Abstract
We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy.
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Basko D.M., Aleiner I.L., Altshuler B.L.: Metal-insulator transition in a weakly interacting many-electron system with localizad single-particle states. Ann. Phys. 321, 1126–1205 (2006)
Bougerol P., Lacroix J.: Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston (1985)
Boumaza H., Stolz G.: Positivity of Lyapunov exponents for Anderson-type models on two coupled strings. Electron. J. Diff. Eq. 2007(47), 1–18 (2007)
Boumaza H.: Localization for a matrix-valued Anderson model. Math. Phys. Anal. Geom. 12(3), 225–286 (2009)
Boumaza, H., Marin, L.: Absence of absolutely continuous spectrum for random scattering zippers (2013, Preprint, arXiv:1303.3116)
Bravyi S., König R.: Disorder-assisted error correction in Majorana chains. Comm. Math. Phys. 316, 641–692 (2012)
Carmona R., Klein A., Martinelli F.: Anderson localization for Bernoulli and other singular potentials. Comm. Math. Phys. 108, 41–66 (1987)
Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Probability Theory and its Applications. Birkhäuser, Boston (1990)
Chapman, J.: Spectral Properties of Random Block Operators. Ph.D. Thesis, University of Alabama at Birmingham (2013), electronically available at http://gradworks.umi.com/3561259
Chapman, J., Stolz, G.: Dynamical localization for the quantum Ising model in random field (2014, in preparation)
Craig W., Simon B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Comm. Math. Phys. 90, 207–218 (1983)
Craig W., Simon B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50(2), 551–560 (1983)
Elgart, A., Schmidt, D.: Eigenvalue statistics for random block operators. Preprint, arXiv:1306.3459
Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators (2012, Preprint, arXiv:1201.2211)
Gebert M., Müller P.: Localization for random block operators. Oper. Theor. Adv. Appl. 232, 229–246 (2013)
Germinet F., Klein A.: Bootstrap multiscale analysis and localization in random media. Comm. Math. Phys. 222, 415–448 (2001)
Germinet, F., Klopp, F.: Spectral statistics for random Schrödinger operators in the localized regime (2010, Preprint, arXiv:1011.1832)
Gol’dsheid I., Margulis G.: Lyapunov indices of a product of random matrices. Russ. Math. Surv. 44:5, 11–71 (1989)
Hamza E., Sims R., Stolz G.: Dynamical Localization in Disordered Quantum Spin Systems. Commun. Math. Phys. 315, 215–239 (2012)
Kirsch, W.: Random Schrödinger operators. Schrödinger operators. Proc. Nord. Summer Sch. Math., Sandbjerg Slot, Sonderborg/Denmark 1988, Lect. Notes Phys. vol. 345, pp. 264–370 (1989)
Kirsch, W.: An invitation to random Schrödinger operators. Panor. Synthésis vol. 25, Random Schrödinger operators, pp. 1–119, Soc. Math. France, Paris (2008)
Kirsch W., Metzger B., Müller P.: Random block operators. J. Stat. Phys. 143(6), 1035–1054 (2011)
Kitaev, A.Yu.: Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001, see also arXiv:cond-mat/0010440)
Klein, A.: Multiscale analysis and localization of random operators. Random Schrödinger operators, Panor. Synthésis, vol. 25, pp. 121–159, Soc. Math. France, Paris (2008)
Klein A., Lacroix J., Speis A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94, 135–155 (1990)
Kotani S., Simon B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119, 403–429 (1988)
Lieb E., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)
Oganesyan V., Huse D.A.: Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007)
Pal A., Huse D.A.: The many-body localization phase transition. Phys. Rev. B 82, 174411 (2010)
Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Springer-Verlag, Berlin (1992)
Pfeuty P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970)
Schulz-Baldes H.: Rotation numbers for Jacobi matrices with matrix entries. Math. Phys. Electron. J. 13(5), 40 (2007)
Schulz-Baldes H.: Geometry of Weyl theory for Jacobi matrices with matrix entries. J. Anal. Math. 110, 129–165 (2010)
Stolz, G.: An introduction to the mathematics of Anderson localization. Entropy and the quantum II. Contemp. Math. vol. 552, pp. 71–108, Amer. Math. Soc., Providence (2011)
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 Anton Bovier.
Both authors were supported in part by NSF grant DMS-1069320 (PI G. Stolz).
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Chapman, J., Stolz, G. Localization for Random Block Operators Related to the XY Spin Chain. Ann. Henri Poincaré 16, 405–435 (2015). https://doi.org/10.1007/s00023-014-0328-2
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DOI: https://doi.org/10.1007/s00023-014-0328-2