Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases

The modified algebraic Bethe ansatz, introduced by Cramp\'e and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to $N$ the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with two upper triangular boundaries.Comment: V2: published version, some typos were corrected and one remark (5.2) on scalar product was adde

Slavnov and Gaudin-Korepin Formulas for Models without U(1){\rm U}(1) Symmetry: the Twisted XXX Chain

We consider the XXX spin-$\frac{1}{2}$ Heisenberg chain on the circle with an arbitrary twist. We characterize its spectral problem using the modified algebraic Bethe anstaz and study the scalar product between the Bethe vector and its dual. We obtain modified Slavnov and Gaudin-Korepin formulas for the model. Thus we provide a first example of such formulas for quantum integrable models without ${\rm U}(1)$ symmetry characterized by an inhomogenous Baxter T-Q equation

Drinfeld J Presentation of Twisted Yangians

We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra

Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric RR-Matrix

We study quantum integrable models with GL(3) trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix

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

The spectral problem of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to $N$, the length of the chain, and which satisfies a set of Bethe equations with an additional term.Comment: V2 published versio

Applications de l'ansatz de Bethe Algébrique et au-delà

Dans cette thèse, nous discuterons des systèmes intégrables quantiques et des chaînes de spins. Nous présenterons la notion d'intégrabilité quantique ainsi que des structures mathématiques, les groupes quantiques, reliées à cette dernière. Cela nous permettra d'introduire les chaînes de spins universelles étudiées par le groupe d' Annecy depuis plusieurs années. Ces chaînes universelles ont la particularité d'englober l'ensemble des chaînes de spins préalablement étudiées dans la littérature. La question posée pour cette thèse était d'utiliser l'ansatz de Bethe algébrique pour déterminer les valeurs propres et les vecteurs propres de ces chaînes de spins universelles . Nous discuterons donc cette méthode pour les chaînes de spins périodiques et avec bords. Cette étude mettra en évidence les limites de l'ansatz de Bethe algébrique pour certaines chaînes avec bords et nous présenterons un nouveau cadre mathématique qui permettrait d'obtenir le spectre dans ces cas. Nous discuterons aussi le problème du produit scalaire des vecteurs propres obtenus grâce à l'ansatz de Bethe algébrique.In this thesis, we will discuss quantum integrable systems and spin chains. We will present the notion of quantum integrability and a related algebraic structure, the quantum group. This study allows us to introduce the universal spin chains used by the Annecy group few years ago. These universal chains encompass all the spins chains studied in the literature. The purpose of this thesis is to evaluate, with the algebraic Bethe ansatz (ABA), the eigenvalues and eigenvectors of these universal spins chains. We will discuss the case of closed and open spin chains. This study will highlight the limit of the ABA for open spins chains and we will present a new mathematical framework that may allow to find the spectral problem in this case. We will also discuss the computation of the scalar product between two eigenvectors obtained with the ABA.CHAMBERY -BU Bourget (730512101) / SudocSudocFranceF

An attractive basis for the q−Onsager algebra

14 pagesLet A,A∗ be the fundamental generators of the q−Onsager algebra. A linear basis for the q−Onsager algebra is known as the `zig-zag' basis [IT09]. In this letter, an attractive basis for the q−Onsager algebra is conjectured, based on the relation between the q−Onsager algebra and a quotient of the infinite dimensional algebra Aq introduced in [BK05]