O(N) algorithms for disordered systems
arXiv:cond-mat/0312046
Abstract
The past thirteen years have seen the development of many algorithms for approximating matrix functions in O(N) time, where N is the basis size. These O(N) algorithms rely on assumptions about the spatial locality of the matrix function; therefore their validity depends very much on the argument of the matrix function. In this article I carefully examine the validity of certain O(N) algorithms when applied to hamiltonians of disordered systems. I focus on the prototypical disordered system, the Anderson model. I find that O(N) algorithms for the density matrix function can be used well below the Anderson transition (i.e. in the metallic phase;) they fail only when the coherence length becomes large. This paper also includes some experimental results about the Anderson model's behavior across a range of disorders.
12 pages, 5 figures. All code and configuration files necessary to reproduce the results will be made available at http://www.sacksteder.com . Accepted by Numerical Linear Algebra with Applications. Changes in version two include correction of a substantial error in the error estimates section, changes of both the error estimates section and the numerical results section to give for the first time a good way of estimating the errors incurred by an O(N) algorithm, and other less important refinements
References in corpus (6)
- Exponential decay properties of Wannier functions and related quantities
- Numerical verification of universality for the Anderson transition
- Locality properties and Wannier functions for interacting systems
- Sublocalization, superlocalization, and violation of standard single parameter scaling in the Anderson model
- Spatial decay of the single-particle density matrix in tight-binding metals: +AFwAXA- analytic results in two dimensions
- Comparative numerical study of Anderson localization in disordered electron systems