A deformed analogue of Onsager’s symmetry in the XXZ open spin chain

Baseilhac, Pascal; Koizumi, Kozo

doi:10.1088/1742-5468/2005/10/p10005

Cited by 82 publications

(149 citation statements)

References 30 publications

Supporting

Mentioning

146

Contrasting

Order By: Relevance

“…To conclude, let us mention that finite dimensional modules of the form (3.17) have found applications in the solution of the open XXZ spin chain with generic integrable boundary conditions [59]. For the family of tridiagonal pairs discussed in [79], up to a normalization, the eigenvectors in each basis have been determined explicitly.…”

Section: Finite Dimensional Modulesmentioning

confidence: 99%

“…For instance, in the literature, the two-dimensional Ising and superintegrable Potts models have been studied in details for q = 1 using the explicit relation between the Onsager algebra and a fixed-point subalgebra of sl 2 (under the action of a certain automorphism of sl 2 [47]). For q = 1, the explicit relation between the q−Onsager algebra and a certain coideal subalgebra of U q ( sl 2 ) has been used to analyze the finite open XXZ spin chain [59] or its thermodynamic limit analog for various types of boundary conditions [58,85]. According to the size of the system -finite or semi-infinite -, finite or infinite dimensional representations (q−vertex operators) of the q−Onsager algebra have been considered.…”

Section: The Q−dolan-grady Hierarchy and Spectral Problem Revisitedmentioning

confidence: 99%

“…Based on these results, the spectral problem for the q−Dolan-Grady hierarchy of mutually commuting quantities (1.2) is reconsidered in Section 4, where the eigenfunctions are shown to admit an expansion in terms of Gasper-Rahman polynomials. In particular, subspaces that are invariant under the action of the Abelian subalgebra generated by the elements (1.2) are identified, extending the analysis of [59]. These subspaces are characterized by certain relations that already appeared in the literature on boundary integrable models, named 'Nepomechie's relations'.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

Abstract. For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q = 1).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

show abstract

Section: Finite Dimensional Modulesmentioning

confidence: 99%

Section: The Q−dolan-grady Hierarchy and Spectral Problem Revisitedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…This deformed DG-algebra plays an important role in theory of quantum XXZ Heisenberg model, Azbel-Hofstadter model etc [3,2]. It would be interesting to study what is the meaning of the exact solvability of the Heisenberg picture for the q-deformed DG-relations (7.10) in corresponding exactly solvable models.…”

Section: Beyond the Aw-algebramentioning

confidence: 99%

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Vinet

Zhedanov²

2008

SIGMA

View full text Add to dashboard Cite

Abstract. We study Poisson and operator algebras with the "quasi-linear property" from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of "time" t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems. Classical versionAssume that there is a classical Poisson manifold with the Poisson brackets (PB) {x, y} defined for all dynamical variables x, y belonging to this manifold. Of course, it is assumed that the PB satisfy the standard conditions: (i) {x, α 1 y + α 2 z} = α 1 {x, y} + α 2 {x, z} -linearity (α 1 , α 2 are arbitrary constants);(ii) {x, y} = −{y, x} -antisymmetricity; (iii) {x, yz} = z{x, y} + y{x, z} -the Leibnitz rule; (iv) {{x, y}, z} + {{y, z}, x} + {{z, x}, y} = 0 -the Jacobi identity.If one chooses the "Hamiltonian" H (i.e. some dynamical variable belonging to this manifold), then we have the standard Hamiltonian dynamics: all variables x(t) become depending on an additional time variable t and the equation of motion is defined aṡ x(t) = {x(t), H}.(1.1) Of course, the Hamiltonian H is independent of t, becauseḢ = {H, H} = 0. More generally, a dynamical variable Q is called the integral of motion ifQ = 0. Clearly, this is equivalent to the condition {Q, H} = 0. From (1.1) we haveẍ = {{x, H}, H} and more generally d n x dt n = {. . . {x, H}, H}, . . . H} = {x, H (n) }, This paper is a contribution to the

show abstract

“…This is so for instance in the case of the two-dimensional Ising model [O44], of the superintegrable Potts model [GeR85] at q = 1 or of the open XXZ spin chain for q = 1 [BK07,BB12]. For the integrable models that fall in this class, finding the spectrum and eigenstates of the Hamiltonian relies on the construction of explicit finite or infinite dimensional representations of the q−Onsager algebra.…”

Section: Introductionmentioning

confidence: 99%

Theq-Onsager algebra and multivariableq-special functions

Baseilhac¹,

Vinet²,

Zhedanov³

2017

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract. Two sets of mutually commuting q−difference operators xi and yj, i, j = 1, ..., N such that xi and yi generate a homomorphic image of the q−Onsager algebra for each i are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in N variables and N + 3 parameters. Under certain conditions on the parameters, they form two 'dual' bases of polynomials in N variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of q−Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual q−Krawtchouk polynomials. Finally, the 'split' basis in which the two families of operators act as block bidiagonal matrices is also provided.MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

show abstract

A deformed analogue of Onsager’s symmetry in the XXZ open spin chain

Cited by 82 publications

References 30 publications

A bispectral q-hypergeometric basis for a class of quantum integrable models

A bispectral q-hypergeometric basis for a class of quantum integrable models

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Theq-Onsager algebra and multivariableq-special functions

Contact Info

Product

Resources

About