A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

Abstract

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/2} (weight of contractible loops) and α(weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.

Replacement of "The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians", 31 pages

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• Eigenvalue amplitudes of the Potts model on a torus
• Combinatorial aspects of boundary loop models
• Standard Modules, Induction and the Temperley-Lieb Algebra
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Cited by in corpus (10)

• Logarithmic Conformal Field Theory: a Lattice Approach
• Modular invariant partition function of critical dense polymers
• Koornwinder polynomials and the XXZ spin chain
• Jordan cells of periodic loop models
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• All local conserved quantities of the XXZ model
• Tate cycles on some quaternionic Shimura varieties mod p
• Spin chains as modules over the affine Temperley-Lieb algebra
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• Cyclic representations of the periodic Temperley Lieb algebra, complex Virasoro representations and stochastic processes