1. Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0. Plateaux in Hall resistivity  =h/(ne 2 ) with integer n correspond to the minima 1 (From Datta page 25) discovery: 1980 Nobel prize: 1985 K. v. Klitzing

2. Origin of the Oscillations of longitudina resistivity =Shubnikov-deHaas 2  without H   with H  resistivity minima close to 0

3. Rectangular conductor very thin in z direction uniform in x direction confined in y direction with B in z direction. Assuming for the sake of argument that H is separable, the transverse dimensions yield infinite solutions that are called subbands: let us assume that only one z subband is occupied and the confinement along y is described by U(y).

4. 4 This is a paradox: one would expect minimum resistance when LL is at Fermi energy, but it is maximum; how does the sample carry the current when E F is between LL and there are no states at E F ? Reply: there are 1d states at the eges of the sample ( hedge states ) that carry the current! The minimum resistivity is very low because of the suppression of relaxation. Carriers that go to opposite directions are far away and never meet.  with H   Due to impurities, the DOS is not like this but more like this Due to disorder and impurities, it is possible to find the Fermi level away from the LL (otherwise it is unlikely to find it where the DOS is small and a small charge can move EF). The conduction takes place through the M hedge states that have very small resistance. For  =integer,..filled or empty LL  gap, no scattering  xx resistivity =0 For =0.5,1.5,2.5,.. Half filled LL  maximum xx resistivity

5. 5 Origin of zero resistance (see also Datta page 175) Including confining potential in first order, states with k in x direction in the LL n have energy : In the middle of the sample, bulk-like eigenvalues and eigenvectors prevail, but near the edges the levels are shifted by U yielding a quasi-continuum of levels, also at fermi energy. Current in edge state can be evaluated by the group velocity: We try to include confining potential U(y) along y as a perturbation, which is nearly constant over the extent of the LL wavefunction. y kxkx

6. 6 No backscattering takes place.This situation when E F is between two LL, otherwise the LL at Fermi level carries current within the sample with scattering and maximum resistance (not an explanation but a description)

7. Could be measured i n balistic conductors within a few % in QHE with better than ppb accuracy ! in QHE mm-sized electron mean free paths because curent carrying states in opposite directions are localized on opposite sides  no backscattering 7

8. 8 Quantized resistance

9. 9 Elaborated from a seminar by Michael Adler Convincing explanation ? NO! Drude theory ia rough, the result is extremely precise, and why the plateaux?

10. Due to the applied bias, a current flows in tha sample; this produces the Hall field in the y direction. So the upper edge states, where electrons go to the left, have the chemicel potential  R of the right electrode, the lower ones have the chemical potential  L of the left electrode. The potential drop V Longitudina along x for both is zero. Since the potential drop along y is V H,  R -  L =V H. Good, but a serious doubt remains. We did several crude approximations. Why is the result so precise)? 10

11. 11 Why the plateaux? Why so exact? Consider a Metal ribbon long side along x, magnetic field along z. 11 x y

12. Laughlin does not even insert confining potential U(y) which plays no role in his argument. Now add an electric field along y.

13. 13

14. Consider closing it as a ring pierced by a flux, with opposite sides connected to charge reservoirs of infinite capacity, each as the same potential as the side to which it is connected. A current I flows around, the Hall potential V H exists between reservoirs.

15. 15 Next replace E by a time-dependent flux  inside; it produces a e.m.f that excites a current I around (along x) but the magnetic field along z then produces a Hall electric field V H, along the ribbon, that will transfer charge q from one reservoir to the other, contributing qV H to the energy. If the flux is one fluxon, the system is completely restored in previous state.

16. 16 In order to have the ribbon in same state as before the fluxon is applied, q=ne with n integer. Hence, Laughlin argues that the same holds true even in the presence of interactions. Laughlin’s argument has been criticized on the grounds that different cycles of the pump may transport different amounts of charge, since q is not a conserved quantity. It is the mean transferred charge that must be quantized. It appears to me that the criticism is rather sophistic, because if the average is an exact integer that does imply that every measurement gives an integer. One could envisage a situation where two exactly equally likely outcomes are 0.80000 and 1.20000 (sharp!) and so the average is 1,0000 without having integer outcomes each time. One should discover fractionally charged real electrons before accepting this explanation. However some authors of the above criticisms have produced a remarkable alternative explanation.

17. 17 According to the Authors, Laughlin’s argument is short in one important step, namely, the inclusion of topological quantum numbers. A

18. 18

19. 19

20. 20 Gauss and Charles Bonnet formula Chern formula

21. http://www.riken.go.jp/lab-www/theory/colloquium/furusaki.pdf Bi 2 Se 3 is a 3d topological insulator Topological Insulators (band) insulator with a nonzero gap to excitated states topological number stable against any (weak) perturbation gapless edge mode Low-energy effective theory of topological insulators = topological field theory (Chern-Simons) The QHE is the prototype topological insulator http://wwwphy.princeton.edu/~yazdaniweb/Research_TopoInsul.php

22. 22 Fractional Quantum Hall effect D.C. Tsui, H.L. Störmer and A.C. Gossard, prl (1982): quantization of Hall conductance at = 1/3 and 2/3 below 1 Kelvin The fractional quantum Hall effect (FQHE) is a physical phenomenon in which a certain strongly correlated system at T under a very strong magnetic field behaves as if it were composed of particles with fractional charge (1998 Nobel Prize).

23. 23 experiments performed on gallium arsenide heterostructures gallium arsenideheterostructures

24. Denominators almost ever odd. The FQHE is a different phenomenon, requiring a different explanation. It is believed that the effect is due to the Coulomb interaction. All electrons in LLL treating interactions as a perturbation that tends to lower the symmetry.

25. 25 Wave functions for the LLL in polar coordinates

26. 26 Laughlin sought thee Many-Body wave function for LLL in the form We want it to be an eigenfunction of total angular momentum

27. 27 Physical Picture: Outer electron encloses l max fluxons R. Laughlin

28. 28 In terms of such wavefunctions one can estimate correlations and compute successfully the relevant quantities. Quasiparticles of fractional charge e/q obey anyon statistics (exchange of two quasiparticles brings a phase , which can be calculated as a Berry phase). Let us evaluate q. Expanding the product in Laughlin ansatz the m aximum power of z of any particle evidently turns out to be : l max = N el q Very accurate wave function when compared with numerical estimates. q=1 yields integer QHE wavefunction

29. 29 From a seminar by Michael Adler

30. 30 Some CONCLUSIONS The dimensionality conditions the many-body behaviour already at the classical level- e.g. phase transitions At the quantum level, the transport properties are very different when nanoscopic objects are considered, and this requires new intriguing concepts and funny mathematical methods, many of which involve the Berry phase too. Here I just recall some. The subject is in rapid evolution, and new applications are also under way. Ballistic conduction, nonlinear magnetic behaviour, possibility of various kinds of pumping. The role of correlation effects is much more critical in 2d (e.g. QHE, charge fractionalization, anyons, Kosterlitz-Thouless ) and above all in 1d (Peierls transition and charge fractionalization, ) and there was no time to introduce others, that would require the Luttinger Liquid formalism….

31. 31 Charge-spin separation spins are assumed to jump freely to empty sites spinonholon

32. 32 nearby sites are assumed to exchange spins spinon propagation holon Fermi particle becomes a pair of boson excitations that can propagate with different speeds (in 1d only) Bosonization Fermion operators can be expressed in terms of bosons: spinons and holons and problem is solved spinonholon holon also travels

33. 33 Separation Realized experimentally!