Skip to main content
SpringerLink
Log in

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H xy , indexed by \({x,y \in \Lambda \subset \mathbb{Z}^d}\), are independent, uniformly distributed random variables if \({\lvert{x-y}\rvert}\) is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales \({t\ll W^{d/3}}\). We also show that the localization length of the eigenvectors of H is larger than a factor W d/6 times the band width. All results are uniform in the size \({\lvert{\Lambda}\rvert}\) of the matrix.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrahams E., Anderson P.W., Licciardello D.C., Ramakrishnan T.V.: Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979)

    Article  ADS  Google Scholar 

  2. Anderson G., Zeitouni O.: A CLT for a band matrix model. Probab. Theory Related Fields 134(2), 283–338 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Aizenman M., Sims R., Warzel S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264(2), 371–389 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Anderson P.: Absences of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)

    Article  ADS  Google Scholar 

  6. Bachmann S., De Roeck W.: From the Anderson model on a strip to the DMPK equation and random matrix theory. J. Stat. Phys. 139, 541–564 (2010)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bai Z.D., Yin Y.Q.: Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21(3), 1275–1294 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bourgain, J.: Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena. Lecture Notes in Mathematics, Vol. 1807, Berlin-Heidelberg-New York: Springer, 2003, pp. 70–99

  9. Chen T.: Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys. 120(1–2), 279–337 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Denisov S.A.: Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not. 2004(74), 3963–3982 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Disertori M., Pinson H., Spencer T.: Density of states for random band matrices. Commun. Math. Phys. 232, 83–124 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Disertori, M., Spencer, T.: Anderson localization for a supersymmetric sigma model. http://arxiv.org/abs/0910.3325v1 [math-ph], 2009

  13. Disertori M., Spencer T., Zirnbauer M.: Quasi-diffusion in a 3D Supersymmetric Hyperbolic Sigma Model. Commun. Math. Phys. 300, 435–486 (2010)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Efetov K.B.: Supersymmetry in Disorder and Chaos. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  15. Elgart A.: Lifshitz tails and localization in the three-dimensional Anderson model. Duke Math. J. 146(2), 331–360 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Erdős, L., Knowles, A.: Quantum diffusion and delocalization for band matrices with general distribution. http://arxiv.org/abs/1005.1838v3 [math-ph], 2010

  17. Erdős L., Péché G., Ramírez J., Schlein B., Yau H.-T.: Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63(7), 895–925 (2010)

    MathSciNet  Google Scholar 

  18. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200(2), 211–277 (2008)

    Article  MathSciNet  Google Scholar 

  19. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Commun. Math. Phys. 271, 1–53 (2007)

    Article  ADS  Google Scholar 

  20. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion for the Anderson model in scaling limit. Ann. Inst. H. Poincare 8(4), 621–685 (2007)

    Article  ADS  Google Scholar 

  21. Erdős L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287, 641–655 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  22. Erdős, L., Schlein, B., Yau, H.-T.: Universality of Random Matrices and Local Relaxation Flow. http://arxiv.org/abs/0907.5605v4 [math-ph], 2010

  23. Erdős, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. http://arxiv.org/abs/0911.3687v5 [math-ph], 2010

  24. Erdős L., Yau H.-T.: Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation. Commun. Pure Appl. Math. LIII, 667–735 (2000)

    Article  Google Scholar 

  25. Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. http://arxiv.org/abs/1001.3453v6 [math-ph], 2010

  26. Feldheim, O., Sodin, S.: A universality result for the smallest eigenvalues of certain sample covariance matrices. http://arxiv.org/abs/0812.1961v4 [math-ph], 2009

  27. Froese R., Hasler D., Spitzer W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230(1), 184–221 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fröhlich, J., de Roeck, W.: Diffusion of a massive quantum particle coupled to a quasi-free thermal medium in dimension d ≥ 4. http://arxiv.org/abs/0906.5178v3 [math-ph], 2010

  29. Fröhlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)

    Article  MATH  ADS  Google Scholar 

  30. Fröhlich J., Martinelli F., Scoppola E., Spencer T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101(1), 21–46 (1985)

    Article  MATH  ADS  Google Scholar 

  31. Fyodorov Y.V., Mirlin A.D.: Scaling properties of localization in random band matrices: a σ-model approach. Phys. Rev. Lett. 67, 2405–2409 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Gradshteyn I.S., Ryzhik I.M.: Tables of integrals, series, and products. Academic Press, New York (2007)

    Google Scholar 

  33. Guionnet A.: Large deviation upper bounds and central limit theorems for band matrices. Ann. Inst. H. Poincaré Probab. Stat. 38, 341–384 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Kirsch W., Krishna M., Obermeit J.: Anderson model with decaying randomness: mobility edge. Math. Z. 235, 421–433 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Klein A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)

    MATH  MathSciNet  Google Scholar 

  36. Krasikov I.: Uniform bounds for Bessel functions. J. Appl. Anal. 12(1), 83–91 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mehta M.L.: Random Matrices. Academic Press, New York (1991)

    MATH  Google Scholar 

  38. Minami N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177(3), 709–725 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Rodnianski I., Schlag W.: Classical and quantum scattering for a class of long range random potentials. Int. Math. Res. Not. 5, 243–300 (2003)

    Article  MathSciNet  Google Scholar 

  40. Schenker J.: Eigenvector localization for random band matrices with power law band width. Commun. Math. Phys. 290, 1065–1097 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Schlag W., Shubin C., Wolff T.: Frequency concentration and location lengths for the Anderson model at small disorders. J. Anal. Math. 88, 173–220 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  42. Sodin, S.: The spectral edge of some random band matrices. http://arxiv.org/abs/0906.4047v4 [math-ph], 2010

  43. Spencer, T.: Lifshitz tails and localization. Preprint, 1993

  44. Spencer, T.: Random banded and sparse matrices (Chapter 23). To appear in “Oxford Handbook of Random Matrix Theory”, edited by G. Akemann, J. Baik, P. Di Francesco, Oxford Univ. Press, 2010

  45. Spohn H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17(6), 385–412 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  46. Tao, T., Vu, V.: Random matrices: Universality of the local eigenvalue statistics. http://arxiv/abs/0906.0510v10 [math.PR], 2010, to appear Acta Math.

  47. Valkó, B., Virág, B.: Random Schrödinger operators on long boxes, noise explosion and the GOE. http://arxiv.org/abs/0912.0097v2 [math.PR], 2009

  48. Wigner E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62, 548–564 (1955)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333, Munich, Germany

    László Erdős

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Antti Knowles

Authors
  1. László Erdős
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antti Knowles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antti Knowles.

Additional information

Communicated by H.-T. Yau

Partially supported by SFB-TR 12 Grant of the German Research Council.

Partially supported by U.S. National Science Foundation Grant DMS 08–04279.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Erdős, L., Knowles, A. Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model. Commun. Math. Phys. 303, 509–554 (2011). https://doi.org/10.1007/s00220-011-1204-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1204-2

Keywords

Search

Navigation