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 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. Besides the defining parameters β = u2 + u−2 with u = eiλ/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 TN(λ, ν) depends on the anisotropy ν and the spectral parameter λ that fixes the model. (The thermodynamic limit of TN is believed to describe conformal field theory of central charge c = 1 − 6λ2/(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 between these link representations and the eigenspaces of Sz 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 fails to be an isomorphism are given.