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arXiv:hep-th/0003276 · Journal of Physics A Mathematical and General

Difference equations for correlation functions of Belavin's ℤn-symmetric model with boundary reflection

Yas-Hiro Quano

Published Nov 9, 2000• 3 citations
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Abstract

Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.

DOI
10.1088/0305-4470/33/46/310

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Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.
Integral solutions to the quantum Knizhnik-Zamolodchikov equations for the correlation functions of both the bulk and boundary XXZ models in the anti-ferromagnetic regime are presented. The difference equations can be derived from Smirnov-type master equations for correlation functions on the basis of the CTM bootstrap. Our integral solutions with an appropriate choice of the integral kernel reproduce the formulae previously obtained by using the bosonization of the vertex operators of the quantum affine algebra Uq ().
Belavin's (ℤ/ n ℤ)-symmetric model is considered on the basis of bosonization of vertex operators in the A (1) n −1 model and vertex–face transformation. The corner transfer matrix (CTM) Hamiltonian of the (ℤ/ n ℤ)-symmetric model and tail operators are expressed in terms of bosonized vertex operators in the A (1) n −1 model. Correlation functions of the (ℤ/ n ℤ)-symmetric model can be obtained by using these objects, in principle. In particular, we calculate spontaneous polarization, which reproduces the result we obtained in 1993.
In this paper we compute the most general non-diagonal reflection matrices of the RSOS/SOS models and the hard hexagon model using the boundary Yang-Baxter equations. We find a new one-parameter family of the reflection matrices for the RSOS model in addition to the previous result obtained in Ahn C and Koo W M 1996 Nucl. Phys. B 468 [FS] 461. We also find three classes of the reflection matrices for the SOS model, which has one or two free parameters. For the hard-hexagon model which can be mapped to the RSOS(5) model by folding four RSOS heights into two, the solutions can be obtained similarly with a main difference in the boundary unitarity conditions. Due to this, the reflection matrices can have two free parameters. We show that these extra terms can be identified with the `decorated' solutions. We also generalize the hard hexagon model by `folding' the RSOS heights of the general RSOS( p ) model and show that they satisfy the integrability conditions such as the Yang-Baxter and boundary Yang-Baxter equations. These models can be solved using the results for the RSOS models.