Diagonalization of the XXZ Hamiltonian by Vertex Operators

TL;DR

This paper presents a novel algebraic method to diagonalize the anti-ferroelectric XXZ Hamiltonian directly in the thermodynamic limit using quantum affine algebra representation theory and vertex operators.

Contribution

It introduces a new approach that bypasses finite lattice and Bethe Ansatz methods by employing quantum affine algebra and vertex operators for diagonalization.

Findings

Successfully diagonalized the XXZ Hamiltonian in the thermodynamic limit.

Validated results against Bethe Ansatz and scaling limit analyses.

Provided explicit construction of eigenvectors using vertex operators.

Abstract

We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of affine U_q( sl(2) ). Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit --- the $su(2)$-invariant Thirring model.