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Yang-Baxter equations for systems with boundaries and defects

Mattias Andersson

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

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.

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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.
We explore the reflection–transmission quantum Yang–Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow us to establish, for the first time, a direct relationship with the different previous works on the subject and make evident the advantages of the reflection–transmission algebra as a universal approach to integrable systems with impurities.
Recently, researchers have proposed the Asymmetric Bethe ansatz method - a theoretical tool that extends the scope of Bethe ansatz-solvable models by "breaking" partial mirror symmetry via the introduction of a fully reflecting boundary. Within this framework, the integrability conditions which were originally put forward by Gaudin have been further generalized. In this work, building on Gaudin's generalized kaleidoscope model, we present a detailed investigation of the relationship between DN symmetry and its integrability. We demonstrate that the mathematical essence of integrability in this class of models is characterized by a newly proposed Kaleidoscope Yang-Baxter Equation. Furthermore, we show that the solvability of a model via the coordinate Bethe ansatz depends not only on the consistency relations satisfied by scattering matrices, but also on the model's boundary conditions and the symmetry of the subspace where solutions are sought. Through finite element method based numerical studies, we further confirm that Bethe ansatz integrability arises in a specific symmetry sector. Finally, by analyzing the algebraic structure of the Kaleidoscope Yang-Baxter Equation, we derive a series of novel quantum algebraic identities within the framework of quantum torus algebra.
We establish that the Yang-Baxter equations in the presence of an impurity can in general only admit solutions of simultaneous tranmission and reflection when the transmission and reflection amplitudes commute in the defect degrees of freedom with an additional exchange of the corresponding rapidities. In the absence of defect degrees of freedom we show in complete generality, that the only exceptions to this are theories which possess rapidity independent bulk scattering matrices. In particular bulk theories with diagonal scattering matrices, can only be the free Boson and Fermion, the Federbush model and their generalizations. These anyonic solutions do not admit the possibility of excited impurity states.
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.