Advertisement

Journal of Statistical Physics

, Volume 156, Issue 6, pp 1199–1220 | Cite as

Dynamic Lattice Supersymmetry in \(\mathfrak {gl}\left( {n}|{m}\right) \) Spin Chains

  • David Meidinger
  • Vladimir Mitev
Article

Abstract

Supersymmetry operators that change a spin chain’s length have appeared in numerous contexts, ranging from the AdS/CFT correspondence to statistical mechanics models. In this article, we present, via an analysis of the Bethe equations, all homogeneous, rational and trigonometric, integrable \(\mathfrak {gl}( n|m)\) spin chains for which length-changing supersymmetry can be present. Furthermore, we write down the supercharges explicitly for the simplest new models, namely the \(\mathfrak {sl}( n|1)\) spin chains with the \((n-1)\)-fold antisymmetric tensor product of the fundamental representation at each site and check their compatibility with integrability.

Keywords

Supersymmetry Spin chains Integrable models 

Notes

Acknowledgments

The authors thank Rouven Frassek, Sergey Frolov, Nils Kanning, Yumi Ko, Matthias Staudacher, Zengo Tsuboi, Matthias Wilhelm and especially Christian Hagendorf for inspiring discussions and insightful suggestions. Furthermore VM is grateful to the Kavli IPMU, Université Catholique de Louvain and Sogang University for the kind hospitality while working on parts of the manuscript.

