Determinant formula for the six - vertex model with reflecting end Osamu Tsuchiya

Using the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six vertex model with reflecting end. E-mail address: [email protected] 1 1 Introducntion Determinant representations of correlation functions of one dimensional quantum integrable models were studied extensively[1]. Especially Essler et. al. studied the correlation functions of XXZ model with periodic boundary conditions [4]. In the calculation of scalar products of the XXZ model, the relation between the scalar products of the model and the partition function of the six-vertex with domain wall boundary conditions is essential [2]. Izergin et. al. used this relation to obtain the determinant formula for the partition function of the six-vertex model with domain wall boundary conditions [5]. On the other hand, integrable models with open boundary conditions are interesting. Integrable model with open boundary conditions are related to the BN type Weyl group [6] and in condenced matter physics they are related to the impurity problems in the Luttinger liquid [7]. Correlation function of the 1 + 1 dimensional delta function interacting Bose gas with open boundary conditions were studied in [8]. For the XXZ model with open boundary conditions, the algebraic Bethe ansatz of the model can be used to calculate the partition function of the six-vertex model with reflecting boundary condition at one boundary and domain wall boundary conditions at other three boundaries. (In this paper we call six vertex model with this type of boundary conditions six vertex model with reflecting end.) In this paper, we obtain the determinant formula for the above six-vertex model using above relation. 2 Quantum Inverse ScatteringMethod for the Open XXZ Model Let us start with a brief review of the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions [9]. The trigonometrical solution of the Yang-Baxter equation is given by R00′(λ) = 1 2 {[ sinh(λ− 2iη)− sinh(2iη) sinh(λ) ] 10 ⊗ 10′ + σ x 0 ⊗ σ x 0 + σ y 0 ⊗ σ y 0 (1) + [ sinh(λ− 2iη) + sinh(2iη) sinh(λ) ]