SciPost Thesis Link

Title:  A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains
Author:  Stephen Gerard Mc Ateer
As Contributor:   Stephen Mc Ateer
Type: Ph.D.
Discipline: Physics
Subject area: Mathematical Physics
Approach: Theoretical
Degree granting institution:  The University of Melbourne
Supervisor(s): Omar Foda
Defense date:  2015-10-20


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(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 su_2 F-matrix equivalent to the algebraic expression of Maillet and Sanchez de Santos. Next we present a fully factorized expression for the su_n F-matrix which is of a similar form to that of Maillet and Sanchez de Santos [18] for the su_2 F-matrix and equivalent to the unfactorized expression of Albert, Boos, Flume and Ruhlig [2] for the su_n F-matrix. Using a diagrammatic description of the nested algebraic Bethe Ansatz, we present an expression for the eigenvectors of the sun transfer matrix as components of appropriately selected sun F-matrices. Finally, we present expressions for the sun elementary matrices (and therefore the local spin operators in the case of su_2) in terms of components of the sun monodromy matrix.

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