A quest for Pfaffian

1. A quest for Pfaffian Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty, Belgrade, 2010)

2. Hall experiment:

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

4. 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?

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

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

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

8. Pfaffian for 4 particles: p-wave superconductor (p-ip) pairing function wave function of a pair

9. Effective theory of a p-wave superconductor N. Read and D.Green, PRB 61,10267(2000) i.e. BCS mean field theory for eigenfunction of rotations in for eigenvalue

10. Excitations by Bogoliubov: a gapped system Ground state “weak pairing”

11. should not be too large: If large: (a) local maximum local minimum then likely: (b) i.e. Fermi liquid phase

12. FQHE systems • 5/2: numerics favorable for Pfaffian in 2nd 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)

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

14. theory (mathematical identity) two-component:

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

16. BCS formalism of : with tunneling chemical potentials of parts: even: grows with tunneling! odd:

17. BCS formalism of : likely outcome: Fermi liquid If i.e. an open system then we may have a path: with Pfaffian outcome

18. How to recognize Pfaffian? Pfaffian makes a topological phase! What are the signatures of a topological phase? (a) gap (b) characteristic degeneracy of ground state on higher genus surfaces like torus X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992)

19. Torus Create a qp-qh pair, separate and drag in opposite directions along one of the two distinct paths of torus and annihilate: a global process

20. Cylinder: To go to the other side requires energy (gapped excitations) and we may not end up with the same ground state but a new sector

21. 3 – number connected with quasiparticles of Pfaffian: neutral fermions and vortices of the underlying superconductor, M. Milovanovic and N.Read, PRB53, 13559 (1996)

22. Numerics with tunneling, Z. Papic et al., arxiv:0912.3103 in a bilayer Sphere; overlap with tunneling: Sphere is biased for Pfaffian.

23. Torus; ground states with tunneling: No (clear) signatures of Pf degeneracy (2 – trivial degeneracy in a translatory invariant QH system at ½)

24. The quest for Pfaffian goes on!