Skip to main content
SpringerLink
Log in

The topological basis expression of Heisenberg spin chain

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, it is shown that the Heisenberg XY, XXZ, XXX, and Ising model all can be constructed from the Braid group algebra generator and the Temperley–Lieb algebra generator. And a new set of topological basis expression is presented. Through acting on the different subspaces, we get the new nontrivial six-dimensional and four-dimensional Braid group matrix representations and Temperley–Lieb matrix representations. The eigenstates of Heisenberg model can be described by the combination of the set of topological bases. It is worth mentioning that the ground state is closely related to parameter q which is the meaningful topological parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bethe, H.: Eigenvalues and eigenfunctions of the linear atom chain. Z. Phys. 71, 205–226 (1931)

    Article  ADS  Google Scholar 

  2. Hulthen, L.: Ark. Mat. Astron. Fys. A 26, 1 (1939)

    Google Scholar 

  3. Faddeev, L., Takhtajan, L.: Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model. J. Sov. Math. 24, 241 (1984)

    Article  MATH  Google Scholar 

  4. Temperley, H.N.V., Lieb, E.H.: Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proc. R. Soc. Lond. Ser. A 322, 251 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Wadati, M., Deguchi, T., Akutsu, Y.: Exactly solvable models and knot theory. Phys. Rep. 180, 247 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Owczarek, A.L., Baxter, R.J.: A class of interaction-round-a-face models and its equivalence with an ice-type model. J. Stat. Phys. 49, 1093 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Pasquier, V.: Two-dimensional critical systems labelled by Dynkin diagrams. Nucl. Phys. B 285, 162 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  8. Pasquier, V.: Lattice derivation of modular invariant partition functions on the torus. J. Phys. A 20, L1229 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  9. Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers–Ramanujan-type identities. J. Stat. Phys. 35, 193 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Klümper, A.: New results for q-state vertex models and the pure biquadratic spin-1 Hamiltonian. Europhys. Lett. 9, 815 (1989)

    Article  ADS  Google Scholar 

  11. Kulish, P.P.: On spin systems related to the Temperley–Lieb algebra. J. Phys. A Math. Gen. 36, L489 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Parkinson, J.B.: On the integrability of the S=1 quantum spin chain with pure biquadratic exchange. J. Phys. C Solid State Phys. 20, L1029 (1987)

    Article  ADS  Google Scholar 

  13. Parkinson, J.B.: The S=1 quantum spin chain with pure biquadratic exchange. J. Phys. C Solid State Phys. 21, 3793 (1988)

    Article  ADS  Google Scholar 

  14. Barber, M., Batchelor, M.T.: Spectrum of the biquadratic spin-1 antiferromagnetic chain. Phys. Rev. B 40, 4621 (1989)

    Article  ADS  Google Scholar 

  15. Aufgebauer, B., Klümper, A.: Quantum spin chains of Temperley–Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature. J. Stat. Mech. 2010, 05018 (2010)

    Article  MathSciNet  Google Scholar 

  16. Sun, C.F., Xue, K., Wang, G.C., Zhou, C.C., Du, G.J.: The topological basis realization and the correspongding XXX spin chain. EPL 94, 50001 (2011)

    Article  ADS  Google Scholar 

  17. Prasolov, V.V., Sossinsky, A.B.: Knots, Links, Braids and 3-Manifolds, American Mathematical Society-Translations of Mathematical Monographs, vol. 154 (1997)

  18. Akutsu, Y., Deguchi, T., Wadati, M.: The Yang–Baxter relation: a new tool for knot theory. In: Yang, C.N., Ge, M.L. (eds.) Braid Group, Knot Theroy and Statistical Mechanics, p. 151. World Scientific Publ Co Ltd, Singapore (1989)

    Google Scholar 

  19. Kauffman, L.H., Lomonaco Jr, S.J.: Braiding operators are universal quantum gates. New J. Phys 6, 134 (2004)

    Article  ADS  Google Scholar 

  20. Yang, C.N., Ge, M.L. (eds): Braid Group, knot Theory and Statistical Mechanics (I and II). World Scientific Publ Co Ltd., Singapore (1989 and 1994)

  21. Franko, J.M., Rowell, E.C., Wang, Z.: Extraspecial 2-groups and images of braid group representations. J. Knot Theory Ramif. 15, 413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, Y., Kauffman, L.H., Ge, M.L.: Universal quantum gate, Yang-Baxterization and Hamiltonian. Int. J. Quantum. Inf. 3, 669 (2005)

    Article  MATH  Google Scholar 

  23. Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Hu, S.W., Xue, K., Ge, M.L.: Optical simulation of the Yang–Baxter equation. Phys. Rev. A 78, 022319 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  26. Wang, G.C., Xue, K., Sun, C.F., Zhou, C.C., Du, G.J.: Quantum Tunneling Effect and Quantum Zeno Effect in a Topological System, e-print arXiv:1012.1474v2

  27. Ardonne, E., Schoutens, K.: Wavefunctions for topological quantum registers. Ann. Phys. (N.Y.) 322, 201 (2007)

    Google Scholar 

  28. Feiguin, A., Trebst, S., Ludwig, A.W.W., Troyer, M., Kitaev, A., Wang, Z.H., Freedman, M.H.: Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett 98, 160409 (2007)

    Article  ADS  Google Scholar 

  29. Hikami, K.,: Skein theory and topological quantum registers: braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states. Ann. Phys. (N.Y.) 323, 1729–1769 (2008)

    Google Scholar 

  30. Chen, Y., Ge, M.L., Xue, K.: Yang baxterization of braid group representations. Commun. Math. Phys. 136, 195 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Temperley, H.N.V., Lieb, E.H.: Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proc. R. Soc. Lond. A 322, 251–280 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Kauffman, L.H.: Knots in Physics. World Scientific Pub, Singapore (1991)

    MATH  Google Scholar 

  33. Jimbo, M.: Yang–Baxter Equations on Integrable Systems. World Scientific, Singapore (1990)

    Book  MATH  Google Scholar 

  34. Jing, N.H., Ge, M.L., Wu, Y.S.: A new quantum group associated with a ‘nonstandard’ braid group representation. Lett. Math. Phys. 21, 193 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Lu, P., Müller, G., Karbach, M.: Quasiparticles in the XXZ model. Condens. Matter Phys. 12, 381–398 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by NSF of China (Grants No. 11175043) and NSF of China (Grants No. 11305033) and the Fundamental Research Funds for the Central Universities (Grants No. 11QNJJ012)

Author information

Authors and Affiliations

  1. School of Physics, Northeast Normal University, Changchun , 130024, People’s Republic of China

    Taotao Hu & Kang Xue

  2. Department of Physics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun , 130033, People’s Republic of China

    Hang Ren

Authors
  1. Taotao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hang Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kang Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taotao Hu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, T., Ren, H. & Xue, K. The topological basis expression of Heisenberg spin chain. Quantum Inf Process 13, 401–414 (2014). https://doi.org/10.1007/s11128-013-0658-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-013-0658-x

Keywords

Search

Navigation