1204.6490 (Emil Prodan)

Quantum Transport in Disordered Systems under Magnetic Fields: A Study
based on Operator Algebras    [PDF]

Emil Prodan

The noncommutative theory of charge transport in aperiodic solids is analyzed from a computational point of view. The linear conductivity tensor for these systems was formulated as a noncommutative Kubo formula [Bellissard et al, 1994, 1998], defined directly in the thermodynamic limit within the framework of $C^*$-algebras and noncommutative calculi over the infinite space. The present work defines an approximate $C^*$-algebra and an approximate noncommutative calculus over a finite real-space torus, which then naturally leads to an approximate finite-volume Kubo formula amenable on a computer. For finite temperatures, it is shown that this approximate formula converges exponentially fast to its thermodynamic limit. The approximate noncommutative Kubo formula is then deconstructed to a form that is implementable on a computer and simulations of the quantum transport in a 2-dimensional disordered lattice gas in a magnetic field are presented. These simulations have direct relevance for the Integer Quantum Hall Effect.

View original: http://arxiv.org/abs/1204.6490