Abstract
Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness, are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO*(2L), and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely continuous distribution.
Article PDF
Similar content being viewed by others
References
Bougerol P., Lacroix J.: Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston (1985)
Boumaza, H.: Localization for a matrix-valued Anderson model. http://arxiv.org/abs/0902.1628v1[math-ph], 2009
de Bievre, S., Pulé, J.V.: Propagating edge states for a magnetic Hamiltonian. Math. Phys. Elect. J. 5, Paper 3, 17 pages (1999)
Evers F., Mirlin A.D.: Anderson transitions. Rev. Mod. Phys. 80, 1355 (2008)
Fröhlich J., Graf G.M., Walcher J.: On the extended nature of edge states of Quantum Hall Hamiltonians. Ann. H. Poincaré 1(3), 405–442 (2000)
Gesztesy F., Tsekanovskii E.: On matrix-valued Herglotz functions. Math. Nachr. 218, 61–138 (2000)
Goldsheid I., Margulis G.: Lyapunov indices of a product of random matrices. Russ. Math. Surv. 44, 11–71 (1989)
Guivarch Y., Raugi A.: Fronière de Furstenberg, propriétés de contraction et théorèmes de convergence. Z. f. Wahrs. Verw. G. 69, 187–242 (1987)
Hinton D.B., Schneider A.: On the Titchmarsh-Weyl coefficients for singular S-Hermitian Systems I. Math. Nachr. 163, 323–342 (1993)
Jaksic V., Last Y.: Spectral structure of Anderson type Hamiltonians. Invent. Math. 141, 561–577 (2000)
Jitormiskaya S., Schulz-Baldes H., Stolz G.: Delocalization in random polymer models. Commun. Math. Phys. 233, 27–48 (2003)
Kellendonk J., Richter Th., Schulz-Baldes H.: Edge channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14, 87–119 (2002)
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. Commun. Math. Phys. 119, 403–429 (1988)
Lesch M., Malamud M.: On the number of square integrable solutions and self-adjointness of symmetric first order systems of differential equations. J. Diff. Eq. 189, 556–615 (2003)
Schulz-Baldes, H.: Rotation numbers for Jacobi matrices with matrix entries. Math. Phys. Elect. Journal 13, Paper , 40 pages (2007)
Schulz-Baldes, H.: Geometry of Weyl theory for Jacobi matrices with matrix entries. to appear in J. d’Analyse Mathématique. http://arxiv.org/abs/0804.3746v1[math-ph], 2008
Sun F.: Kotani theory for stochastic Dirac operators. Northeast. Math. J 9, 49–62 (1993)
Acknowledgments
We thank M. Zirnbauer for raising our interest in the time reversal invariant stochastic Dirac operators and the Newton Institute for hospitality and support during our stay in Cambridge. We also thank the Cambridge Philosophical Society for supporting the stay of Christian Sadel at the Newton Institute. This work was funded by the DFG.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Simon
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Sadel, C., Schulz-Baldes, H. Random Dirac Operators with Time Reversal Symmetry. Commun. Math. Phys. 295, 209–242 (2010). https://doi.org/10.1007/s00220-009-0956-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0956-4