Propagation of Correlations in Quantum Lattice Systems
First Online: 06 July 2006 Received: 24 March 2006 Accepted: 31 May 2006 DOI:
10.1007/s10955-006-9143-6 Cite this article as: Nachtergaele, B., Ogata, Y. & Sims, R. J Stat Phys (2006) 124: 1. doi:10.1007/s10955-006-9143-6 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 References
O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Volume 2., 2nd Edn. (Springer Verlag, 1997).
S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and toplogical quantum order, arXiv:quant-ph/0603121.
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.
J. Eisert and T. J. Osborne, General entanglement scaling laws from time evolution, arXiv:quant-phys/0603114.
M. B. Hastings, Locality in Quantum and Markov Dynamics on Lattices and Networks.
Phys. Rev. Lett
, 140402 (2004).
M. B. Hastings and T. Koma, Spectral Gap and Exponential Decay of Correlations, to appear in
Commun. Math. Phys., arXiv:math-ph/0507008.
T. Matsui, Markov semigroups on UHF algebras.
Rev. Math. Phys
, 587–600 (1993).
MATH CrossRef MathSciNet
E. H. Lieb and D. W. Robinson, The Finite Group Velocity of Quantum Spin Systems.
Commun. Math. Phys
, 251–257 (1972).
CrossRef ADS MathSciNet
B. Nachtergaele and R. Sims, Lieb-Robinson Bounds and the Exponential Clustering Theorem.
Commun. Math. Phys. 265, 119–130 (2006), arXiv:math-ph/0506030
N. Schuch, J. I. Cirac and M. M. Wolf, Quantum states on harmonic lattices, arXiv:quant-ph/0509166.
B. Simon, The Statistical Mechanics of Lattice Gases, Volume I, (Princeton University Press, 1993).
© Springer Science + Business Media, Inc. 2006