Journal of Statistical Physics

, Volume 124, Issue 1, pp 1–13 | Cite as

Propagation of Correlations in Quantum Lattice Systems

Article

Abstract

We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.

Keywords

Lieb-Robinson bounds quantum spin systems correlations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Volume 2., 2nd Edn. (Springer Verlag, 1997).Google Scholar
  2. 2.
    S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and toplogical quantum order, arXiv:quant-ph/0603121.Google Scholar
  3. 3.
    M. Cramer and J. Eisert, Correlations and spectral gap in harmonic quantum systems on generic lattices. New J. Phys. 871, (2006), arXiv:quant-ph/0509167.Google Scholar
  4. 4.
    J. Eisert and T. J. Osborne, General entanglement scaling laws from time evolution, arXiv:quant-phys/0603114.Google Scholar
  5. 5.
    M. B. Hastings, Locality in Quantum and Markov Dynamics on Lattices and Networks. Phys. Rev. Lett. 93, 140402 (2004).CrossRefADSGoogle Scholar
  6. 6.
    M. B. Hastings and T. Koma, Spectral Gap and Exponential Decay of Correlations, to appear in Commun. Math. Phys., arXiv:math-ph/0507008.Google Scholar
  7. 7.
    T. Matsui, Markov semigroups on UHF algebras. Rev. Math. Phys. 5, 587–600 (1993).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    E. H. Lieb and D. W. Robinson, The Finite Group Velocity of Quantum Spin Systems. Commun. Math. Phys. 28, 251–257 (1972).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    B. Nachtergaele and R. Sims, Lieb-Robinson Bounds and the Exponential Clustering Theorem. Commun. Math. Phys. 265, 119–130 (2006), arXiv:math-ph/0506030Google Scholar
  10. 10.
    N. Schuch, J. I. Cirac and M. M. Wolf, Quantum states on harmonic lattices, arXiv:quant-ph/0509166.Google Scholar
  11. 11.
    B. Simon, The Statistical Mechanics of Lattice Gases, Volume I, (Princeton University Press, 1993).Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Bruno Nachtergaele
    • 1
  • Yoshiko Ogata
    • 1
    • 2
  • Robert Sims
    • 1
  1. 1.Department of MathematicsUniversity of California at DavisDavisUSA
  2. 2.Department of Mathematical SciencesUniversity of TokyoTokyoJapan

Personalised recommendations