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arXiv:solv-int/9710024 · Journal of Physics A Mathematical and General

Complete non-diagonal reflection matrices of RSOS/SOS and hard hexagon models

Changrim Ahn, Choong-Ki You

Published Mar 6, 1998• 12 citations
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Abstract

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.

DOI
10.1088/0305-4470/31/9/003

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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.
The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.
In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular RRR matrices containing several free-parameters. The corresponding reflection KKK matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular KKK matrices, regardless of what RRR matrix we choose.
The S-matrix bootstrap is extended to a 1+1d theory with O(N) symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.
A bstract We propose explicit expressions for the boundary reflection matrices of the Am+(r,s){\mathcal{A}}_{m}+(r, s)Am​+(r,s) series of massive scattering theories, obtained by perturbing the Am{\mathcal{A}}_{m}Am​ unitary minimal models with ( r, s ) boundary conditions with both bulk and boundary ϕ 1 , 3 operators. We identify the vacua that live on the boundary with the allowed edges of the ( r, s ) conformal boundary conditions of the A m Andrews-Baxter-Forrester model. The boundary reflection matrices are then “direct sums” of certain pairs of A m− 1 Behrend-Pearce solutions of the boundary Yang-Baxter equation and are consistent with the boundary bootstrap and the recently-introduced crossing, as well as the Z 2 (height-reversal), Kac table and non-invertible symmetries.
On the multiparametric Uq[Dn+1(2)]{\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]Uq​[Dn+1(2)​] vertex model