References

  1. 1.
    Minahan, J., Zarembo, K.: The Bethe ansatz for N=4 superYang-Mills. JHEP. 0303, 013 (2003). arXiv:hep-th/0212208
  2. 2.
    Beisert, N., Ahn, C., Alday, L. F., Bajnok, Z, Drummond, J. M. et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3–32 (2012). arXiv:1012.3982
  3. 3.
    Beisert, N.: The su(2\(|\)3) dynamic spin chain. Nucl. Phys. B682, 487–520 (2004). arXiv:hep-th/0310252
  4. 4.
    Beisert N.: The dilatation operator of N=4 super Yang-Mills theory and integrability. Phys. Rept. 405, 1–202 (2004). arXiv:0407277
  5. 5.
    Zwiebel, B. I.: Iterative structure of the N=4 SYM spin chain. JHEP. 0807, 114 (2008). arXiv:0806.1786
  6. 6.
    Beisert, N.: The SU(2\(|\)2) dynamic S-matrix. Adv. Theor. Math. Phys. 12, 945–979 (2008). arXiv:hep-th/0511082
  7. 7.
    Beisert, N.: The analytic Bethe Ansatz for a chain with centrally extended su(2\(|\)2) symmetry. J. Stat. Mech. 0701, P01017 (2007). arXiv:nlin/0610017
  8. 8.
    Gomez, C., Hernandez, R.: The magnon kinematics of the AdS/CFT correspondence. JHEP. 0611, 021 (2006). arXiv:hep-th/0608029
  9. 9.
    Plefka, J., Spill, F., Torrielli, A.: On the Hopf algebra structure of the AdS/CFT S-matrix. Phys. Rev. D74, 066008 (2006). arXiv:hep-th/0608038
  10. 10.
    Arutyunov, G., Frolov, S., Zamaklar, M.: The Zamolodchikov–Faddeev algebra for AdS\(_5\) \(\times \) S\(^5\) superstring. JHEP. 0704, 002 (2007). arXiv:hep-th/0612229
  11. 11.
    Beisert, N., Staudacher, M.: Long-range psu(\(2,2|4\)) Bethe Ansätze for gauge theory and strings. Nucl. Phys. B727, 1–62 (2005). arXiv:hep-th/0504190
  12. 12.
    Gromov, N., Kazakov, V., Kozak, A., Vieira, P.: Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang–Mills Theory: TBA and excited states. Lett. Math. Phys. 91, 265–287 (2010). arXiv:0902.4458
  13. 13.
    Gromov, N., Kazakov, V., Vieira, P.: Exact spectrum of planar \({\cal N}=4\) supersymmetric Yang–Mills theory: Konishi dimension at any coupling. Phys. Rev. Lett. 104, 211601 (2010). arXiv:0906.4240
  14. 14.
    Arutyunov, G., Frolov, S., Suzuki, R.: Five-loop Konishi from the mirror TBA. JHEP. 1004, 069 (2010). arXiv:1002.1711
  15. 15.
    Fendley, P., Schoutens, K., de Boer J.: Lattice models with N=2 supersymmetry. Phys. Rev. Lett. 90, 120402 (2003). arXiv:hep-th/0210161
  16. 16.
    Fendley, P., Schoutens, K., Nienhuis, B.: Lattice fermion models with supersymmetry. J. Phys. A36, 12399–12424 (2003). arXiv:cond-mat/0307338
  17. 17.
    Yang, X., Fendley, P.: Non-local space-time supersymmetry on the lattice. J. Phys. A. 37, 8937 (2004). arXiv:cond-mat/0404682
  18. 18.
    Hagendorf, C., Fendley, P.: The Eight-vertex model and lattice supersymmetry. J. Statist. Phys. 146, 1122–1155 (2012). arXiv:1109.4090
  19. 19.
    Hagendorf, C.: Spin chains with dynamical lattice supersymmetry. J. Stat. Phys. 150, 609–657 (2013). arXiv:1207.0357
  20. 20.
    Beisert, N., Zwiebel, B. I.: On symmetry enhancement in the psu(1,1|2) sector of N=4 SYM. JHEP. 0710, 031 (2007). arXiv:0707.1031
  21. 21.
    Zwiebel B. I.: Two-loop integrability of planar N=6 superconformal Chern-Simons theory. J. Phys. A42, 495402 (2009). arXiv:0901.0411
  22. 22.
    Perk, J.H.H., Schultz, C.L.: New families of commuting transfer matrices in q state vertex models. Phys. Lett. A84, 407–410 (1981)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Belliard, S., Ragoucy, E.: Nested Bethe ansatz for ’all’ closed spin chains. J. Phys. A41 295202, (2008). arXiv:0804.2822
  24. 24.
    Frassek, R., Lukowski, T., Meneghelli, C., Staudacher, M.: Baxter operators and Hamiltonians for ’nearly all’ integrable closed gl(n) spin chains. arXiv:1112.3600
  25. 25.
    Ragoucy, E., Satta, G.: Analytical Bethe Ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin diagrams. JHEP. 0709, 001 (2007). arXiv:0706.3327
  26. 26.
    Arnaudon, A., Crampe, N., Doikou, A., Frappat, L., Ragoucy, E.: Analytical Bethe Ansatz for closed and open gl(n)-spin chains in any representation. J. Stat. Mech. 0502, P02007 (2005). arXiv:math-ph/0411021
  27. 27.
    Arnaudon, D., Crampe, N., Doikou, A., Frappat, L.: Spectrum and Bethe ansatz equations for the U(q) (gl(N)) closed and open spin chains in any representation. arXiv:math-ph/0512037
  28. 28.
    Avdeev, L., Vladimirov, A.: On exceptional solutions of the Bethe Ansatz equations. Theor. Math. Phys. 69, 1071 (1987)CrossRefGoogle Scholar
  29. 29.
    Hao W., Nepomechie R. I., Sommese A. J.: Singular solutions, repeated roots and completeness for higher-spin chains. arXiv:1312.2982
  30. 30.
    Baxter R. J., Completeness of the Bethe ansatz for the six and eight vertex models. J. Statist. Phys. 108, 1–48 (2002). arXiv:cond-mat/0111188
  31. 31.
    Frappat L., Sorba P., Sciarrino A.: Dictionary on Lie superalgebras. arXiv:hep-th/9607161
  32. 32.
    Zabrodin A.: Discrete Hirota’s equation in quantum integrable models. arXiv:hep-th/9610039
  33. 33.
    Kulish, P., Reshetikhin, N.Y., Sklyanin, E.: Yang–Baxter equation and representation theory. I. Lett. Math. Phys. 5, 393–403 (1981)ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Ferro, L., Lukowski, T., Meneghelli, C., Plefka, J., Staudacher, M.: Spectral parameters for scattering amplitudes in N=4 Super Yang–Mills Theory. arXiv:1308.3494
  35. 35.
    Bargheer, T., Beisert, N., Loebbert, F.: Boosting nearest-neighbour to long-range integrable spin chains. J. Stat. Mech. 0811, L11001 (2008). arXiv:0807.5081
  36. 36.
    Bargheer, T., Beisert, N., Loebbert, F.: Long-range deformations for integrable spin chains. J. Phys. A42, 285205 (2009). arXiv:0902.0956

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations