Why scalar products in the algebraic Bethe ansatz have determinant representation

Belliard, S.; Slavnov, N. A.

doi:10.1007/jhep10(2019)103

Cited by 35 publications

(38 citation statements)

References 49 publications

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“…However, the fact that the transfer matrix eigenstates can be re-expressed as Bethe-type states involving the multiple action of an element of the monodromy matrix as in (31)-(32) is not completely general in the SoV approach, but rather a specificity of models for which the Q-functions have the same functional form as the transfer matrix eigenfunctions of the model: for instance, it is not true in the anti-periodic XXZ model, for which the Q-functions have a double periodicity with respect to the transfer matrix eigenfunctions of the model [110,131]. So as to remain as general as possible, it is therefore better to start directly from (76) and 15- (17), (22).…”

Section: Left Action On Separate Statesmentioning

confidence: 99%

“…In particular, the obtention of a generalization of the Slavnov's formula [30] for the scalar products of Bethe states, and more generally of a similar determinant representation for the matrix elements of local operators, as in [17], may be a very difficult problem if the combinatorial structure of the Bethe states is too involved. This is for instance the case in the XYZ model, for which first results about scalar products within ABA were obtained only very recently in [73] (but for which the obtention of a compact formula for matrix elements of local operators in finite volume remains an open problem 3 ), using the fact that the scalar products of on-shell/off-shell Bethe vectors can be characterized as solutions to a system of linear equations, as initially proposed in [76]. This is also the case for models based on higher rank algebras, see for instance the works [77][78][79][80][81][82][83][84][85].…”

Section: Introductionmentioning

confidence: 99%

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Correlation functions by separation of variables: the XXX spin chain

2021

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We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

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Section: Left Action On Separate Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Correlation functions by separation of variables: the XXX spin chain

2021

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“…Recently, a new method was proposed in [49], which avoids all combinatorial difficulties and allows one to reduce the calculation of scalar products of on-shell and off-shell Bethe vectors in models with the 6-vertex R-matrix to solving a system of linear equations. This method explains why the scalar products of on-shell and off-shell Bethe vectors have determinant representations.…”

Section: Jhep06(2020)123mentioning

confidence: 99%

“…The lack of compact determinant representations for scalar products in the 8-vertex model (or the XY Z chain) is a serious obstacle for the study of the correlation functions within the framework of the QISM. At the same time, the method of [49] relies solely on the formula for the action of the transfer matrix on the Bethe vectors. For the XY Z chain, this formula was obtained in pioneer work [34].…”

Section: Jhep06(2020)123mentioning

confidence: 99%

Scalar products of Bethe vectors in the 8-vertex model

Slavnov

Zabrodin

Zotov

2020

J. High Energ. Phys.

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We obtain a determinant representation of normalized scalar products of onshell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solvability and solve it in terms of determinants of explicitly known matrices.

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“…An important progress for the unrestricted cases was achieved by the introduction of the offdiagonal Bethe ansatz [CYSW13], a method that proposes an inhomogeneous Baxter T-Q equation as solution of the spectral problem for integrable models without U (1) symmetry [WYCS]. Beyond the computation of the spectrum, a modification of the algebraic Bethe ansatz was developed in [BC13,Be15,C14,BP15,ABGP15] providing the construction of the associated off-shell Bethe states 6 , which in particular allows the computation of scalar products between on-shell/off-shell Bethe states, see [BP15a,BP15b,BS19a,BS19b] and references therein. The main feature of the modified algebraic Bethe ansatz are the off-shell relations satisfied by the 'creation operators', see e.g.…”

Section: Introductionmentioning

confidence: 99%

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

Baseilhac

Pimenta

2019

Nuclear Physics B

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An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of 'symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.MSc: 81R50; 81R10; 81U15.

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Why scalar products in the algebraic Bethe ansatz have determinant representation

Cited by 35 publications

References 49 publications

Correlation functions by separation of variables: the XXX spin chain

Correlation functions by separation of variables: the XXX spin chain

Scalar products of Bethe vectors in the 8-vertex model

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

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