arXiv:2409.19775

R-matrix for the XX spin chain in a staggered magnetic field

P. N. Bibikov

Published Sep 29, 2024• 0 citations

Abstract

We suggest the elliptic $16\times16$ R-matrix for the XX spin chain in a staggered magnetic field. This integrable model is a special case of the more general one, studied long ago within the alternative approach by V. M. Kontorovich and V. M. Tsukernik. The Yang-Baxter equation is verified with the full rigorous only in the two special degenerate trigonometric cases. For the general one, studying the first terms of the Taylor expansions for the $R$-matrix entries, we give the strong evidence for implementation of the Yang-Baxter equation.

DOI: 10.48550/arxiv.2409.19775

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arXiv:2409.19775

R-matrix for the XX spin chain in a staggered magnetic field

P. N. Bibikov

Published Sep 29, 2024• 0 citations

arXiv HTML PDF

Abstract

We suggest the elliptic

16\times16

R-matrix for the XX spin chain in a staggered magnetic field. This integrable model is a special case of the more general one, studied long ago within the alternative approach by V. M. Kontorovich and V. M. Tsukernik. The Yang-Baxter equation is verified with the full rigorous only in the two special degenerate trigonometric cases. For the general one, studying the first terms of the Taylor expansions for the

R

-matrix entries, we give the strong evidence for implementation of the Yang-Baxter equation.

DOI: 10.48550/arxiv.2409.19775

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We suggest the elliptic

16\times16

R

-matrix entries, we give the strong evidence for implementation of the Yang-Baxter equation.

It is pointed, that the

16\times16

Hamiltonian density matrix, corresponding to XX spin chain in a staggered magnetic field, satisfies the Reshetikhin condition as well, as the two higher ones. All of them are necessary for the existence of the corresponding

R

-matrix. At the special imaginary value of the staggered field, the

R

-matrix is obtained explicitly. It is shown that the corresponding Hamiltonian has purely imaginary spectrum.

Matrix representations of

{\rm BC}_n

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

{\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.

An elliptic generalization of the

D^{(2)}_{2}

Spin chain

An integrable elliptic generalization of the closed

D^{(2)}_2

quantum spin chain is constructed. Taking the trigonometric limit, we have shown that the corresponding R-matrix reduces to that of the

D^{(2)}_2

one, up to a similarity transformation. Basing on the very fact of the factorization for the R-matrix and the corresponding transfer matrix which indeed ensures the integrability of the model, we give the exact solution of the model which can be expressed in terms of a homogeneous T-Q relation.