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arXiv:1205.2101

Riemann-Hilbert Approach to the Six-Vertex Model

Pavel Bleher, Karl Liechty

Published May 9, 2012• 0 citations
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Abstract

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $n\times n$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $n$ asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.

DOI
10.48550/arxiv.1205.2101

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Le modèle d'Izergin-Korepin

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•Sep 16, 2015•0 citations
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite nnn by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an n×nn\times nn×n Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large nnn asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta 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.