Scalar Product of Bethe Vectors from Functional Equations

Galleas, W.

doi:10.1007/s00220-014-1976-2

Cited by 15 publications

(53 citation statements)

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“…Scalar products of Bethe vectors can also be tackled through the algebraic-functional method. This was demonstrated in [Gal14] for the six-vertex model with boundary twists, and in [Gal15] for the case of open boundaries. In particular, in the works [Gal14,Gal15] we have obtained a system of two functional equations determining such off-shell scalar products.…”

Section: Introductionmentioning

confidence: 80%

“…In the work [Gal14] we have presented functional equations determining the scalar product (2.7). Among the results of [Gal14] we have also shown that the specialization (2.9) satisfy a simpler functional relation exhibiting the same structure as the equation satisfied by the partition function of the six-vertex model with domain-wall boundaries [Gal13b,Gal13a]. This simplified equation has also been studied in [GL15] and it is given as follows.…”

Section: Bethe Vectors and Scalar Productsmentioning

confidence: 99%

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Continuous representations of scalar products of Bethe vectors

Galleas

2017

Journal of Mathematical Physics

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We present families of single determinantal representations of on-shell scalar products of Bethe vectors. Our families of representations are parameterized by a continuous complex variable which can be fixed at convenience. Here we consider Bethe vectors in two versions of the six-vertex model: the case with boundary twists and the case with open boundaries.Comment: 21 pages; v2: minor stylistic changes, missing factor in 3.15 and 3.16 adde

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Section: Introductionmentioning

confidence: 80%

Section: Bethe Vectors and Scalar Productsmentioning

confidence: 99%

Continuous representations of scalar products of Bethe vectors

Galleas

2017

Journal of Mathematical Physics

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“…To date all cases in which the functional method has been used to obtained a closed expression were previously tackled using the Korepin-Izergin method. For the six-vertex model the domainwall partition function [8,32] was of course first found by Korepin and Izergin [23,24,28], the reflecting-end partition function [20] by Tsuchiya [43], and the corresponding (off-/on-shell) scalar products of Bethe vectors [13,14] by Slavnov [40], Wang [44] and Kitanine et al [25]. In the dynamical (sos) case the domain-wall partition function [10,12] was found by Rosengren [38], and the reflecting-end partition function [31,32] by Filali and Kitanine [4,5,6].…”

Section: Discussionmentioning

confidence: 99%

“…[40], via the functional method [13,14]. On the other hand, for the domain-wall partition function of the Izergin-Korepin nineteen-vertex model, which was computed exactly at a particular root of unity [21], ingredient (ii b ) seems to fail because the analogue of (3.4) mixes the two creation operators B 1 and B 2 for that model.…”

Section: Further Examples 41 Comments On Applicabilitymentioning

confidence: 99%

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The Functional Method for the Domain-Wall Partition Function

Lamers¹

2018

SIGMA

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Abstract. We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domainwall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation we outline its analysis. The result is a closed expression in the form of a symmetrized sum -or, equivalently, multiple-integral formula -that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a 'crossing-symmetrized' sum with 2 L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.

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A new integral representation for the scalar products of Bethe states for the XXX spin chain

Kazama

Komatsu

Nishimura

2013

J. High Energ. Phys.

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Based on the method of separation of variables due to Sklyanin, we construct a new integral representation for the scalar products of the Bethe states for the SU(2) XXX spin 1/2 chain obeying the periodic boundary condition. Due to the compactness of the symmetry group, a twist matrix must be introduced at the boundary in order to extract the separated variables properly. Then by deriving the integration measure and the spectrum of the separated variables, we express the inner product of an on-shell and an off-shell Bethe states in terms of a multiple contour integral involving a product of Baxter wave functions. Its form is reminiscent of the integral over the eigenvalues of a matrix model and is expected to be useful in studying the semi-classical limit of the product. †

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Scalar Product of Bethe Vectors from Functional Equations

Abstract: In this work the scalar product of Bethe vectors for the six-vertex model is studied by means of functional equations. The scalar products are shown to obey a system of functional equations originated from the Yang-Baxter algebra and its solution is given as a multiple contour integral.

Cited by 15 publications

References 44 publications

Continuous representations of scalar products of Bethe vectors

Continuous representations of scalar products of Bethe vectors

The Functional Method for the Domain-Wall Partition Function

A new integral representation for the scalar products of Bethe states for the XXX spin chain

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