Modified algebraic Bethe ansatz for XXZ chain on the segment – II – general cases

Belliard, S.; Pimenta, Rodrigo A.

doi:10.1016/j.nuclphysb.2015.03.016

Cited by 78 publications

(104 citation statements)

References 56 publications

(133 reference statements)

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“…. , u i }), using the notaion of [6]. It remains to find the overall constant and to prove the conjecture.…”

Section: ω|C (U) (28)mentioning

confidence: 97%

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model.…”

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confidence: 99%

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Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Belliard¹,

Pimenta²

2016

J. Phys. A: Math. Theor.

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Abstract. We consider the isotropic spin− 1 2Heisenberg chain with the most general integrable boundaries. The scalar product between the on-shell Bethe vector and its off-shell dual, obtained by means of the modified algebraic Bethe ansatz, is given by a modified Slavnov formula. The corresponding Gaudin-Korepin formula, i.e., the square of the norm, is also obtained.Introduction. The algebraic Bethe ansatz (ABA) [31,30] is a powerful technique to study the spectral problem of quantum integrable models, as well as to construct their correlation functions and compute physical quantities [23]. The possibility of expressing scalar products in a compact form [17,18,22,32] is a crucial aspect of the method. Nevertheless, the application of the usual ABA to obtain the spectrum and the eigenvectors of quantum integrable models becomes problematic for some important cases, in particular in the presence of the non-diagonal boundaries. In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model. We remember that such inhomogeneous term was firstly proposed in the context of the off-diagonal Bethe ansatz (ODBA) method (see [33] for a review) and recovered in the separation of variables (SoV) framework [24]. Let us also remark that the SoV basis [13] has been used to prove the off-shell equation for the creation operator in the MABA context [2] as well as to obtain the on-shell Bethe vector in the ODBA method [36,37].Once the spectral level is understood, the next natural step is to consider the evaluation of scalar products between Bethe vectors obtained in the MABA framework. The calculation of scalar products within this context is a primordial step to access the physical behavior of systems with general integrable open boundaries, since it paves the way to consider form factors and correlation functions. This task has been recently initiated in the case of the twisted XXX spin chain [7], a prototype model that can be described by the MABA. Modified Slavnov and Gaudin-Korepin formulas, i.e., compact expressions for the scalar product between an on-shell and an off-shell Bethe state and the square of the norm, have been conjectured. We propose in this note similar formulas for the XXX spin chain on the segment. They are given in t...

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“…. , u i }), using the notaion of [6]. It remains to find the overall constant and to prove the conjecture.…”

Section: ω|C (U) (28)mentioning

confidence: 97%

mentioning

confidence: 99%

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Belliard¹,

Pimenta²

2016

J. Phys. A: Math. Theor.

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“…On one hand, recall that for certain regimes of boundary parameters or generic parameters, through the Bethe ansatz [98,89,90,91], its modified versions [92,93,94,95], functional alternatives [96,97,99] or the separation of variables approach [87,100], eigenstates and eigenvalues are expressed in terms of (Bethe) roots of highly transcendental Bethe equations. It appears that an explicit characterization of the Hamiltonian's eigenfunctions as polynomials offers the possibility of studying the correspondence between Bethe roots and solutions of algebraic equations [101].…”

Section: Perspectivesmentioning

confidence: 99%

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

Baseilhac

Martín

2018

Journal of Mathematical Physics

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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.

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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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Modified algebraic Bethe ansatz for XXZ chain on the segment – II – general cases

Cited by 78 publications

References 56 publications

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

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

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

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