Abstract
We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.
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References
Affleck I., Kennedy T., Lieb E. H., Tasaki H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477–528 (1988)
Aizenman M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994)
Aizenman, M., Klein, A., Newman, C.M.: Percolation methods for disordered quantum Ising models. In: Phase transitions: Mathematics, Physics, Biology, Kotecky, R. ed., Singapore: World Scientific, 1993
Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies. Commun. Math. Phys. 157, 245–278 (1993)
Aizenman M., Warzel S.: Localization bounds for multiparticle systems. Commun. Math. Phys. 290, 903–934 (2009)
Amour L., Levy-Bruhl P., Nourrigat J.: Dynamics and Lieb-Robinson estimates for lattices of interacting anharmonic oscillators. Colloq. Math. 118(2), 609–648 (2010)
Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309, 835–871 (2012)
Basko D. M., Aleiner I. L., Altshuler B. L.: Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006)
Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics 2, 2nd ed., New York, NY: Springer Verlag, 1997
Bravyi S., Hastings M.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609–627 (2011)
Bravyi S., Hastings M., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)
Burrell C. K., Eisert J., Osborne T. J.: Information propagation through quantum chains with fluctuating disorder. Phys. Rev. A 80, 052319 (2009)
Burrell C.K., Osborne T.J.: Bounds on the speed of information propagation in disordered quantum spin chains. Phys. Rev. Lett. 99, 167201 (2007)
Campanino M., Klein A., Perez J.F.: Localization in the ground state of the Ising model with a random transverse field. Commun. Math. Phys. 135, 499–515 (1991)
Chulaevsky V., Suhov Y.: Multi-particle Anderson localisation: induction on the number of particles. Math. Phys. Anal. Geom. 12, 117–139 (2009)
Chulaevsky V., Suhov Y.: Eigenfunctions in a two-particle Anderson tight binding model. Commun. Math. Phys. 289, 701–723 (2009)
Cramer, M., Serafini, A., Eisert, J.: Locality of dynamics in general harmonic quantum systems. In: Quantum Information and Many Body Quantum Systems. Ericsson, M., Montangero, S. eds., Pisa: Edizioni della Normale ISBN 78-88-7642-307-9, 2008
Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin-Heidelberg: Springer, 1987
Damanik, D.: A short course on one-dimensional random Schrödinger operators. http://arxiv.org/abs/1107.1094v1 [math.SP], 2011
Fisher D. S.: Random antiferromagnetic quantum spin chains. Phys. Rev. B 50, 3799–3821 (1994)
Hamza E., Sims R., Stolz G.: A note on fractional moments for the one-dimensional continuum Anderson model. J. Math. Anal. Appl. 365, 435–446 (2010)
Hastings M.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)
Hastings M.: An area law for one dimensional quantum systems. JSTAT 2007, P08024 (2007)
Hastings, M.: Quasi-adiabatic continuation of disordered systems: applications to correlations, Lieb-Schultz-Mattis, and Hall conductance. http://arxiv.org/abs/1001.5280v2 [math-ph], 2010
Hastings, M.: Locality in Quantum Systems. http://arxiv.org/abs/1008.5137v1 [math-ph], 2010
Hastings M., Koma T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265(3), 781–804 (2006)
Hastings, M., Michalakis, S.: Quantization of hall conductance for interacting electrons without averaging assumptions. http://arxiv.org/abs/0911.4706v1 [math-ph], 2009
Kirsch, W.: An invitation to random Schrödinger operators. With an appendix by Frédéric Klopp. Panor. Synthèses, 25, Random Schrödinger operators, Paris: Soc. Math. France, 2008, pp. 1–119
Kirsch W., Metzger B., Müller P.: Random block operators. J. Stat. Phys. 143(6), 1035–1054 (2011)
Klein A., Perez J. F.: Localization in the ground-state of the one dimensional X-Y model with a random transverse field. Commun. Math. Phys. 128, 99–108 (1990)
Klein A., Perez J. F.: Localization in the ground state of a disordered array of quantum rotators. Commun. Math. Phys. 147, 241–252 (1992)
Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, 201–246 (1980/81)
Lieb E. H., Robinson D. W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)
Lieb E. H., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)
Minami N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996)
Nachtergaele B., Ogata Y., Sims R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124(1), 1–13 (2006)
Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb-Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009)
Nachtergaele B., Schlein B., Sims R., Starr S., Zagrebnov V.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22(2), 207–231 (2010)
Nachtergaele B., Sims R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265(1), 119–130 (2006)
Nachtergaele B., Sims R.: A multi-dimensional Lieb-Schultz-Mattis theorem. Commun. Math. Phys. 276, 437–472 (2007)
Nachtergaele, B., Sims, R.: Locality estimates for quantum spin systems. In: New trends in mathematical physics. Selected contributions of the XVth international congress on mathematical physics. Berlin-Heidelberg-New York: Springer-Verlag, 2009, pp. 591–614
Nachtergaele, B., Sims, R.: Lieb-Robinson bounds in quantum many-body physics. In: Entropy and the Quantum (Tucson, AZ, 2009), Contemp. Math. 529, Providence, RI: Amer. Math. Soc., 2010, pp. 141 – 176
Nachtergaele, B., Vershynina, A., Zagrebnov, V.: Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics. In: Entropy and the Quantum II (Tucson, AZ, 2010), Contemp. Math. 552, Providence, RI: Amer. Math. Soc., 2011, pp. 161 – 175
Poulin D.: Lieb-Robinson bound and locality for general Markovian quantum dynamics. Phys. Rev. Lett. 104, 190401 (2010)
Prémont-Schwarz I., Hamma A., Klich I., Markopoulou-Kalamara F.: Lieb-Robinson bounds for commutator-bounded operators. Phys. Rev. A. 81(4), 040103(R) (2010)
Simon, B.: The statistical mechanics of lattice gases. Princeton Series in Physics, Vol, 1, Princeton, NJ: Princeton University Press, 1993
Sims, R.: Lieb-Robinson bounds and quasi-locality for the dynamics of many-body quantum systems. In: Mathematical results in quantum physics. Proceedings of the QMath 11 Conference, Exner, P. ed., River Edge, NJ: World Scientific, 2011, pp, 95–106
Stolz, G.: An introduction to the mathematics of Anderson localization. In: Entropy and the Quantum II (Tucson, AZ, 2010), Contemp. Math. 552, Providence, RI: Amer. Math. Soc., 2011, pp. 71–108
Znidaric M., Prosen T., Prelovsek P.: Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008)
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Communicated by H. Spohn
R. S. was supported in part by NSF grants DMS-0757424 and DMS-1101345.
G. S. was supported in part by NSF grants DMS-0653374 and DMS-1069320.
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Hamza, E., Sims, R. & Stolz, G. Dynamical Localization in Disordered Quantum Spin Systems. Commun. Math. Phys. 315, 215–239 (2012). https://doi.org/10.1007/s00220-012-1544-6
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DOI: https://doi.org/10.1007/s00220-012-1544-6