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.

Sep 16, 20150 citations

Properties of eigenstates of the six-vertex model with twisted and open boundary conditions

Heng Fan, Bo-Yu Hou, Kangjie Shi

We use the method proposed by Izergin and Korepin to discuss the Bethe ansatz equations of the six-vertex model with twisted and open boundary conditions. Let two parameters of the Bethe wave functions be equal, then an additional Bethe ansatz equation arises. For the open boundary condition case, we also have discussed the dual pseudovacuum state and have written out the simplest scalar products of the Bethe wave function.

Journal of Mathematical PhysicsJul 1, 19970 citations