On correlation functions in models related to the Temperley-Lieb algebra

TL;DR

This paper explores correlation functions in quantum spin chains related to the Temperley-Lieb algebra, proposing conjectures that connect these functions to known results in the XXZ model, with validation in specific representations.

Contribution

It introduces two conjectures linking Temperley-Lieb algebra-based spin chain correlations to XXZ model formulas, extending understanding of these models.

Findings

Conjectures relate mean values to XXZ correlation functions.

Factorization of product mean values into sums of current mean values.

Validation of conjectures in golden chain, Potts model, and trace representations.

Abstract

We deal with quantum spin chains whose Hamiltonian arises from a representation of the Temperley-Lieb algebra, and we consider the mean values of those local operators which are generated by the Temperley-Lieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of Temperley-Lieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are non-degenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the so-called golden chain, the $Q$-state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.