Electron Monopole Duality in Quantum Hall Effect25

2

=, =

22

2

xy

xx

xxxy .

xxy xxxy

(28)

for classical Hall effect. In quantum mechanics, the Ham-

iltonian is ( along the X- direction)

2

1

=

2.

gB

pg

mc

H

(29)

Now we choose the Landu gauge in which the vector

potential is indep e ndent of Y— coordinate as

=0, ,0.

x

AE (30)

Let us take a wave function which has a plane-wave

dependence on the Y— coordinate as

,=

ik y

y

ye x

(31)

and substituting the Equation (31) into the Schroedinger

equation, we get

22

22

2

1=

22

cmc y

dmxIkgHxxx

mdx

(32)

where Ic is the classical cyclotron orbit radius .

=

c

c

I.

H

(33)

So, the eigan values and eigan states are described as

2

2

1

=22

iccy

cc

gH

Ei gHIkm

(34)

20

2

0

,= exp2

ik y

y

ic

ci

x

xyex xIHI

(35)

where 2

02

=cy

c

E

xIkm

.

4. Summary and Conclusions

The precision of the quantization in the Hall effect is re-

markable in that it takes place in systems that are impre-

cisely characterized on the microscopic scale. Different

samples may have dierent distributions of impurities,

different geometry and different concentrations of elec-

trons. Nevertheless, whenever their Hall conductances

are quantized, the quantized values mutually agree with

great precision. The quantum Hall effect may also be

interpreted [11] as a measurement of the fine structure

constant so that the Hall conductance may have topo-

logical signicance. Since magnetic monopoles have the

topological origin [12] and the quantum Hall effect is

described in terms of strong magetic field, we have dis-

cussed here the theory of classical and quantum Hall ef-

fect in presence of magnetic monopole so that the mag-

netic field can be obtained directly instead of rotating an

electric charge. In order to seek the existence of magnetic

monopoles in condensed matter physics [6,7], particu-

larly in case of Hall effect, in the foregoing analysis, we

have discussed the manifestly covariant theory of mag-

netic monopoles and established the connections among

the various parameters of classical and quantum Hall

effect in terms of electron monopole duality invariance.

5. Acknowledgements

One of us OPSN is thankful to Professor H. Dehnen,

Universitt Konstanz, Fachbereich Physik, Postfach-M

677, D-78457 Konstanz, Germany for his hospitality at

Universitt Konstanz. He is also grateful to German Aca-

demic Exchange Service (Deutscher Akademischer

Austausch Dienst), Bonn for financial support under

DAAD re-invitation programme.

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