Boundary Quantum Knizhnik-Zamolodchikov Equation

Kasatani, Masahiro

doi:10.1142/9789814324373_0009

Cited by 10 publications

(14 citation statements)

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“…Indeed, extracting interesting physical data -except in some special cases -from integral representations of correlation functions is a rather complicated problem. In this direction, the remarkable connection between the q-Onsager algebra and the theory of special functions [42] may be promising, as well as the link between solutions of the reflection quantum Knizhnik-Zamolochikov equations [6] and Koornwinder polynomials [43,44] (see also [45]).…”

Section: Resultsmentioning

confidence: 99%

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Baseilhac

Kojima

2014

Nuclear Physics B

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The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime.Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values σ a 1 with a = z, ± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.Beyond the Ising model, a large class of solvable lattice models have been discovered. In the context of quantum integrable systems on the lattice, spin chains are among the most studied examples with applications which range from condensed matter to high energy physics. Given a Hamiltonian, finding analytical expressions for the exact spectrum, identifying the structure of the space of the eigenvectors and deriving explicit expressions for correlation functions are essential steps in the non-perturbative characterization of the system's behavior which can be compared with experimental data.Among the simplest examples considered in the literature, the XXZ spin chain with different boundary conditions has received particular attention. Over the years, different approaches have been proposed in order to understand the Hamiltonian's spectral problem and derive the correlation functions. For models with periodic boundary conditions, the spectral problem can be handled by methods such as the Bethe ansatz (BA) [1], or the corner transfer matrix method (CTM) in the thermodynamic limit [2]. The computation of the correlation functions, however, is a much more difficult problem in general.Apart from the simplest example -namely the XXZ spin chain with periodic boundary condition -for which correlation functions have been proposed by the quantum inverse scattering method (QISM) [4,5] arising from the BA, the generalization of this result to models with higher symmetries requires a better understanding of mathematical structures, for instance, of determinant formulae of scalar products that involve the Bethe vectors [7] (see some recent progress in [8]). However, in the thermodynamic limit, this problem can be alternatively tackled using the q-vertex operator approach (VOA) [3] arising from the CTM. The space of states is identified with the irreducible highest weight representation of U q ( sl 2 ) or higher rank quantum algebras. Correlation functions can be obtained using bosonizations of the q-vertex operators for U q ( sl 2 ) or higher rank quantum algebras [3,9,10,11,12,13]. Either within the QISM or the VOA, correlation functions are obtained in the form of integrals of meromorphic functions in the thermodynamic limit. The situation for integrable spin chains with open boundaries is more difficult. On one hand, for the finite XXZ open chain with diagonal boundaries [14], related non-diagonal boundaries [15, 16] or q a root of unity [17], the BA makes it possible to deriv...

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Section: Resultsmentioning

confidence: 99%

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Baseilhac

Kojima

2014

Nuclear Physics B

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“…In view of the previous proposition, it suffices to note that for a given m ∈ Z we have w J 0 ζ = γ m if and only if (5.6) holds true. Different examples of W 0 -invariant Laurent polynomial solutions of reflection quantum KZ equations are given in [34,76,12,22,54].…”

Section: Solutions Of Reflection Quantum Kz Equationsmentioning

confidence: 99%

Koornwinder polynomials and the XXZ spin chain

Stokman

Vlaar

2015

Journal of Approximation Theory

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Abstract. Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin-1 2 chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.

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“…These solutions give rise to correlation functions of the semi-infinite XXZ spin-1 2 chain. See [25,26,38,65,4] for other constructions of solutions for special classes of K-matrices K(z) and K(z).…”

Section: 2mentioning

confidence: 99%

“…Secondly, special cases of the quantum affine KZ equations for principal series modules arise as compatibility conditions for correlation functions and form factors of (semi-)infinite XXZ spin chains [34,35,69]. In these cases vertex operator and algebraic Bethe ansatz techniques have been employed to construct solutions of the quantum affine KZ equations (see, e.g., [27,68,54,34] and references therein for type A, and [35,69,25,26,38,30,40,55,65,4] for type C). These methods have the drawback that they require additional assumptions on the associated XXZ spin chains, such as the existence of pseudo-vacuum vectors.…”

Section: Introductionmentioning

confidence: 99%

Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Stokman

2015

Commun. Math. Phys.

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Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In the case of a principal series module, we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions.For the spin representation of the affine Hecke algebra of type C, the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1 2 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter's 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations.

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Boundary Quantum Knizhnik-Zamolodchikov Equation

Cited by 10 publications

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Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Koornwinder polynomials and the XXZ spin chain

Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

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