Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

• 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
• DOI: 10.1007/s00220-011-1204-2
• ISSN: 0010-3616

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.