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Bussei Kenkyu

Multivariable Vertex Models, the Temperley-Lieb Algebra and Link Polynomials

Tetsuo Deguchi

Published Feb 20, 1992• 0 citations
OpenAlex

Abstract

We present a family of multivariable solvable vertex models associated with representations of the Temperley-Lieb algebra. We discuss free energy and correlation lengths for the multivariable models and also for the fusion models. We also discuss connection of the multivariable vertex models to Jones polynomials and link polynomials given by Akutsu-Wadati, which are associated with representations of U_q(sl(2, C)).

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We present a family of multivariable solvable vertex models associated with representations of the Temperley-Lieb algebra. We discuss free energy and correlation lengths for the multivariable models and also for the fusion models. We also discuss connection of the multivariable vertex models to Jones polynomials and link polynomials given by Akutsu-Wadati, which are associated with representations of U_q(sl(2, C)).
This work concerns studies of boundary integrability of the vertex models from representations of the Temperley–Lieb algebra associated with the quantum group 𝒰 q [ X n ] for the affine Lie algebras X n = , , and . A systematic computation method is used to construct solutions of the boundary Yang–Baxter equations. We find a 2 n 2 + 1 free parameter solution for spin-( n − 1/2) and vertex models. It turns out that for spin- n , and vertex models, the solution has 2 n 2 + 2 n + 1 free parameters.
This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group Uq[Xn]{\cal U}_{q}[X_{n}]Uq​[Xn​] for the affine Lie algebras XnX_{n}Xn​ = A1(1)A_{1}^{(1)}A1(1)​, Bn(1)B_{n}^{(1)}Bn(1)​, Cn(1)C_{n}^{(1)}Cn(1)​ and Dn(1)D_{n}^{(1)}Dn(1)​. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a 2n2+12n^{2}+12n2+1 free parameter solution for $A_{1}^{(1)} $ spin-(n−1/2)(n-1/2)(n−1/2) and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-nnn, $ B_{n}^{(1)}$ and Dn(1)D_{n}^{(1)}Dn(1)​ vertex models, the solution has 2n2+2n+12n^{2}+2n+12n2+2n+1 free parameters.
The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley--Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra [1]. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physical properties related to the generalised Onsager algebras, hinting at a general framework that includes families of quantum lattice models possessing the structure of the generalised Onsager algebras.
Abstract The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.