Communications in Mathematical Physics

, Volume 151, Issue 1, pp 89–153 | Cite as

Diagonalization of theXXZ Hamiltonian by vertex operators

  • Brian Davies
  • Omar Foda
  • Michio Jimbo
  • Tetsuji Miwa
  • Atsushi Nakayashiki


We diagonalize the anti-ferroelectricXXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(2))\). 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—thesu(2)-invariant Thirring model.


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Note added in proof. The subject of the present article is developed further in the following papers

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Brian Davies
    • 1
    • 3
  • Omar Foda
    • 2
  • Michio Jimbo
    • 4
  • Tetsuji Miwa
    • 5
  • Atsushi Nakayashiki
    • 6
  1. 1.Mathematics Department, the FacultiesThe Australian National UniversityCanberraAustralia
  2. 2.Institute for Theoretical PhysicsUniversity of NijmegenNijmegenThe Netherlands
  3. 3.Department of MathematicsUniversity of MelbourneParkvilleAustralia
  4. 4.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan
  5. 5.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  6. 6.The Graduate School of Science and TechnologyKobe UniversityKobeJapan

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