Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 091, 13 pages      arXiv:1008.5285

One-Dimensional Vertex Models Associated with a Class of Yangian Invariant Haldane-Shastry Like Spin Chains

Bireswar Basu-Mallick a, Nilanjan Bondyopadhaya b and Kazuhiro Hikami c
a) Theory Group, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India
b) Integrated Science Education and Research Centre, Siksha-Bhavana, Visva-Bharati, Santiniketan 731 235, India
c) Department of Mathematics, Naruto University of Education, Tokushima 772-8502, Japan

Received September 06, 2010, in final form November 30, 2010; Published online December 10, 2010

We define a class of Y(sl(m|n)) Yangian invariant Haldane-Shastry (HS) like spin chains, by assuming that their partition functions can be written in a particular form in terms of the super Schur polynomials. Using some properties of the super Schur polynomials, we show that the partition functions of this class of spin chains are equivalent to the partition functions of a class of one-dimensional vertex models with appropriately defined energy functions. We also establish a boson-fermion duality relation for the partition functions of this class of supersymmetric HS like spin chains by using their correspondence with one-dimensional vertex models.

Key words: Haldane-Shastry spin chain; vertex model; Yangian quantum group; boson-fermion duality relation.

pdf (318 Kb)   ps (249 Kb)   tex (134 Kb)


  1. Sutherland B., Beautiful models. 70 years of exactly solved quantum many-body problems, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  2. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  3. Essler F.H.L., Frahm H., Göhmann F., Klümper A., Korepin V.E., The one-dimensional Hubbard model, Cambridge University Press, Cambridge, 2005.
  4. Sutherland B., Two-dimensional hydrogen bonded crystals without the ice rule, J. Math. Phys. 11 (1970), 3183-3186.
  5. Baxter R.J., One-dimensional anisotropic Heisenberg chain, Phys. Rev. Lett. 26 (1971), 834-834.
  6. Haldane F.D.M., Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r2 exchange, Phys. Rev. Lett. 60 (1988), 635-638.
  7. Shastry B.S., Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions, Phys. Rev. Lett. 60 (1988), 639-642.
  8. Polychronakos A.P., Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993), 2329-2331, hep-th/9210109.
  9. Frahm H., Spectrum of a spin chain with inverse square exchange, J. Phys. A: Math. Gen. 26 (1993), L473-L479, cond-mat/9303050.
  10. Polychronakos A.P., Exact spectrum of SU(n) spin chain with inverse-square exchange, Nuclear Phys. B 419 (1994), 553-566.
  11. Haldane F.D.M., "Fractional statistics" in arbitrary dimensions: a generalization of the Pauli principle, Phys. Rev. Lett. 67 (1991), 937-940.
  12. Ha Z.N.C., Quantum many-body systems in one dimension, Series on Advances in Statistical Mechanics, Vol. 12, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  13. Bernevig B.A., Gurarie V., Simon S.H., Central charge and quasihole scaling dimensions from model wavefunctions: towards relating Jack wavefunctions to W-algebras, J. Phys. A: Math. Theor. 42 (2009), 245206, 30 pages, arXiv:0903.0635.
  14. Bernevig B. A., Haldane F.D.M., Clustering properties and model wavefunctions for non-Abelian fractional quantum Hall quasielectrons, Phys. Rev. Lett. 102 (2009), 066802, 4 pages, arXiv:0810.2366.
  15. Beisert N., Dippel V., Staudacher M., A novel long-range spin chain and planar N = 4 super Yang-Mills, J. High Energy Phys. 2004 (2004), no. 7, 075, 49 pages, hep-th/0405001.
  16. Serban D., Staudacher M., Planar N=4 gauge theory and the Inozemtsev long range spin chain, J. High Energy Phys. 2004 (2004), no. 6, 001, 31 pages, hep-th/0401057.
  17. Beisert N., Staudacher M., Long-range psu(2,2|4) Bethe ansätze for gauge theory and strings, Nuclear Phys. B 727 (2005), 1-62, hep-th/0504190.
  18. Haldane F.D.M., Ha Z.N.C., Talstra J.C., Benard D., Pasquier V., Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory, Phys. Rev. Lett. 69 (1992), 2021-2025.
  19. Benard D., Gaudin M., Haldane F.D.M., Pasquier V., Yang-Baxter equation in long-range interacting systems, J. Phys. A: Math. Gen. 26 (1993), 5219-5236.
  20. Hikami K., Yangian symmetry and Virasoro character in a lattice spin system with long-range interactions, Nuclear Phys. B 441 (1995), 530-548.
  21. Hikami K., Basu-Mallick B., Supersymmetric Polychronakos spin chain: motif, distribution function, and character, Nuclear Phys. B 566 (2000), 511-528, math-ph/9904033.
  22. Bazhanov V.V., Frassek R., Lukowski T., Meneghelli C., Staudacher M., Baxter Q-operators and representations of Yangians, arXiv:1010.3699.
  23. Haldane F.D.M., Physics of the ideal semion gas: spinons and quantum symmetries of the integrable Haldane-Shastry spin chain, in Proc. 16th Taniguchi Symp. (Kashikojima, Japan, 1993), Editors A. Okiji and N. Kawakami, Springer, 1994, 3-20, cond-mat/9401001.
  24. Basu-Mallick B., Bondyopadhaya N., Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain, Nuclear Phys. B 757 (2006), 280-302, hep-th/0607191.
  25. Basu-Mallick B., Ujino H., Wadati M., Exact spectrum and partition function of SU(m|n) supersymmetric Polychronakos model, J. Phys. Soc. Japan 68 (1999), 3219-3226, hep-th/9904167.
  26. Basu-Mallick B., Bondyopadhaya N., Hikami K., Sen D., Boson-fermion duality in SU(m|n) supersymmetric Haldane-Shastry spin chain, Nuclear Phys. B 782 (2007), 276-295, hep-th/0703240.
  27. Kirilov A.N., Kuniba A., Nakanishi T., Skew Young diagram method in spectral decomposition of integrable lattice models, Comm. Math. Phys. 185 (1997), 441-465, q-alg/9607027.
  28. Hikami K., Exclusion statistics and chiral partition function, in Physics and Combinatorics 2000 (Nagoya), Editors A.N. Kirilov and N. Liskova, World Sci. Publ., River Edge, NJ, 2001, 22-48.
  29. Barba J.C., Finkel F., González-López A., Rodríguez M.A., Inozemtsev's hyperbolic spin model and its related spin chain, Nuclear Phys. B 839 (2010), 499-525, arXiv:1005.0487.
  30. Frahm H., Inozemstsev V.I., New family of solvable 1D Heisenberg models, J. Phys. A: Math. Gen. 27 (1994), L801-L808, cond-mat/9405038.
  31. Basu-Mallick B., Bondyopadhaya N., Sen D., Low energy properties of the SU(m|n) supersymmetric Haldane-Shastry spin chain, Nuclear Phys. B 795 (2008), 596-622, arXiv:0710.0452.
  32. Enciso A., Finkel F., González-López A., On the level density of spin chains of Haldane-Shastry type, arXiv:1005.3202.
  33. Finkel F., González-López A., Global properties of the spectrum of the Haldane-Shastry spin chain, Phys. Rev. B 72 (2005), 174411, 6 pages, cond-mat/0509032.
  34. Barba J.C., Finkel F., González-López A., Rodríguez M.A., The Berry-Tabor conjecture for spin chains of Haldane-Shastry type, Europhys. Lett. 83 (2008), 27005, 6 pages, arXiv:0804.3685.

Previous article   Next article   Contents of Volume 6 (2010)