On the Two-Point Correlation Function for the $U_q[SU(2)]$ Invariant Spin One-Half Heisenberg Chain at Roots of Unity

TL;DR

This paper investigates two-point correlation functions in a quantum spin chain with $U_q[SU(2)]$ symmetry at roots of unity, using tensor calculus, algebraic elements, and diagrammatic methods, revealing connections to known models and special cases.

Contribution

It introduces a tensor calculus approach to compute two-point functions in the $U_q[SU(2)]$ invariant spin chain at roots of unity, linking algebraic elements to physical correlation functions.

Findings

Correlation functions are trivial at $q=e^{i \\pi/3}$.

At $q=e^{i \\pi/4}$, correlation functions relate to Majorana fields.

At $q=e^{i \\pi/6}$, they connect to parafermions of the three-state Potts model.

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 $q=e^{i \pi/3}$, all correlation functions are (trivially) zero, for $q=e^{i \pi/4}$, 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 $q=e^{i \pi/6}$, 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.