quantum inverse scattering method, is a consistent tool based on the YangBaxter equation which allows to construct Bethe Ansatz exact solutions.One of the most important objects in algebraic Bethe Ansatz is a monodromy matrix M , which is dened as an appropriate product of so-called Lax operators L (local transition operators).Monodromy matrix as well as each of Lax operators acts in the tensor product of the quantum space H with an auxiliary space C 2 .Thus M , when written in the standard basis of auxiliary space, consists of four elements A, B, C, D, which are the operators acting in quantum space H, where B and C are step operators and the remaining generate all constants of motion.In this work a consistent method of construction of the Bethe Ansatz eigenstates in terms of objects a, b, c, d i.e. matrix elements of the Lax operators in the auxiliary space is proposed.
In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2 . The form of this matrix in the basis of orbits of the translation group C 4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.
The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as “algebraic Bethe circuits”, have been developed to prepare Bethe states for the spin- 1/2 1 / 2 XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the “F-basis” known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin- 1/2 1 / 2 XXZ model.
We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix.We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices.Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates.We obtain determinant representations for these matrix elements.Thus, we solve the inverse scattering problem in a weak sense.
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.