Journal of Statistical Physics

, Volume 116, Issue 1–4, pp 79–95

Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture

  • B.-Q. Jin
  • V. E. Korepin
Article

Abstract

We consider the one-dimensional quantum spin chain, which is called the XX model (XX0 model or isotropic XY model) in a transverse magnetic field. We are mainly interested in the entropy of a block of Lneighboring spins at zero temperature and of an infinite system. We represent the entropy in terms of a Toeplitz determinant and calculate the asymptotic analytically. We derive the first two terms of the asymptotic decomposition. Interestingly, these two terms of decomposition clearly show a length scale related to the field h.

quantum spin chain XX0 model entropy Toeplitz determinant quantum entanglement 

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REFERENCES

  1. 1.
    C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53:2046 (1996).Google Scholar
  2. 2.
    A. Rényi, Probability Theory(North-Holland, Amsterdam, 1970).Google Scholar
  3. 3.
    S. Abe and A. K. Rajagopal, Phys. Rev. A 60:3461 (1999).Google Scholar
  4. 4.
    E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16:407 (1961).Google Scholar
  5. 5.
    E. Barouch and B. M. McCoy, Phys. Rev. A 3:786 (1971)Google Scholar
  6. 6.
    E. Barouch, B. M. McCoy, and M. Dresden, Phys. Rev. A 2:1075 (1970)Google Scholar
  7. 7.
    D. B. Abraham, E. Barouch, G. Gallavotti, and A. Martin-Löf, Phys. Rev. Lett. 25:1449 (1970); Studies in Appl. Math. 50:121 (1971); ibid. 51:211 (1972).Google Scholar
  8. 8.
    C. Tsallis, J. Stat. Phys. 52:479(1988).Google Scholar
  9. 9.
    N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin, Quantum Inverse Scattering Method and Correlation Functions(Cambridge University Press, Cambridge, 1993).Google Scholar
  10. 10.
    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90:227902 (2003).Google Scholar
  11. 11.
    J. I. Latorre, E. Rico, and G. Vidal, arXiv: quant-ph/0304098.Google Scholar
  12. 12.
    C. A. Cheong and C. L. Henley, arXiv: cond-mat/0206196.Google Scholar
  13. 13.
    T. T. Wu, Phys. Rev. 149:380 (1966).Google Scholar
  14. 14.
    M. E. Fisher and R. E. Hartwig, Adv. Chem. Phys. 15:333 (1968).Google Scholar
  15. 15.
    E. L. Basor, Indiana Math. J. 28:975 (1979).Google Scholar
  16. 16.
    E. L. Basor and C. A. Tracy, Physica A 177:167 (1991).Google Scholar
  17. 17.
    A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators(Springer-Verlag, Berlin, 1990).Google Scholar
  18. 18.
    F. Franchini, private communication.Google Scholar
  19. 19.
    V. E. Korepin, Phys. Rev. Lett. 92:096402 (2004).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • B.-Q. Jin
    • 1
  • V. E. Korepin
    • 1
  1. 1.C. N. Yang Institute for Theoretical PhysicsState University of New York at Stony BrookStony Brook

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