Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H xy , indexed by , are independent, uniformly distributed random variables if is less than the band width W , and zero otherwise. We prove that the time evolution of a quantum particle subject to the...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 303; no. 2; pp. 509 - 554
Main Authors: Erdős, László, Knowles, Antti
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.04.2011
Springer
Subjects:
Algebra
Classical and Quantum Gravitation
Complex Systems
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Theoretical
Quantum Diffusion
Heat Kernel
Chebyshev Polynomial
Anderson Model
Band Matrice
Band matrix
Symmetric matrix
Hamiltonians
Eigenfunctions
Eigenvectors
Localization
Random variable
Mathematical physics
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H xy , indexed by , are independent, uniformly distributed random variables if 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 . 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 of the matrix.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-011-1204-2