1. Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty, Belgrade, 2010)

2. Hall experiment:

3. Classical picture in 2D:

4. T= 85mKPlateaus ! Rigidity !filling factor = J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)

5. Quantum Mechanics: a particle in 2D in perpendicular B Velocity: Landau levels

6. act in a fixed Landau level

7. Summary: guiding center coordinate cyclotron velocity

8. Degeneracy of Landau level: unit of flux “flux quantum”

9. quantum Hall effect: onedimensional problem localized states in bulk no backscattering!

10. localized states Insertion of one flux quantum does not change spectrum! and expect transfer of integer (i) number of electrons integer quantum Hall effect (Laughlin argument)

11. In rotationally symmetric gauge in two dimensions: Single particle wave functions: Orbits at radius: Imagine that we are at the middle of the plateau at 1/3 - How the ground state of the system would look like?

12. Laughlin answer: antisymmetry and in the cases of other “hierarchical constructions” odd denominator expected! R.B. Laughlin, PRL 50, 1995 (1983)

13. W. Pan et al.,PRL 83, 3530,1999. FQHE at 5/2 ! R. Willet et al., PRL 59, 1776, 1987

14. Moore-Read answer : Pfaffian Pfaffian part describes a pairing among particles as in a superconductor = BCS pairing of spinless fermions G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)

15. Pfaffian for 4 particles:

16. Pfaffian (p-wave superconductor) otherwise we would have Fermi-liquid-like state

17. (a) 5/2 : numerics favorable for Pfaffian in 2 nd LL Pfaffian is the most simple ansatz if not only explanation of plateau R.H. Morf, PRL 80, 1505 (1998), E.H. Rezayi and F.D.M. Haldane, PRL 84, 4685 (2000) (b) 1/2 : exps. and numerics find Fermi-liquid-like phase (no plateau) E. Rezayi and N. Read, PRL 72, 900 (1994)

18. at 1/2 (1/4) in WQWs (wide quantum wells): signatures of FQHE – minima in ! likely nature of these states is multi-component (two-component) J. Shabani et al., Phys. Rev. Lett. 103, 256802 (2009)

19. theory (mathematical identity) two-component:

20. Pf state can lead to a first topological quantum computer! We want to know how to make Pfaffian!

21. The quest for Pfaffian begins!