Correspondence Between the XXZ Model in Roots of Unity and the One-Dimensional Quantum Ising Chain with Different Boundary Conditions

  • R. A. Usmanov
Part of the NATO Science Series book series (NAII, volume 35)


We consider the integrable XXZ model with special open boundary conditions that renders its Hamiltonian U q (sl 2) symmetric, and the one-dimensional quantum Ising model with four different boundary conditions. We show that for each boundary condition the Ising quantum chain is exactly given by the Minimal Model of integrable lattice theory LM(3, 4). This last theory is obtained as the result of the quantum group reduction of the XXZ model at anisotropy Δ(q+q −1)/2=√/2, with a number of sites in the latter defined by the type of boundary conditions.


Ising Model Transfer Matrix Configuration Space Transfer Matrice Mixed Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Behrend, R.E., Peaxce, P.A. and O’Brien, D.L. (1995) Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy. Preprint University of Melbourne, hep-th/9507118.Google Scholar
  2. 2.
    O’Brien, D.L., Pearce, P.A. and Warnaar, S.O. (1996) Finitized Conformal Spectrum of the Ising Model on the Cylinder and Torus. Physica A 228 63–77.MathSciNetADSGoogle Scholar
  3. 3.
    Alcaraz, F.C., Barber, M.N., and Batchelor, M.T. (1987) Phys. Rev. Lett. 58, 771; Alcaraz, F.C., Barber, M.N., Batchelor, M.T., Baxter, R.J. and Quispel, G. R. W. (1987) J. Phys. A20, 6397.MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Pasqier, V. and Saleur, H. (1990) Nucl.Phys. B330, 523.ADSCrossRefGoogle Scholar
  5. 5.
    Belavin, A. and Stroganov, Yu. Minimal Models of Integrable Lattice Theory and Truncated Functional Equations, preprint hep-th/9908050.Google Scholar
  6. 6.
    Lusztig, E. (1989) Contemp. Math. 82, 59.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hamer, C.J. (1981) J. Phys. A14, 2981.MathSciNetADSGoogle Scholar
  8. 8.
    Sklyanin, E.K. (1988) Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21, 2375–2389.MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Kulish, P.P. and Sklyanin, E.K. (1991) The general U q[sl(2)] invariant XXZ integrable quantum spin chain, J. Phys. A: Math. Gen. 24, L435–L439.MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Martin, P.P. (1991) Potts models and related problems in statistical mechanics., World Scientific, Singapore.CrossRefGoogle Scholar
  11. 11.
    Alcaraz, F.C., Baake, M., Grimm, U. and Rittenberg, V. (1989) The modified XXZ Heisenberg chain, conformal invariance and the surface exponents of c < 1 systems, J. Phys. A: Math. Gen. 22, L5-L11.Google Scholar
  12. 12.
    Alcaraz, FC, Belavin, A.A. and Usmanov, R.A. (2000) Correspondence between the XXZ model in roots of unity and the one-dimensional quanatum Ising chain with different boundary conditions, preprint hep-th/0007151.Google Scholar
  13. 13.
    Belavin, A.A. and Usmanov, R.A. (2001) Minimal lattice model LM(3, 4) and two-dimentional Ising model with cilindrical boundary conditions, Teor. Mat. Fiz. 126, No 1, 63.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • R. A. Usmanov
    • 1
  1. 1.Landau Institute for Theoretical PhysicsChernogolovka, Moscow regionRussia

Personalised recommendations