Hidden Grassmann Structure in the XXZ Model II: Creation Operators

Abstract

In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space ${\mathcal W_\alpha}$ of all quasi-local operators, which are products of the disorder field ${q^{\alpha \sum_{j=-\infty}^0\sigma ^3_j}}$ with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.