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

Yang-Baxter Equation in

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

We consider the su(n) spin chains with long range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities and we diagonalise them.

Related papers

Yang-Baxter equation in spin chains with long range interacting systems

Denis Bernard, M. Gaudin, F. D. M. Haldane +1 more

We consider the $ su(n) $ spin chains with long range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities and we diagonalize them.

•Jan 1, 1993•1 citation

Yang-Baxter equation in long-range interacting systems

Denis Bernard, M. Gaudin, F. D. M. Haldane +1 more

We consider the su(p) spin chains with long-range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities of the dynamical models and we diagonalise them. In the spin chain case, we establish the connection between the degeneracies of the spectrum and the representation theory of the Yangians. We use a correspondence with the dynamical models to diagonalise the Hamiltonian. Finally, we extend the previous results to the case of a trigonometric R-matrix.

arXiv:hep-th/9301084•Journal of Physics A Mathematical and General•Oct 21, 1993•288 citations

Matrix representations of ${\rm BC}_n$ associative Yang--Baxter equations and related long-range spin chains

M. Matushko, A. Mostovskii, A. Zotov

We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of ${\rm BC}_n$ type. In the elliptic case, two sets of generators are represented in terms of the Baxter--Belavin quantum $R$-matrix in the fundamental representation of ${\rm GL}_N$, while the remaining generators are represented by the elliptic $K$-matrix found by Komori and Hikami. We revisit its derivation using the Felder--Pasquier procedure and correct the unitarity relation for odd $N$. We then use these matrix representations to construct examples of long-range quantum spin chains from $R$-matrix-valued Lax pairs for the classical Calogero--Inozemtsev system associated with the ${\rm BC}_n$ root system. We also construct $R$-matrix-valued Lax pairs for Calogero--Moser models in an external field, leading to Haldane--Shastry and Polychronakos--Frahm chains with modified spin positions.

arXiv:2610.33913•Oct 24, 2026•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

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.

SciPost Physics•Jan 28, 2025•2 citations

Conservation laws of the Haldane-Shastry type spin chains

Cao, Junpeng, Peng cheng He, Yuzhu Jiang +1 more

A systematic method to construct the complete set of conserved quantities of the Haldane-Shastry type spin chains is proposed. The hidden relationship between the Yang-Baxter relation and the conservation laws of the long-range interacting integrable models is exposed explicitly. An integrable anisotropic Haldane-Shastry model is also constructed.

arXiv:0811.4476•Nov 27, 2008•0 citations
We consider the su(p) spin chains with long-range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities of the dynamical models and we diagonalise them. In the spin chain case, we establish the connection between the degeneracies of the spectrum and the representation theory of the Yangians. We use a correspondence with the dynamical models to diagonalise the Hamiltonian. Finally, we extend the previous results to the case of a trigonometric R-matrix.
Matrix representations of BCn{\rm BC}_nBCn​ associative Yang--Baxter equations and related long-range spin chains
We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of BCn{\rm BC}_nBCn​ type. In the elliptic case, two sets of generators are represented in terms of the Baxter--Belavin quantum RRR-matrix in the fundamental representation of GLN{\rm GL}_NGLN​, while the remaining generators are represented by the elliptic KKK-matrix found by Komori and Hikami. We revisit its derivation using the Felder--Pasquier procedure and correct the unitarity relation for odd NNN. We then use these matrix representations to construct examples of long-range quantum spin chains from RRR-matrix-valued Lax pairs for the classical Calogero--Inozemtsev system associated with the BCn{\rm BC}_nBCn​ root system. We also construct RRR-matrix-valued Lax pairs for Calogero--Moser models in an external field, leading to Haldane--Shastry and Polychronakos--Frahm chains with modified spin positions.
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.
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.