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A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains

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Show simple item record McAteer, Stephen Gerard 2016-02-26T02:13:37Z 2016-02-26T02:13:37Z 2015
dc.description © 2015 Dr. Stephen Gerard McAteer
dc.description.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(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. en_US
dc.subject mathematical physics en_US
dc.subject F-matrix en_US
dc.subject Drinfel'd twist en_US
dc.subject quantum spin chains en_US
dc.subject Bethe Ansatz en_US
dc.title A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains en_US
dc.type PhD thesis en_US
melbourne.affiliation.department Mathematics and Statistics
melbourne.affiliation.faculty Science
melbourne.type Thesis
melbourne.accessrights Open Access

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