Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Le modèle d'Izergin-Korepin

Alexandr Garbali

Published Sep 16, 2015• 0 citations
Publisher

Abstract

Among the models of interacting classical statistical mechanics the Yang–Baxter (YB) integrable systems play a special role. The central model in the theory of yb integrable systems is the six vertex model. many powerful techniques were developed to study the six vertex model. The model under consideration is the Izergin-Korepin (IK) nineteen vertex model, which can be viewed as a generalization of the six vertex model. Our aim is to understand the physics of the ik model using the extensions of the methods which were applied to the six vertex model. We review the algebraic bethe ansatz for the IK model based on the nineteen-vertex R-matrix and propose a new presentation for the eigenstate of the relevant transfer matrix. We also address the question of the calculation of the scalar products of the IK model. an important object in the theory of scalar products is the domain wall boundary partition function. for this partition function defined for the IK model we derive a recurrence relation and solve it in a special case. We move on to the representation theory of the underlying quantum group (U_q(A^{(2)}_{2})), for which we compute all higher dimensional irreducible representations which are relevant for the IK model (Kirillov-Reshetikhin (KR) modules). The latter is accomplished in the so-called drinfeld presentation of quantum groups. This presentation has technical advantages for computations of the R-matrices by means of the Khoroshkin-Tolstoy (KT) formula. We use this to compute the R-matrix in a tensor product of the fundamental representation and a generic higher dimensional kr module. On the other hand, the Drinfeld presentation makes apparent the connection between the borel subalgebras of the quantum group U_q(A^{(2)}_{2}) and the Q-deformed oscillator algebras (oscq). The latter algebras are closely related to the representation theoretic definition of special transfer matrices: the Q-operators; these operators are central in the theory of functional relations of integrable models. we use the oscq type algebras in the KT formula to compute some L-matrices which are used to build the Q-operators. finally, we consider a special case of the ground state of the IK model when the deformation parameter Q is equal to a root of unity. In this case we compute explicitly the ground state eigenvalues of various transfer matrices including the Q-operator. we use the latter result to compute the components of the ground state of the IK model for small systems.

DOI
10.70675/e0a78b25zf773z4e93zbae0ze5411adc95cd

Related papers

The Izergin-Korepin model

Alexandr Garbali

Le modèle d'Izergin-Korepin Parmi les modèles de mécanique statistique classique avec interaction les systèmes intégrables de Yang–Baxter (YB) jouent un rôle particulier. Le modèle central dans la théorie des systèmes intégrables YB est le modèle à six vertex. Plusieurs méthodes ont été développées pour étudier le modèle à six vertex. Notre but est de comprendre la physique du modèle à dix-neuf vertex d’Izergin–Korepin (IK), qui peut être vu comme une généralisation du modèle à six vertex. On donne une vue d’ensemble de l’Ansatz algébrique de Bethe pour le modèle IK basé sur la matrice R à dix-neuf vertex et on propose une nouvelle présentation pour les états propres de la matrice de transfert associée. On adresse aussi la question du calcul des produits scalaires pour le modèle IK. Un objet important dans la théorie des produits scalaires est la fonction de partition avec des conditions aux bords de domaine. Pour cette fonction de partition, définie pour le modèle IK, on obtient une relation de récurrence pour laquelle on trouve la solution dans un cas particulier. La théorie de la représentation du groupe quantique (U_q(A^{(2)}_{2})) associé au modèle IK nous permet d’obtenir toutes les représentations de dimension plus élevée pertinentes pour ce modèle (les modules de Kirillov–Reshetikhin (KR)). Ceci est réalisé dans la présentation de Drinfeld des groupes quantiques. Cette présentation a des avantages techniques quand on calcule les matrices R par la formule de Khoroshkin–Tolstoy (KT). On l’utilise pour calculer la matrice R évaluée sur le produit tensoriel de la représentation fondamentale et d’un module KR de dimension plus élevée. D’un autre côté, la présentation de Drinfeld montre la connexion entre les sous-algèbres de Borel du groupe quantique U_q(A^{(2)}_{2})) et les algèbres d’oscillateurs q-déformés (Oscq). Ces algèbres sont étroitement liées à la définition(par la théorie de la représentation) d’un certain type de matrices de transfert : les opérateurs Q; ces opérateurs jouent un rôle central dans la théorie des relations fonctionnelles des modèles intégrables. On utilise les algèbres de type Oscq dans la formule KT pour calculer quelques matrices L, qui sont utilisées pour construire les opérateurs Q. Finalement,on considère un cas particulier de l’état fondamental du modèle IK avec paramètre de déformation q égal à une racine de l’unité. Dans ce cas, on calcule explicitement les valeurs propres de différentes matrices de transfert, y compris de l’opérateur Q. On utilise ce dernier résultat pour obtenir l’état fondamental du modèle IK pour des petites tailles.

