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

arXiv:hep-th/0201001

Integrable Heisenberg-van Vleck chains with variable range exchange

V. I. Inozemtsev

Published Jan 1, 2002• 32 citations
arXivHTMLPDF

Abstract

The review of recent results in the s=1/2 quantum spin chains with $1/\sinh^2(κr$ exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension.

DOI
10.48550/arxiv.hep-th/0201001

Related papers

Bethe-ansatz equations for quantum Heisenberg chains with elliptic exchange

V. I. Inozemtsev

The eigenvectors of the Hamiltonian ${\cal H}_{N}$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.

arXiv:math-ph/9911022•Nov 17, 1999•2 citations

Classical facets of quantum integrability

A. Zabrodin

This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational $GL(n)$-invariant $R$-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.

arXiv:2501.18557•Jan 30, 2025•0 citations

Bethe Ansatz inside Calogero-Sutherland models

Gwenaël Ferrando, Jules Lamers, Fedor Levkovich-Maslyuk +1 more

We study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura-Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest.

arXiv:2308.16865•Aug 31, 2023•2 citations

Solution to three-magnon problem for S=1/2 periodic quantum spin chains with elliptic exchange

V. I. Inozemtsev

The method of solving the three-particle quantum elliptic Calogero–Moser problem is applied to the description of three-magnon wave functions for the S=1/2 quantum Heisenberg chains with the exchange interaction given by the Weierstrass d function. The Bethe-like algebraic equations for the three-magnon case are presented in the explicit form.

Journal of Mathematical Physics•Jan 1, 1996•8 citations

Topics in quantum integrable systems

Kazuhiro Hikami, Miki Wadati

Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.

Journal of Mathematical Physics•Jul 24, 2003•14 citations

Specific heat calculations for quantum spin chains with Heisenberg-biquadratic exchange in a magnetic field

H -U Desgranges, John B. Parkinson

The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.

Journal of Physics C Solid State Physics•Jan 10, 1988•2 citations
The review of recent results in the s=1/2 quantum spin chains with 1/sinh⁡2(κr1/\sinh^2(κr1/sinh2(κr exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension.
The eigenvectors of the Hamiltonian HN{\cal H}_{N}HN​ of NNN-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational GL(n)GL(n)GL(n)-invariant RRR-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.
We study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura-Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest.
Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.
The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.