A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains (PhD thesis)
Suporvisor: Omar Foda (based on publications with Michael Wheeler)
Abstract: The aim of this thesis is a better understanding of certain mathematical structures which arise in integrable spin chains. Specifically, we are concerned with XXZ and XXX Heisenberg spin-1/2 chains and their generalizations.
The mathematical structures in question are the \( F \)-matrix (a symmetrising, change-of-basis operator) and the Bethe eigenvectors (the eigenvectors of the transfer matrix of integrable spin chains). A diagrammatic tensor notation represents these operators in a way which is intuitive and allows easy manipulation of the relations involving them. The sun F-matrix is a representation of a Drinfel’d twist of the \( R \)-matrix of the quantum algebra \( U_q(\mathfrak{su}_n) \) and its associated Yangian \( Y(su_n) \). The \( F \)-matrices of these algebras have proven useful in the calculation of scalar products and domain wall partition functions in the spin-1/2 XXZ model. In this thesis we present a factorized diagrammatic expression for the \( \mathfrak{su}_2 \) \( F \)-matrix equivalent to the algebraic expression of Maillet and Sanchez de Santos. Next we present a fully factorized expression for the \( \mathfrak{su}_n \) and \( F \)-matrix which is of a similar form to that of Maillet and Sanchez de Santos for the \( \mathfrak{su}_2 \) \( F \)-matrix and equivalent to the unfactorized expression of Albert, Boos, Flume and Ruhlig for the \( \mathfrak{su}_n \) \( F \)-matrix.
Using a diagrammatic description of the nested algebraic Bethe Ansatz, we present an expression for the eigenvectors of the \( \mathfrak{su}_n \) transfer matrix as components of appropriately selected \( \mathfrak{su}_n \) \( F \)-matrices. Finally, we present expressions for the sun elementary matrices (and therefore the local spin operators in the case of \( \mathfrak{su}_2 \) in terms of components of the \( \mathfrak{su}_n \) monodromy matrix.