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Annales Henri Poincaré

, Volume 16, Issue 2, pp 405–435 | Cite as

Localization for Random Block Operators Related to the XY Spin Chain

  • Jacob Chapman
  • Günter StolzEmail author
Article

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.

Keywords

Lyapunov Exponent Dynamical Localization Anderson Model Jacobi Matrice Canonical Anticommutation Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    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)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Bougerol P., Lacroix J.: Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetGoogle Scholar
  4. 4.
    Boumaza H.: Localization for a matrix-valued Anderson model. Math. Phys. Anal. Geom. 12(3), 225–286 (2009)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Boumaza, H., Marin, L.: Absence of absolutely continuous spectrum for random scattering zippers (2013, Preprint, arXiv:1303.3116)Google Scholar
  6. 6.
    Bravyi S., König R.: Disorder-assisted error correction in Majorana chains. Comm. Math. Phys. 316, 641–692 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Carmona R., Klein A., Martinelli F.: Anderson localization for Bernoulli and other singular potentials. Comm. Math. Phys. 108, 41–66 (1987)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Probability Theory and its Applications. Birkhäuser, Boston (1990)Google Scholar
  9. 9.
    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
  10. 10.
    Chapman, J., Stolz, G.: Dynamical localization for the quantum Ising model in random field (2014, in preparation)Google Scholar
  11. 11.
    Craig W., Simon B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Comm. Math. Phys. 90, 207–218 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Craig W., Simon B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50(2), 551–560 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Elgart, A., Schmidt, D.: Eigenvalue statistics for random block operators. Preprint, arXiv:1306.3459Google Scholar
  14. 14.
    Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators (2012, Preprint, arXiv:1201.2211)Google Scholar
  15. 15.
    Gebert M., Müller P.: Localization for random block operators. Oper. Theor. Adv. Appl. 232, 229–246 (2013)Google Scholar
  16. 16.
    Germinet F., Klein A.: Bootstrap multiscale analysis and localization in random media. Comm. Math. Phys. 222, 415–448 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Germinet, F., Klopp, F.: Spectral statistics for random Schrödinger operators in the localized regime (2010, Preprint, arXiv:1011.1832)Google Scholar
  18. 18.
    Gol’dsheid I., Margulis G.: Lyapunov indices of a product of random matrices. Russ. Math. Surv. 44:5, 11–71 (1989)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hamza E., Sims R., Stolz G.: Dynamical Localization in Disordered Quantum Spin Systems. Commun. Math. Phys. 315, 215–239 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    Kirsch, W.: An invitation to random Schrödinger operators. Panor. Synthésis vol. 25, Random Schrödinger operators, pp. 1–119, Soc. Math. France, Paris (2008)Google Scholar
  22. 22.
    Kirsch W., Metzger B., Müller P.: Random block operators. J. Stat. Phys. 143(6), 1035–1054 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kitaev, A.Yu.: Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001, see also arXiv:cond-mat/0010440)Google Scholar
  24. 24.
    Klein, A.: Multiscale analysis and localization of random operators. Random Schrödinger operators, Panor. Synthésis, vol. 25, pp. 121–159, Soc. Math. France, Paris (2008)Google Scholar
  25. 25.
    Klein A., Lacroix J., Speis A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94, 135–155 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Kotani S., Simon B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119, 403–429 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Lieb E., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Oganesyan V., Huse D.A.: Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007)ADSCrossRefGoogle Scholar
  29. 29.
    Pal A., Huse D.A.: The many-body localization phase transition. Phys. Rev. B 82, 174411 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Pastur L., Figotin A.: Spectra of Random and Almost-Periodic Operators. Springer-Verlag, Berlin (1992)CrossRefzbMATHGoogle Scholar
  31. 31.
    Pfeuty P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970)ADSCrossRefGoogle Scholar
  32. 32.
    Schulz-Baldes H.: Rotation numbers for Jacobi matrices with matrix entries. Math. Phys. Electron. J. 13(5), 40 (2007)MathSciNetGoogle Scholar
  33. 33.
    Schulz-Baldes H.: Geometry of Weyl theory for Jacobi matrices with matrix entries. J. Anal. Math. 110, 129–165 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    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)Google Scholar
  35. 35.
    Znidaric M., Prosen T., Prelovsek P.: Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsWilliam Carey UniversityHattiesburgUSA
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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