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


SIGMA 6 (2010), 024, 13 pages      arXiv:1003.2147      http://dx.doi.org/10.3842/SIGMA.2010.024
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Ordering of Energy Levels for Extended SU(N) Hubbard Chain

Tigran Hakobyan
Yerevan State University, Alex Manoogian 1, Yerevan, Armenia
Yerevan Physics Institute, Alikhanian Br. 2, Yerevan, Armenia

Received November 03, 2009, in final form March 15, 2010; Published online March 20, 2010

Abstract
The Lieb-Mattis theorem on the antiferromagnetic ordering of energy levels is generalized to SU(N) extended Hubbard model with Heisenberg exchange and pair-hopping terms. It is proved that the minimum energy levels among the states from equivalent representations are nondegenerate and ordered according to the dominance order of corresponding Young diagrams. In particular, the ground states form a unique antisymmetric multiplet. The relation with the similar ordering among the spatial wavefunctions with different symmetry classes of ordinary quantum mechanics is discussed also.

Key words: Lieb-Mattis theorem; SU(N) Hubbard model; ground state; dominance order; Schur-Weyl duality.

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References

  1. Marshall W., Antiferromagnetism, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 232 (1955), 48-68.
  2. Lieb E.H., Schultz T.D., Mattis D.C., Two soluble models of an antiferromagnetic chain, Ann. Physics 16 (1961), 407-466.
  3. Lieb E.H., Mattis D., Ordering energy levels of interacting spin systems, J. Math. Phys. 3 (1962), 749-751.
  4. Lieb E.H., Two theorems on the Hubbard model, Phys. Rev. Lett. 62 (1989), 1201-1204.
  5. Lieb E.H., Mattis D., Theory of ferromagnetism and the ordering of electronic energy levels, Phys. Rev. 125 (1962), 164-172.
    Lieb E.H., The Hubbard model: some rigorous results and open problems, in Proceedings of XIth International Congress of Mathematical Physics (Paris, 1994), Editor D. Iagolnitzer, International Press, 1995, 392-412, cond-mat/9311033.
  6. Munro R.G., Extension of a theorem on antiferromagnetism to include biquadratic exchange, Phys. Rev. B 13 (1976), 4875-4876.
    Parkinson J.B., The eigenstates of short chains of spin-1 atoms with Heisenberg and biquadratic exchange, J. Phys. C: Solid State Phys. 10 (1977), 1735-1739.
  7. Xiang T., d'Ambrumenil N., Energy-level ordering in the one-dimensional tJ model: a rigorous result, Phys. Rev. B 46 (1992), 599-602.
  8. Xiang T., d'Ambrumenil N., Theorem on the one-dimensional interacting-electron system on a lattice, Phys. Rev. B 46 (1992), 11179-11181.
  9. Shen S.-Q., Strongly correlated electron systems: spin-reflection positivity and some rigorous results, Internat. J. Modern Phys. B 12 (1998), 709-779.
  10. Hakobyan T., The ordering of energy levels for su(n) symmetric antiferromagnetic chains, Nuclear Phys. B 699 (2004), 575-594, cond-mat/0403587.
  11. Hakobyan T., Energy-level ordering and ground-state quantum numbers for a frustrated two-leg spin-1/2 ladder, Phys. Rev. B 75 (2007), 214421, 14 pages, cond-mat/0702148.
    Hakobyan T., Antiferromagnetic ordering of energy levels for a spin ladder with four-spin cyclic exchange: generalization of the Lieb-Mattis theorem, Phys. Rev. B 78 (2008), 012407, 4 pages, arXiv:0802.2392.
  12. Nachtergaele B., Spitzer W., Starr Sh., Ferromagnetic ordering of energy levels, J. Statist. Phys. 116 (2004), 719-738, math-ph/0308006.
    Nachtergaele B., Starr Sh., Ferromagnetic Lieb-Mattis theorem, Phys. Rev. Lett. 94 (2005), 057206, 4 pages, math-ph/0408020.
  13. Greiter M., Rachel S., Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap, Phys. Rev. B 75 (2007), 184441, 25 pages, cond-mat/0702443.
  14. Rachel S., Thomale R., Führinger M., Schmitteckert P., Greiter M., Spinon confinement and the Haldane gap in SU(n) spin chains, Phys. Rev. B 80 (2009), 180420(R), 4 pages, arXiv:0904.3882.
  15. Aguado M., Asorey M., Ercolessi E., Ortolani F., Pasini S., Density-matrix renormalization-group simulation of the SU(3) antiferromagnetic Heisenberg model, Phys. Rev. B 79 (2009), 012408, 4 pages, arXiv:0801.3565.
  16. Honerkam C., Hofstetter W., Ultracold fermions and the SU(N) Hubbard model, Phys. Rev. Lett. 92 (2004), 170403, 4 pages, cond-mat/0309374.
  17. Gorshkov A.V., Hermele M., Gurarie V., Xu C., Julienne P.S., Ye J., Zoller P., Demler E., Lukin M.D., Rey A.M., Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms, Nature Phys., to appear, arXiv:0905.2610.
  18. Haldane F.D.M., Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model, Phys. Lett. A 93 (1983), 464-468.
    Haldane F.D.M., Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50 (1983), 1153-1156.
  19. Marston J.B., Affleck I., Large-n limit of the Hubbard-Heisenberg model, Phys. Rev. B 39 (1989), 11538-11558.
  20. Li Y.-Q., Tian G.-S., Ma M., Lin H.-Q., Ground state and excitations of a four-component fermion model, Phys. Rev. B 70 (2004), 233105, 4 pages, cond-mat/0407601.
  21. Affleck I., Lieb E.H., A proof of part of Haldane's conjecture on spin chains, Lett. Math. Phys. 12, (1986) 57-69.
  22. Angelucci A., Sorella S., Some exact results for the multicomponent tJ model, Phys. Rev. B 54 (1996), R12657-R12660, cond-mat/9609107.
  23. Li Y.Q., Rigorous results for a hierarchy of generalized Heisenberg models, Phys. Rev. Lett. 87 (2001), 127208, 4 pages, cond-mat/0201060.
  24. Lancaster P., Theory of matrices, Academic Press, New York - London, 1969.
  25. Lieb E.H., Schupp P., Ground state properties of a fully frustrated quantum spin system, Phys. Rev. Lett. 83 (1999), 5362-5365, math-ph/9908019.
    Lieb E.H., Schupp P., Singlets and reflection symmetric spin systems, Phys. A 279 (2000), 378-385, math-ph/9910037.
  26. Sutherland B., Model for a multicomponent quantum system, Phys. Rev. B 12 (1975), 3795-3805.
  27. Hamermesh M., Group theory and its application to physical problems, Dover, New York, 1989.
  28. Macdonald I.G., Symmetric functions and Hall polynomials, Oxford University Press, New York, 1979.
  29. Mattis D., The theory of magnetism made simple, World Scientific, 2004.
  30. Landau L.D., Lifshitz L.M., Quantum mechanics non-relativistic theory, Elsevier Science, 1977.

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