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arXiv:hep-th/9310119 · Nuclear Physics B

On the two-point correlation functions for the Uq[SU(2)] invariant spin one-half Heisenberg chain at roots of unity

Haye Hinrichsen B, Paul Martin, V. Rittenberg, Manfred Scheunert

Published Mar 1, 1994• 16 citations
arXivHTMLPDF
DOI
10.1016/0550-3213(94)90302-6

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We consider the U q SU(2) invariant spin-1/2 XXZ quantum spin chain at the roots of unity q=exp(i pi /(m+1)), corresponding to different minimal models of conformal field theory. We conduct a numerical investigation of the correlation functions of U q SU(2) scalar two-point operators in order to find which operators in the minimal models they correspond to. Using graphical representations of the Temperley-Lieb algebra we are able to deal with chains of up to 28 sites. Depending on q, the correlation functions show different characteristics and finite-size behaviour. For m=2/3, which corresponds to the Lee-Yang edge singularity, we find the surface and bulk critical exponent -1/5. Together with the known result in the case m=3 (Ising model) this indicates that in the continuum limit the two-point operators involve conformal fields of spin-m-1/m+1. For other roots of unity q the chains are too short to determine the surface and bulk critical exponents.
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We demonstrate that the transfer matrix of the inhomogeneous N -state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with N th root-of-unity and representation parameter for odd N . The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ (2) models. In particular, the spin-( N −1)/2 XXZ chain model with and the homogeneous N -state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q 72 operator of the root-of-unity eight-vertex model, we construct the Q R , Q L and Q operators of a superintegrable τ (2) model, then identify them with transfer matrices of the N -state chiral Potts model for a positive integer N . We thus obtain a new method of producing the superintegrable N -state chiral Potts transfer matrix from the τ (2) model by constructing its Q operator.