The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians

Details The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians

2012-03-22

arxiv.org/abs/1203.4996v1

Physics

Mathematical Physics

29 pages

Scientific paper

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has $N$ sites. A new family of link modules of the periodic Temperley-Lieb algebra $\mathcal ETLP_N(\beta, \alpha)$ is introduced. These extend the link modules $\tilde V_N^d$ (standard modules) that are labeled by the numbers of sites $N$ and of defects $d$. Beside the defining parameters $\beta=u^2+u^{-2}$ with $u=e^{i\lambda/2}$ (weight of contractible loops) and $\alpha$ (weight of non-contractible loops), this family also depends on a {\em twist parameter} $v$ that keeps track of how the defects wind around the cylinder. The transfer matrix $T_N(\lambda, \nu)$ depends on an anisotropy parameter $\nu$ and the number $\lambda$ that fixes the model. (The thermodynamic limit of $T_N$ is believed to describe a conformal field theory of central charge $c=1-6\lambda^2/(\pi(\lambda-\pi))$.) The family of periodic XXZ Hamiltonians is similarly extended to depend on this new parameter and the relationship between this family and the loop models is established. The Gram determinant for a bilinear form on the new link modules is computed and a linear map $\tilde i_N^d$ between these modules and the subspaces of fixed eigenvalue of $S^z$ of the XXZ models is shown to be an isomorphism for generic values of $u$ and $v$. The critical values of the parameters for which $\tilde i_N^d$ fails to be an isomorphism are given.

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