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The bound state S-matrix of the deformed Hubbard chain

  • Marius de Leeuw
  • Takuya Matsumoto
  • Vidas Regelskis
Article

Abstract

In this work we use the q-oscillator formalism to construct the atypical (short) supersymmetric representations of the centrally extended \( {\mathcal{U}_q}\left( {\mathfrak{s}\mathfrak{u}\left( {2|2} \right)} \right) \) algebra. We then determine the S-matrix describing the scattering of arbitrary bound states. The crucial ingredient in this derivation is the affine extension of the aforementioned algebra.

Keywords

Quantum Groups AdS-CFT Correspondence Exact S-Matrix Integrable Field Theories 

References

  1. [1]
    J. Hubbard, Electron correlations in narrow energy bands, Proc. Roy. Soc. London A 276 (1963) 238.ADSGoogle Scholar
  2. [2]
    M. Rasetti, The Hubbard modelrecent results, World Scientific, Singapore (1991).Google Scholar
  3. [3]
    A. Montorsi, The Hubbard model, World Scientific, Singapore (1992).Google Scholar
  4. [4]
    V. Korepin and F. Eßler, Exactly solvable models of strongly correlated electrons, World Scientific, Singapore (1994).zbMATHCrossRefGoogle Scholar
  5. [5]
    F. Eßler, H. Frahm, F. Goehmann, A. Klumper and V. Korepin, The one-dimensional Hubbard model, Cambridge University Press, Cambridge U.K. (2005).zbMATHCrossRefGoogle Scholar
  6. [6]
    J. Spalek, t-J model then and now: a personal perspective from the pioneering times, Acta Phys. Polon. A 111 (2007) 409 [arXiv:0706.4236].ADSGoogle Scholar
  7. [7]
    F.C. Alcaraz and R.Z. Bariev, Interpolation between Hubbard and supersymmetric t-J models: two-parameter integrable models of correlated electrons, J. Phys. A 32 (1999) L483 [cond-mat/9908265].MathSciNetADSGoogle Scholar
  8. [8]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    N. Beisert, The analytic Bethe ansatz for a chain with centrally extended su(2|2) symmetry, J. Stat. Mech. (2007) P01017 [nlin/0610017] [INSPIRE].
  10. [10]
    N. Beisert and P. Koroteev, Quantum deformations of the one-dimensional Hubbard model, J. Phys. A 41 (2008) 255204 [arXiv:0802.0777] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    M. Martins and C. Melo, The Bethe ansatz approach for factorizable centrally extended S-matrices, Nucl. Phys. B 785 (2007) 246 [hep-th/0703086] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    B. Shastry, Exact integrability of the one-dimensional Hubbard model, Phys. Rev. Lett. 56 (1986) 2453 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    D. Uglov and V. Korepin, The Yangian symmetry of the Hubbard model, Phys. Lett. A 190 (1994) 238 [hep-th/9310158] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    N. Beisert, The S-matrix of AdS/CFT and Yangian symmetry, PoS(SOLVAY)002 [arXiv:0704.0400] [INSPIRE].
  15. [15]
    N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].MathSciNetGoogle Scholar
  16. [16]
    G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS 5 × S 5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
  21. [21]
    G. Arutyunov and S. Frolov, The S-matrix of string bound states, Nucl. Phys. B 804 (2008) 90 [arXiv:0803.4323] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    M. de Leeuw, Bound states, Yangian symmetry and classical r-matrix for the AdS 5 × S 5 superstring, JHEP 06 (2008) 085 [arXiv:0804.1047] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    N. Beisert, The analytic Bethe ansatz for a chain with centrally extended su(2|2) symmetry, J. Stat. Mech. (2007) P01017 [nlin/0610017] [INSPIRE].
  24. [24]
    N. Dorey, Magnon bound states and the AdS/CFT correspondence, J. Phys. A 39 (2006) 13119 [hep-th/0604175] [INSPIRE].MathSciNetGoogle Scholar
  25. [25]
    H.-Y. Chen, N. Dorey and K. Okamura, Dyonic giant magnons, JHEP 09 (2006) 024 [hep-th/0605155] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    H.