Algebraic and Analytic Properties of the One-Dimensional Hubbard Model

TL;DR

This paper explores the algebraic and analytic structures of the one-dimensional Hubbard model, emphasizing the properties of its R-matrix, monodromy matrix, and conserved quantities, with implications for its symmetries and invariances.

Contribution

It provides a detailed analysis of the R-matrix and monodromy matrix of the Hubbard model, including their properties, parametrization, and symmetry transformations, which were previously not fully explored.

Findings

Meromorphic parametrization of the transfer matrix using elliptic functions

Identification of the momentum operator within the transfer matrix expansion

Proof of su(2)⊕su(2) invariance for even-site chains

Abstract

We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that $R$-matrix and monodromy matrix of the Hubbard model, which are known since ten years now, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the $\h$-pairing su(2) Lie algebra. Our results imply su(2)$\oplus$su(2) invariance of the transfer matrix for the model on a chain with an even number of sites.