https://doi.org/10.1140/epjb/e2005-00011-1
The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents
1
Institute of Solid State Physics, University of
Latvia, 8 Kengaraga Street, LV – 1063 RIGA, Latvia
2
Institut für Physikalische und Theoretische Chemie,
Technische Universität Braunschweig, Hans-Sommer-Strasse 10,
38106 Braunschweig, Germany
Corresponding author: a kuzovkov@latnet.lv
Received:
15
June
2004
Revised:
18
October
2004
Published online:
18
January
2005
The method proposed by the present authors to deal analytically
with the problem of Anderson localization via disorder [J. Phys.:
Condens. Matter 14, 13777 (2002)] is generalized for higher
spatial dimensions D. In this way the generalized Lyapunov
exponents for diagonal correlators of the wave function, 
, can be calculated analytically and
exactly. This permits to determine the phase diagram of the
system. For all dimensions D > 2 one finds intervals in the
energy and the disorder where extended and localized states
coexist: the metal-insulator transition should thus be interpreted
as a first-order transition. The qualitative differences permit to
group the systems into two classes: low-dimensional systems
(
), where localized states are always
exponentially localized and high-dimensional systems (
), where states with non-exponential localization are also
formed. The value of the upper critical dimension is found to be
for the Anderson localization problem; this value is also
characteristic of a related problem – percolation.
PACS: 72.15.Rn – Localization effects (Anderson or weak localization) / 71.30.+h – Metal-insulator transitions and other electronic transitions
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2004
