Chapter

SUSY statistical mechanics and random band matrices Thomas Spencer

Thomas Spencer

in Quantum Theory from Small to Large Scales

Published in print May 2012 | ISBN: 9780199652495
Published online September 2012 | e-ISBN: 9780191741203 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199652495.003.0007

Series: Lecture Notes of the Les Houches Summer School

SUSY statistical mechanics and random band matrices Thomas Spencer

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This chapter presents a mathematical approach to the spectral analysis of large random band matrices, using statistical mechanics and the chapter supersymmetric (SUSY) formalism. It emphasizes the simplest aspects of this approach by starting with GUE matrices. Random band matrices are indexed by vertices i, j of a lattice. Their matrix elements are small for large ¦i-j¦ and hence such matrices reflect the geometry of the lattice. The study of random band matrices is motivated by problems arising from disordered quantum systems — especially Anderson localization and delocalization. The chapter concludes with a mathematical discussion of a 3D SUSY sigma model which exhibits a phase transition analogous to the Anderson transition.

Keywords: random band matrix; supersymmetry; delocalization; Anderson model

Chapter.  27105 words.  Illustrated.

Subjects: Atomic, Molecular, and Optical Physics

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