Using the strong localization bounds obtained by the Aizenman–Molcanov method for a particle in a magnetic field and a disordered potential, we show that the zero-temperature Hall conductivity of a gas of such particles is quantized and constant as long as both Fermi energy and disorder coupling parameter vary in a region of strong localization of the corresponding two-dimensional phase diagram.

1.
The Quantum Hall Effect, 2nd Ed., edited by R. Prange and S. Girvin, (Springer-Verlag, Berlin, 1990).
2.
J.
Bellissard
,
A.
van Elst
, and
H.
Schulz-Baldes
, “
The noncommutative geometry of the quantum Hall effect
,”
J. Math. Phys.
35
,
5373
5451
(
1994
).
3.
M.
Aizenman
and
S.
Molchanov
, “
Localization at large disorder and at extreme energies: An elementary derivation
,”
Commun. Math. Phys.
157
,
245
278
(
1993
).
4.
M.
Aizenman
and
G.
Graf
, “
Localization bounds for an electron gas
,”
J. Phys. A
31
,
6783
6806
(
1998
).
5.
M.
Aizenman
, “
Localization at weak disorder: Some elementary bounds
,”
Rev. Math. Phys.
6
,
1163
1182
(
1994
).
6.
M. Aizenman, J. H. Schenker, R. M. Friedrich, and D. Hundertmark, “Finite volume criteria for Anderson localization,” Commun. Math. Phys. m _p.arc 99–389 (to be published).
7.
Y.
Tan
, “
Localization and quantum Hall effect in a two-dimensional periodic potential
,”
J. Phys.: Condens. Matter
6
,
7941
7954
(
1994
).
8.
F.
Wegner
, “
Density of states in disordered systems
,”
Z. Phys. B: Condens. Matter
44
,
9
15
(
1981
).
9.
Y.
Avron
and
L.
Sadun
, “
Fredholm indices and phase diagram for quantum Hall systems
,”
J. Math. Phys.
42
,
1
14
(
2001
).
10.
D. J.
Thouless
,
M.
Kohmoto
,
M. P.
Nightingale
, and
M.
den Nijs
, “
Quantized Hall coinductance in a two-dimensional periodic potential
,”
Phys. Rev. Lett.
49
,
405
408
(
1982
).
11.
S.
Nakamura
and
J.
Bellissard
, “
Low energy bands do not contribute to the quantum Hall effect
,”
Commun. Math. Phys.
131
,
282
305
(
1990
).
12.
D.
Khmelnitsky
, “
Quantization of Hall conductivity
,”
JETP Lett.
38
,
552
556
(
1983
).
13.
H.
Schulz-Baldes
,
J.
Kellendonk
, and
T.
Richter
, “
Simultaneous quantization of the edge and bulk Hall conductivity
,”
J. Phys. A
33
,
L27
L32
(
2000
).
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