Conclusions and Unresolved Issues

  • Martin GreiterEmail author
Part of the Springer Tracts in Modern Physics book series (STMP, volume 244)


In this monograph, we have presented an exact model of a critical spin chain with spin S. If we write the ground state of the Haldane–Shastry model, which is equivalent to the Gutzwiller state obtained by projection of filled bands.


Momentum Spacing Thermodynamic Limit Exact Model Read State Arbitrary Spin 
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.


  1. 1.
    F.D.M. Haldane, Exact Jastrow–Gutzwiller resonant-valence-bond ground state of the spin-\(\frac{1}{2}\) antiferromagnetic Heisenberg chain with \(1/r^2\) exchange. Phys. Rev. Lett. 60, 635 (1988)Google Scholar
  2. 2.
    B.S. Shastry, Exact solution of an \(S = \frac{1}{2}\) Heisenberg antiferromagnetic chain with long-ranged interactions. Phys. Rev. Lett. 60, 639 (1988)Google Scholar
  3. 3.
    M.C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals. Phys. Rev. Lett. 10, 159 (1963)CrossRefADSGoogle Scholar
  4. 4.
    W. Metzner, D. Vollhardt, Ground-state properties of correlated fermions: exact analytic results for the Gutzwiller wave function. Phys. Rev. Lett. 59, 121 (1987)CrossRefADSGoogle Scholar
  5. 5.
    F. Gebhard, D. Vollhardt, Correlation functions for Hubbard-type models: the exact results for the Gutzwiller wave function in one dimension. Phys. Rev. Lett. 59, 1472 (1987)CrossRefADSGoogle Scholar
  6. 6.
    J. Wess, B. Zumino, Consequences of anomalous ward identities. Phys. Lett. 37, 95 (1971)MathSciNetGoogle Scholar
  7. 7.
    E. Witten, Non-Abelian bosonization in two dimensions. Commun. Math. Phys. 92, 455 (1984)CrossRefzbMATHADSMathSciNetGoogle Scholar
  8. 8.
    R. Thomale, S. Rachel, P. Schmitteckert, M. Greiter, manuscript in preparationGoogle Scholar
  9. 9.
    N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267 (2000)CrossRefADSGoogle Scholar
  10. 10.
    C. Nayak, F. Wilczek, 2n-quasihole states realize \(2^{n-1}\)-dimensional spinor braiding statistics in paired quantum Hall states. Nucl. Phys. B 479, 529 (1996)Google Scholar
  11. 11.
    D.A. Ivanov, Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268 (2001)CrossRefADSGoogle Scholar
  12. 12.
    A. Stern, von F. Oppen, E. Mariani, Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics. Phys. Rev. B 70, 205338 (2004)CrossRefADSGoogle Scholar
  13. 13.
    G. Moore, N. Read, Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362 (1991)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    M. Greiter, X.G. Wen, F. Wilczek, Paired Hall state at half filling. Phys. Rev. Lett. 66, 3205 (1991)CrossRefADSGoogle Scholar
  15. 15.
    M. Greiter, X.G. Wen, F. Wilczek, Paired Hall states. Nucl. Phys. B 374, 567 (1992)CrossRefzbMATHADSMathSciNetGoogle Scholar
  16. 16.
    M. Greiter, R. Thomale, Non-Abelian statistics in a quantum antiferromagnet. Phys. Rev. Lett. 102, 207203 (2009)CrossRefADSGoogle Scholar
  17. 17.
    M. Greiter, D. Schuricht, Many-spinon states and the secret significance of Young tableaux. Phys. Rev. Lett. 98, 237202 (2007)CrossRefADSGoogle Scholar
  18. 18.
    N. Kawakami, Asymptotic Bethe-ansatz solution of multicomponent quantum systems with \(1/r^2\) long-range interaction. Phys. Rev. B 46, 1005 (1992)Google Scholar
  19. 19.
    N. Kawakami, SU(N) generalization of the Gutzwiller–Jastrow wave function and its critical properties in one dimension. Phys. Rev. B 46, 3191 (1992)CrossRefADSGoogle Scholar
  20. 20.
    M. Greiter, S. Rachel, Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap. Phys. Rev. B 75, 184441 (2007)CrossRefADSGoogle Scholar
  21. 21.
    Y. Kuramoto, H. Yokoyama, Exactly soluble supersymmetric t–J-type model with a long-range exchange and transfer. Phys. Rev. Lett. 67, 1338 (1991)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Theorie der Kondensierten Materie (TKM)Karlsruhe Institut für Technologie (KIT)KarlsruheGermany

Personalised recommendations