•Sep 16, 2015•1 citation

Functional relations in nineteen-vertex models with domain-wall boundaries

A. Bossart, W. Galléas

This work is concerned with functional properties shared by partition functions of nineteen-vertex models with domain-wall boundary conditions. In particular, we describe both Izergin-Korepin and Fateev-Zamolodchikov models with the aforementioned boundary conditions and show that their partition functions are governed by a system of functional equations originating from the associated Yang-Baxter algebra.

arXiv:1810.06482•Journal of Mathematical Physics•Oct 1, 2019•5 citations

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

Christian Korff

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.

arXiv:math-ph/0404028•Journal of Physics A Mathematical and General•Jul 8, 2004•15 citations

Q-operators in the six-vertex model

Vladimir V. Mangazeev

In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.

arXiv:1406.0662•Nuclear Physics B•Jul 2, 2014•31 citations

On the Yang–Baxter equation for the six-vertex model

Vladimir V. Mangazeev

In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.

arXiv:1401.6494•Nuclear Physics B•Feb 24, 2014•66 citations

Symmetric functions and wavefunctions of XXZ-type six-vertex models and elliptic Felderhof models by Izergin–Korepin analysis

Kohei Motegi

We present a method to analyze the wavefunctions of six-vertex models by extending the Izergin–Korepin analysis originally developed for domain wall boundary partition functions. First, we apply the method to the case of the basic wavefunctions of the XXZ-type six-vertex model. By giving the Izergin–Korepin characterization of the wavefunctions, we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed Grothendieck polynomials. As a second example, we show that the Izergin–Korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions. As a third example, we apply the method to the elliptic Felderhof model which is a face-type version and an elliptic extension of the trigonometric Felderhof model. We show that the wavefunctions can be expressed as one-parameter deformations of an elliptic analog of the Vandermonde determinant and elliptic symmetric functions.

