Desbois.Les.Houches. Impurities and quantum Hall effect
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect 905
Fig. 2. Hall conductivity in unit of e2=h of the random magnetic impurity model at rst order in for = 0:01 as a function of the lling factor = : straight line = classical result, full line = perturbative result.
largely discussed in the literature [10], as well as the persistent current due to a point-like vortex [11].
Let us start with the de nition of the total magnetization
M = 2 |
Z |
d~r (~r h~j(~r)i) ~k + 2h zi |
(44) |
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e |
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where h i means average over Boltzmann or Fermi-Dirac distributions. In
the Boltzmann case, |
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obtains |
the thermal |
magnetization |
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Z Tr e−H |
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Trne−H (~r ~v) ~k + zo |
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M = |
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(45) |
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2Z |
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It is easy to realize that |
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lim |
@ lnZ (B0) |
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M = |
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(46) |
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@B0 |
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B0!0 |
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where Z (B0) is the partition function when an uniform magnetic eld, B0, perpendicular to the plane, is added to the system.
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Topological Aspects of Low Dimensional Systems |
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Dropping the spin term, the orbital part of the magnetization reads |
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Morb = M − |
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z: |
(47) |
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Let us now turn to the persistent current and consider in the plane a semi in nite line D starting at ~r0 and making an angle o with the horizontal x-axis. The orbital persistent current Iorb(~r0; 0) through the line is
Iorb(~r0; 0) ZD dj~r −~r0j |
(~r −~r0) h~j(~r)i |
~k: |
(48) |
j~r −~r0j |
Consider now systems rotationnally invariant around ~r0. Iorb(~r0; 0) no longer depends on 0 and, without loss of generality, can be averaged over0. So
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(~r −~r ) h~j(~r)i |
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Iorb(~r0) = |
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d~r |
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~k |
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j~r −~r0j2 |
and for the Boltzmann distribution
Iorb(~r0) = 2 Z Tr e−H |
j~r −~r0j2 |
~k |
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e |
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(~r |
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~v |
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It is possible to show that [3]: |
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Iorb(~r ) = |
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@ lnZ ( 0) |
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G (~r0;~r0) |
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@ 0 |
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Z |
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2 0!0 |
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z |
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(49)
(50)
(51)
where G is the thermal propagator and Z ( 0) the partition function when a ctitious vortex of strength 0 is added in ~r0. The last term in equation (51) emphasizes the importance of the spin coupling in the Hamiltonian for a correct de nition of persistent currents.
For systems that are both invariant by translation and rotation, we can write, using (45, 50),
Iorb = V |
Z |
d~r0I(~r0) = V Morb: |
(52) |
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Let us now discuss some speci c examples:
i) Point vortex in O + uniform magnetic eld case
Using the corresponding partition function (b !c) |
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V be−b |
e−b |
− e−( −1)b |
sinh b |
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Z (B; ) = |
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+ |
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(53) |
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sinhb |
2 sinhb |
sinhb |
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect
one obtains |
b − cothb |
− 2V |
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− e−( −1)b |
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Morb = 2 |
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b2 |
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sinhb |
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e 1 |
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sinh b |
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b |
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sinh b |
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2e−2 b |
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Iorb(0)~ = |
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2V |
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with, of course, Morb |
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907
(54)
(55)
ii) Magnetic impurities
After averaging over disorder, the system is invariant by translation and rotation, so it is su cient to compute Morb to get the orbital persistent current.
In the Brownian motion approach, we get:
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e− S sin( A)A |
fCg |
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Morb = − |
he− S cos( A)ifCg |
(56) |
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where A, S and A have been de ned previously in equations (9-11). (56) allows for numerical computations.
Morb is actually only a function of and , odd in . Thus, for2 [0;1=2], necessarily
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Morb = (1 − 2 )F( ; (1 − )) = (1 − 2 ) |
( )n |
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amn( (1 − ))m |
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n=1 |
m n |
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(57) |
which can in principle be obtained in perturbation theory. |
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Setting |
hbi0 |
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(58) |
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Morbjmean = (1 − 2 )2 |
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with hbi0 = (1− ) and using the previous example i), one obtains the result (mean eld + one vortex corrections) [3]:
Morb |
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Morbj + e (1 − )(1 − 2 ) |
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mean |
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2hbi0 |
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hbi0 |
1 − e−2hbi0 |
2hbi0 |
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Topological Aspects of Low Dimensional Systems |
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0.05 |
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0.04 |
βρ=1 |
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0.02 |
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orb β |
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M |
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Fig. 3. The orbital magnetization in unit e = 1 in the magnetic impurity problem for = 1: Comparison between the analytical computation, solid curve, and numerical simulations.
Figure 3 ( = 1), shows a rather good agreement between (59) { full curve { and the numerical simulations { points { based on (56). However, the situation becomes less transparent for higher values. Clearly, the perturbative analytical approach needs more and more corrections coming from hBi + two vortices, ...
References
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[7]Janssen M., et al., \Introduction to the theory of the Integer Quantum Hall E ect", edited by J. Hadju (VCH, Weinheim, 1994) and references therein.
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J. Desbois: RandomMagneticImpuritiesandQuantumHallE ect 909
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[10]Levy L.P., Dolan G., Dunsmuir J. and Bouchiat H., Phys. Rev. Lett. 64 (1990) 2074; Cheung H.F., Gefen Y., Riedel E.K. and Shih W.H., Phys. Rev. B 37 (1988) 6050; Cheung H.F., Riedel E.K. and Gefen Y., Phys. Rev. Lett. 62 (1989) 587; Buttiker M., Phys. Scr. T54 (1994) 104; Buttiker M., Imry Y. and Landauer R., Phys. Lett. 96A (1983) 365; Avishai Y. and Kohmoto M., Ben Gurion University Report for a review and references see Narevich R., Technion Haifa Thesis 5757 (July 1997).
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A., Phys. Rev. A 53 (1996) 669; for a review and references see R. Narevich, Technion Haifa Thesis 5757 (july 1997); see also Sitenko Yu.A. and Babansky A.Yu., Bogolyubov Institute Report (1997).