Abstract
We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a proof of the Exponential Clustering Theorem, and discuss a multi-dimensional Lieb-Schultz-Mattis Theorem.
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References
I. Affleck and E. Lieb, A proof of part of Haldane’s conjecture on quantum spin chains. Lett. Math. Phys. 12, 57–69 (1986)
H. Araki, Gibbs states of a one dimensional quantum lattice. Commun. Math. Phys. 14, 120–157 (1969)
H. Araki, K. Hepp, and D. Ruelle, Asymptotic behaviour of Wightman functions. Helv. Phys. Acta 35, 164 (1962)
S. Bravyi, M.B. Hastings, and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006). arXiv:quant-ph/0603121
J. Eisert and T.J. Osborne, General entanglement scaling laws from time evolution. Phys. Rev. Lett. 97, 150404 (2006). arXiv:quant-ph/0603114
K. Fredenhagen, A remark on the cluster theorem. Commun. Math. Phys. 97, 461–463 (1985)
R. Haag, Local Quantum Physics. Springer, Berlin (1992)
F.D.M. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model. Phys. Lett. A 93, 464–468 (1983)
M.B. Hastings, Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)
M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006)
M.B. Hastings and X.-G. Wen, Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005)
M. Keyl, T. Matsui, D. Schlingemann, and R. Werner, Entanglement, Haag-duality and type properties of infinite quantum spin chains. arXiv:math-ph/0604071
E.H. Lieb and D. Mattis, Ordering energy levels in interacting spin chains. J. Math. Phys. 3, 749–751 (1962)
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)
E.H. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)
T.M. Liggett, Interacting Particle Systems. Springer, Berlin (1985)
B. Nachtergaele, The spectral gap for some quantum spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565–606 (1996)
B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006)
B. Nachtergaele and R. Sims, Recent progress in quantum spin systems. Markov Process. Relat. Fields 13, 315–329 (2007)
B. Nachtergaele and R. Sims, A multi-dimensional Lieb-Schultz-Mattis theorem. Commun. Math. Phys. 276, 437–472 (2007)
B. Nachtergaele, Y. Ogata, and R. Sims, Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124, 1–13 (2006)
B. Nachtergaele, H. Raz, B. Schlein, and R. Sims, Lieb-Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009)
D. Ruelle, On the asymptotic condition in quantum field theory. Helv. Phys. Acta 35, 147 (1962)
W. Wreszinski, Charges and symmmetries in quantum theories without locality. Fortschr. Phys. 35, 379–413 (1987)
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Nachtergaele, B., Sims, R. (2009). Locality Estimates for Quantum Spin Systems. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_39
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DOI: https://doi.org/10.1007/978-90-481-2810-5_39
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2809-9
Online ISBN: 978-90-481-2810-5
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