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arXiv:1703.07924 · Journal of Mathematical Physics

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

Kohei Motegi

Published May 1, 2018• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4986534

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On elliptic quantum integrability: vertex models, solid-on-solid models and spin chains

Jules Lamers

In this thesis the topic of quantum integrability is investigated, with the elliptic case constituting the core of the thesis. The first part of the thesis is about the exact computation of the partition function governing a six-vertex or solid-on-solid model on a lattice of arbitrary but fixed size for a specific choice of boundary conditions. Exact expressions for these quantities were already available due to the work of Korepin and Izergin and others. Using the approach put forward by Galleas we study the partition functions from another point of view. We show that this approach contains that of Korepin-Izergin, while offering an algorithm that allows one to construct, rather than guess, a formula for the partition function. This yields a new expression for that function in the case of a reflecting end and domain walls on the three other ends. The shorter second part of the thesis is about the question whether the partially anisotropic (think: XXZ) version of Inozemtsev's elliptic spin chain is exactly solvable. This model, interpolating between the XXZ and Haldane-Shastry spin chains, is very interesting from a theoretical viewpoint. Inozemtsev's exact solution of the original fully isotropic (XXX) model is rather intricate, and it would be very interesting to know whether that is an isolated case or part of a more general pattern. A few other modifications of Inozemtsev's spin chain are known to be exactly solvable, yet, unlike the partially isotropic version, those models are not 'continuously connected' to (deformations of) Inozemtsev's spin chain. Although we have not yet found a satisfying answer this question is very interesting, and our findings so far might be of interest to other researchers. Quantum integrability and exact solvability are beautiful topics in mathematical physics. They come with algebraic and analytic structures that provide powerful tools enabling one to analyse the models in great detail. The unavoidable consequence, however, is that this field can be quite technical, which tends to makes it rather inaccessible. Therefore a fair portion of this thesis is devoted to an introduction aimed at non-experts with the hope of making the remainder more accessible. These pieces may also be useful by themselves as introductions to quantum integrability in vertex models and exact solvability in spin chains.

•Jun 1, 2016•2 citations
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.
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.
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.
Abstract. We consider the trigonometric Felderhof model, of free fermions in an external field, on a finite lattice with domain wall boundary conditions. The vertex weights are functions of rapidities and external fields. We obtain a determinant expression for the partition function in the special case where the dependence on the rapidities is eliminated, but for general external field variables. This determinant can be evaluated in product form. In the homogeneous limit, it is proportional to a 2-Toda τ function. Next, we use the algebraic Bethe ansatz factorized basis to obtain a product expression for the partition function in the general case with dependence on all variables. In [1], Korepin introduced domain wall boundary conditions for the six vertex model on a finite square lattice, and proposed recursion relations that determine the corresponding domain wall partition function. In [2], Izergin obtained a determinant solution of Korepin’s recursion relations. At the free fermion point, the
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author.We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domainwall boundary conditions.At the heart of the method lies a linear functional equation for the partition function.After deriving this equation we outline its analysis.The result is a closed expression in the form of a symmetrized sum -or, equivalently, multiple-integral formula -that can be rewritten to recover Izergin's determinant.Special attention is paid to the relation with other approaches.In particular we show that the Korepin-Izergin approach can be recovered within the functional method.We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end.We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine.Our result takes the form of a 'crossing-symmetrized' sum with 2 L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model.Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.
In this thesis the topic of quantum integrability is investigated, with the elliptic case constituting the core of the thesis. The first part of the thesis is about the exact computation of the partition function governing a six-vertex or solid-on-solid model on a lattice of arbitrary but fixed size for a specific choice of boundary conditions. Exact expressions for these quantities were already available due to the work of Korepin and Izergin and others. Using the approach put forward by Galleas we study the partition functions from another point of view. We show that this approach contains that of Korepin-Izergin, while offering an algorithm that allows one to construct, rather than guess, a formula for the partition function. This yields a new expression for that function in the case of a reflecting end and domain walls on the three other ends. The shorter second part of the thesis is about the question whether the partially anisotropic (think: XXZ) version of Inozemtsev's elliptic spin chain is exactly solvable. This model, interpolating between the XXZ and Haldane-Shastry spin chains, is very interesting from a theoretical viewpoint. Inozemtsev's exact solution of the original fully isotropic (XXX) model is rather intricate, and it would be very interesting to know whether that is an isolated case or part of a more general pattern. A few other modifications of Inozemtsev's spin chain are known to be exactly solvable, yet, unlike the partially isotropic version, those models are not 'continuously connected' to (deformations of) Inozemtsev's spin chain. Although we have not yet found a satisfying answer this question is very interesting, and our findings so far might be of interest to other researchers. Quantum integrability and exact solvability are beautiful topics in mathematical physics. They come with algebraic and analytic structures that provide powerful tools enabling one to analyse the models in great detail. The unavoidable consequence, however, is that this field can be quite technical, which tends to makes it rather inaccessible. Therefore a fair portion of this thesis is devoted to an introduction aimed at non-experts with the hope of making the remainder more accessible. These pieces may also be useful by themselves as introductions to quantum integrability in vertex models and exact solvability in spin chains.