It is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a two-dimensional manifold.
REFERENCES
1.
2.
B.
Simon
, Phys. Rev. Lett.
51
, 2167
(1983
).3.
F.
Wilczek
and A.
Zee
, Phys. Rev. Lett.
52
, 2111
(1984
).4.
D.
Viennot
, J. Math. Phys.
46
, 072102
(2005
).5.
Y. M.
Cho
, Phys. Rev. Lett.
44
, 1115
(1980
).6.
A.
Sinha
, Phys. Rev. D
14
, 2016
(1976
).7.
J.
Moody
, A.
Shapere
, and F.
Wilczek
, Phys. Rev. Lett.
56
, 893
(1986
).8.
9.
10.
11.
A.
Bohm
, A.
Mostafazadeh
, H.
Koizumi
, Q.
Niu
, and J.
Zwanziger
, The Geometric Phase in Quantum Systems
(Springer
, Berlin
, 2003
).12.
A.
Bohm
, Lectures notes
, NATO Summer Institute, Salamaca, 1992
.13.
14.
Y.
Aharonov
and J.
Anandan
, Phys. Rev. Lett.
58
, 1593
(1987
).15.
V.
Rohlin
and D.
Fuchs
, Premiers Cours de Topologie, Chapitres Géométriques
(Mir
, Moscow
, 1977
).16.
N.
Steenrod
, The Topology of Fibre Bundles
(Princeton University Press
, New York
, 1951
).17.
18.
J.
Anandan
and Y.
Aharonov
, Phys. Rev. Lett.
65
, 1697
(1990
).19.
20.
J. P.
Killingbeck
and G.
Jolicard
, J. Phys. A
36
, R105
(2003
).21.
22.
P.
Durand
and I.
Paidarová
, Phys. Rev. A
28
, 3184
(1983
).23.
B.
Doubrovine
, S.
Novikov
, and A.
Fomenko
, Géométrie Contemporaine 1: Géométrie des Surfaces, des Groupes de Transformations et des Champs
(Mir
, Moscow
, 1979
).24.
B. G.
Adams
, J.
Čižek
, and J.
Paldus
, Adv. Quantum Chem.
19
, 1
(1988
).25.
P.
Leboeuf
and A.
Mouchet
, J. Phys. A
36
, 2847
(2003
).26.
27.
D.
Lehmann
and C.
Sacré
, Géométrie et Topologie des Surfaces
(Presses Universitaires de France
, Paris
, 1982
).28.
J. N.
Elgin
, Phys. Lett.
80
, 140
(1980
).29.
F. T.
Hioe
and J. H.
Eberly
, Phys. Rev. Lett.
47
, 838
(1981
).30.
A.
Mostafazadeh
, J. Math. Phys.
37
, 1218
(1996
).31.
Z. F.
Ezawa
and H. C.
Tze
, Phys. Rev. D
15
, 1647
(1977
).32.
H.
Bacry
, Leçons sur la Théorie des Groupes et les Symétries des Particules Élémentaires
(Dunod
, Paris
, 1967
).33.
R.
Ticciati
, Quantum Field Theory for Mathematicians
(Cambridge University Press
, New York
, 1999
).34.
G.
Jolicard
and J. P.
Killingbeck
, J. Phys. A
36
, R411
(2003
).35.
A.
Mostafazadeh
and A.
Bohm
, J. Phys. A
26
, 5473
(1993
).36.
A.
Mostafazadeh
, J. Phys. A
32
, 8157
(1999
).© 2006 American Institute of Physics.
2006
American Institute of Physics
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