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Communications in Theoretical Physics

Solutions to Braid Group, Yang-Baxter Equation and Integrable Fermion Systems

H. Yan, J. Zhang

Published Oct 30, 1996• 0 citations
Publisher

Abstract

Braid group representations are found in ferrnion systems in one space dimension. Explicit baxterization is performed to find corresponding new trigonometric solutions of Yang-Baxter equation. The quantum algebra structures implied in the new solutions are discovered. Algebraic Bethe ansatz method is applied to solving these systems. The relationships between these fermion systems and polaron model, spin- Heisenberg spin chain are discussed.

DOI
10.1088/0253-6102/26/3/295

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Braid group representations are found in ferrnion systems in one space dimension. Explicit baxterization is performed to find corresponding new trigonometric solutions of Yang-Baxter equation. The quantum algebra structures implied in the new solutions are discovered. Algebraic Bethe ansatz method is applied to solving these systems. The relationships between these fermion systems and polaron model, spin- Heisenberg spin chain are discussed.
The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.
In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
Abstract Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang–Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in Maity et al (2025 J. High Energy Phys 5 1–31) In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori–Hecke algebra and the Birman–Murakami–Wenzl algebra. As a result, we can Baxterize the constant Yang–Baxter solutions to yield spectral parameter dependent R -matrices. The regularity of these R -matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are ‘almost’ regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant 4 × 4 invertible Yang–Baxter solutions. Apart from reproducing nearest-neighbor interaction spin 1 2 systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.