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

Abstract

Using U q [SU(2)] tensor calculus we compute the two-point scalar operators (TPSO), their averages on the ground-state give the two-point correlation functions. The TPSOs are identified as elements of the Temperley-Lieb algebra and a recurrence relation is given for them. We have not tempted to derive the analytic expressions for the correlation functions in the general case but got some partial results. For q = e iπ/3 , all correlation functions are (trivially) zero, for q = e iπ/4 , they are related in the continuum to the correlation functions of left-handed and right-handed Majorana fields in the half plane coupled by the boundary condition. In the case q = e iπ/6 , one gets the correlation functions of Mittag's and Stephen's parafermions for the three-state Potts model. A diagrammatic approach to compute correlation functions is also presented.

A DMRG study of the -symmetric Heisenberg chain

“…These values were also obtained directly by relating the three-state Potts model to a Gaussian model [26]. From the explicit form of the parafermion operators (products of local and string operators) [19], it can be seen that they have spinor properties as discussed in [26] with s = 1/3. Thus one would expect a common exponent x = 0.467 for all four g's, whereas the numerical values lie about 10% higher and lower.…”

“…Our main effort involved correlations. The functions which we studied were q-symmetric generalizations of the quantity < σ l σ m > and correspond to fermionic and parafermionic correlators in the Ising and Potts case, respectively [19]. In a previous study with exact diagonalizations of short systems no conclusive results could be obtained for the Potts model [20].…”

PT symmetry on the lattice: the quantum group invariantXXZspin chain

“…This is particularly important for the calculation of physically relevant quantities such as correlation functions where one considers matrix elements of local operators [26]. While the expression (88) is especially convenient for numerical computations, an alternative formulation in terms of the relevant algebras would be desirable in order to apply more powerful mathematical techniques.…”

Numerical investigation of correlation functions for the UqSU(2) invariant spin-1/2 Heisenberg chain

“…For m = 5 there is a quotient of T N (q) which gives a three state Potts model, but the correlation function has not been computed in this case. In this article we present a numerical investigation of the correlation functions in the cases m = 3; 5; 2 3 corresponding to Ising model, three state Potts model, and Lee{Yang edge singularity respectively. The rst case is included to estimate the accuracy of the numerical results.…”

“…For m = 5 we nd four di erent correlation functions hg k;l i, depending on k and l odd or even. In the case m = 2 3 there is only one correlation function. We attempt to nd critical exponents and to identify conformal elds that correspond to the continuum limit of the two-point scalar operators.…”