O ( N ) algorithms for disordered systems

The past 13 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 behaviour across a range of disorders. Copyright © 2005 John Wiley & Sons, Ltd.