-Y. Chen, N. Dorey and K. Okamura, On the scattering of magnon boundstates, JHEP 11 (2006) 035 [hep-th/0608047] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    G. Arutyunov, M. de Leeuw and A. Torrielli, The bound state S-matrix for AdS 5 × S 5 superstring, Nucl. Phys. B 819 (2009) 319 [arXiv:0902.0183] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    T. Matsumoto and S. Moriyama, An exceptional algebraic origin of the AdS/CFT Yangian symmetry, JHEP 04 (2008) 022 [arXiv:0803.1212] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    F. Spill and A. Torrielli, On Drinfelds second realization of the AdS/CFT SU(2|2) Yangian, J. Geom. Phys. 59 (2009) 489 [arXiv:0803.3194] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    G. Arutyunov, M. de Leeuw and A. Torrielli, On Yangian and long representations of the centrally extended SU(2|2) superalgebra, JHEP 06 (2010) 033 [arXiv:0912.0209] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    G. Arutyunov, M. de Leeuw and A. Torrielli, Universal blocks of the AdS/CFT scattering matrix, JHEP 05 (2009) 086 [arXiv:0903.1833] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    T. Matsumoto, S. Moriyama and A. Torrielli, A secret symmetry of the AdS/CFT S-matrix, JHEP 09 (2007) 099 [arXiv:0708.1285] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    J. Drummond, G. Feverati, L. Frappat and É. Ragoucy, Super-Hubbard models and applications, JHEP 05 (2007) 008 [hep-th/0703078] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    N. Beisert, The classical trigonometric r-matrix for the quantum-deformed Hubbard chain, J. Phys. A 44 (2011) 265202 [arXiv:1002.1097] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    N. Beisert, W. Galleas and T. Matsumoto, A quantum affine algebra for the deformed Hubbard chain, arXiv:1102.5700 [INSPIRE].
  36. [36]
    A. Macfarlane, On q-analogs of the quantum harmonic oscillator and the quantum group SU(2)q, J. Phys. A 22 (1989) 4581 [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    L. Biedenharn, The quantum group SU(2)q and a q-analog of the boson operators, J. Phys. A 22 (1989) L873 [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    T. Hayashi, Q-analogs of Clifford and Weyl algebras: spinor and oscillator reprsentations of quantum enveloping algebras, Commun. Math. Phys. 127 (1990) 129 [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    M. Chaichian and P. Kulish, Quantum Lie superalgebras and q-oscillators, Phys. Lett. B 234 (1990) 72 [INSPIRE].MathSciNetADSGoogle Scholar
  40. [40]
    G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].MathSciNetGoogle Scholar
  43. [43]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    B. Hoare and A. Tseytlin, Towards the quantum S-matrix of the Pohlmeyer reduced version of AdS 5 × S 5 superstring theory, Nucl. Phys. B 851 (2011) 161 [arXiv:1104.2423] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    B. Hoare, T.J. Hollowood and J. Miramontes, A relativistic relative of the magnon S-matrix, JHEP 11 (2011) 048 [arXiv:1107.0628] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    V.G. Drinfeld, Quasi Hopf algebras, Alg. Anal. 1 N6 (1989) 114.MathSciNetGoogle Scholar
  47. [47]
    J.-U.H. Petersen, Representations at a root of unity of q-oscillators and quantum Kac-Moody algebras, Ph.D. thesis, University of London, London U.K. (1994) [hep-th/9409079] [INSPIRE].
  48. [48]
    M. Chaichian and A.P. Demichev, Introduction to quantum groups, World Scientific, Singapore (1996).zbMATHCrossRefGoogle Scholar
  49. [49]
    A.N. Kirillov and N.Y. .Reshetikhin, Representations of the algebra U(q)(sl(2)) q-orthogonal polynomials and invariants of links, in New developments in the theory of knots, T. Kohno ed., World Scientific, Singapore (1991), pg. 202.Google Scholar
  50. [50]
    R. Murgan and R.I. Nepomechie, q-deformed SU(2|2) boundary S-matrices via the ZF algebra, JHEP 06 (2008) 096 [arXiv:0805.3142] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    C. Ahn and R.I. Nepomechie, Yangian symmetry and bound states in AdS/CFT boundary scattering, JHEP 05 (2010) 016 [arXiv:1003.3361] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    N. MacKay and V. Regelskis, Yangian symmetry of the Y = 0 maximal giant graviton, JHEP 12 (2010) 076 [arXiv:1010.3761] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    N. MacKay and V. Regelskis, Reflection algebra, Yangian symmetry and bound-states in AdS/CFT, JHEP 01 (2012) 134 [arXiv:1101.6062] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    M. de Leeuw, The S-matrix of the AdS 5 × S 5 superstring, arXiv:1007.4931 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Marius de Leeuw
    • 1
  • Takuya Matsumoto
    • 2
    • 3
  • Vidas Regelskis
    • 4
    • 5
  1. 1.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  4. 4.Department of MathematicsUniversity of YorkYorkUK
  5. 5.Institute of Theoretical Physics and Astronomy of Vilnius UniversityVilniusLithuania

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