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O ( N ) algorithms for disordered systems
Author(s) -
Sacksteder V. E.
Publication year - 2005
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.454
Subject(s) - matrix (chemical analysis) , locality , mathematics , function (biology) , algorithm , anderson impurity model , range (aeronautics) , discrete mathematics , statistical physics , physics , quantum mechanics , philosophy , linguistics , materials science , evolutionary biology , composite material , biology , electron
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.