arXiv:1703.07924•Journal of Mathematical Physics•May 1, 2018•10 citations
Among the models of interacting classical statistical mechanics the Yang–Baxter (YB) integrable systems play a special role. The central model in the theory of yb integrable systems is the six vertex model. many powerful techniques were developed to study the six vertex model. The model under consideration is the Izergin-Korepin (IK) nineteen vertex model, which can be viewed as a generalization of the six vertex model. Our aim is to understand the physics of the ik model using the extensions of the methods which were applied to the six vertex model. We review the algebraic bethe ansatz for the IK model based on the nineteen-vertex R-matrix and propose a new presentation for the eigenstate of the relevant transfer matrix. We also address the question of the calculation of the scalar products of the IK model. an important object in the theory of scalar products is the domain wall boundary partition function. for this partition function defined for the IK model we derive a recurrence relation and solve it in a special case. We move on to the representation theory of the underlying quantum group (U_q(A^{(2)}{2})), for which we compute all higher dimensional irreducible representations which are relevant for the IK model (Kirillov-Reshetikhin (KR) modules). The latter is accomplished in the so-called drinfeld presentation of quantum groups. This presentation has technical advantages for computations of the R-matrices by means of the Khoroshkin-Tolstoy (KT) formula. We use this to compute the R-matrix in a tensor product of the fundamental representation and a generic higher dimensional kr module. On the other hand, the Drinfeld presentation makes apparent the connection between the borel subalgebras of the quantum group U_q(A^{(2)}{2}) and the Q-deformed oscillator algebras (oscq). The latter algebras are closely related to the representation theoretic definition of special transfer matrices: the Q-operators; these operators are central in the theory of functional relations of integrable models. we use the oscq type algebras in the KT formula to compute some L-matrices which are used to build the Q-operators. finally, we consider a special case of the ground state of the IK model when the deformation parameter Q is equal to a root of unity. In this case we compute explicitly the ground state eigenvalues of various transfer matrices including the Q-operator. we use the latter result to compute the components of the ground state of the IK model for small systems.
Le modèle d'Izergin-Korepin Parmi les modèles de mécanique statistique classique avec interaction les systèmes intégrables de Yang–Baxter (YB) jouent un rôle particulier. Le modèle central dans la théorie des systèmes intégrables YB est le modèle à six vertex. Plusieurs méthodes ont été développées pour étudier le modèle à six vertex. Notre but est de comprendre la physique du modèle à dix-neuf vertex d’Izergin–Korepin (IK), qui peut être vu comme une généralisation du modèle à six vertex. On donne une vue d’ensemble de l’Ansatz algébrique de Bethe pour le modèle IK basé sur la matrice R à dix-neuf vertex et on propose une nouvelle présentation pour les états propres de la matrice de transfert associée. On adresse aussi la question du calcul des produits scalaires pour le modèle IK. Un objet important dans la théorie des produits scalaires est la fonction de partition avec des conditions aux bords de domaine. Pour cette fonction de partition, définie pour le modèle IK, on obtient une relation de récurrence pour laquelle on trouve la solution dans un cas particulier. La théorie de la représentation du groupe quantique (U_q(A^{(2)}_{2})) associé au modèle IK nous permet d’obtenir toutes les représentations de dimension plus élevée pertinentes pour ce modèle (les modules de Kirillov–Reshetikhin (KR)). Ceci est réalisé dans la présentation de Drinfeld des groupes quantiques. Cette présentation a des avantages techniques quand on calcule les matrices R par la formule de Khoroshkin–Tolstoy (KT). On l’utilise pour calculer la matrice R évaluée sur le produit tensoriel de la représentation fondamentale et d’un module KR de dimension plus élevée. D’un autre côté, la présentation de Drinfeld montre la connexion entre les sous-algèbres de Borel du groupe quantique U_q(A^{(2)}_{2})) et les algèbres d’oscillateurs q-déformés (Oscq). Ces algèbres sont étroitement liées à la définition(par la théorie de la représentation) d’un certain type de matrices de transfert : les opérateurs Q; ces opérateurs jouent un rôle central dans la théorie des relations fonctionnelles des modèles intégrables. On utilise les algèbres de type Oscq dans la formule KT pour calculer quelques matrices L, qui sont utilisées pour construire les opérateurs Q. Finalement,on considère un cas particulier de l’état fondamental du modèle IK avec paramètre de déformation q égal à une racine de l’unité. Dans ce cas, on calcule explicitement les valeurs propres de différentes matrices de transfert, y compris de l’opérateur Q. On utilise ce dernier résultat pour obtenir l’état fondamental du modèle IK pour des petites tailles.
We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
We present a method to analyze the wavefunctions of six-vertex models by extending the Izergin–Korepin analysis originally developed for domain wall boundary partition functions. First, we apply the method to the case of the basic wavefunctions of the XXZ-type six-vertex model. By giving the Izergin–Korepin characterization of the wavefunctions, we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed Grothendieck polynomials. As a second example, we show that the Izergin–Korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions. As a third example, we apply the method to the elliptic Felderhof model which is a face-type version and an elliptic extension of the trigonometric Felderhof model. We show that the wavefunctions can be expressed as one-parameter deformations of an elliptic analog of the Vandermonde determinant and elliptic symmetric functions.