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A quest for Pfaffian

This talk by Milica V. Milovanović at the Institute of Physics in Belgrade discusses the quest for understanding the Pfaffian state in the context of the fractional quantum Hall effect (FQHE). It covers key concepts such as ground states, plateaus, and the significance of excitations in gapped systems. With references to critical literature and experiments, it explores pairing phenomena akin to superconductivity, the implications for topological quantum computing, and the theoretical frameworks surrounding these intriguing states. Join the discussion about this profound aspect of quantum physics!

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A quest for Pfaffian

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  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!

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