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arXiv:1810.06482 · Journal of Mathematical Physics

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

A. Bossart, W. Galléas

Published Oct 1, 2019• 5 citations
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Abstract

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.

DOI
10.1063/1.5095588

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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.
We introduce and study the domain wall boundary partition function of the integrable six-vertex model with triangular boundary. We first formulate the domain wall boundary partition function with triangular boundary by using the Uq(sl2)U_q(sl_2)Uq​(sl2​) RRR-matrix and a special class of the triangular KKK-matrix. By using its graphical representation, we make the Izergin-Korepin analysis with the help of the Yang-Baxter relation and the reflection equation to give a characterization of the partition function. The explict form of the symmetric function representing the partition function is presented by showing that it satisfies all the required properties.
Abstract We study the domain wall partition function Z N for the (Izergin–Korepin) integrable nineteen-vertex model on a square lattice of size N . Z N is a symmetric function of two sets of parameters: horizontal and vertical rapidities. For generic values of the parameter q we derive the recurrence relation for the domain wall partition function relating Z N +1 to , where P N is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables and . After setting the recurrence relation simplifies and we solve it in terms of a Jacobi–Trudi-like determinant of polynomials generated by P N .
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.