1. 1 The 5/2 Edge IPAM meeting on Topological Quantum Computing February 26- March 2, 2007 MPA Fisher, with Paul Fendley and Chetan Nayak Motivation: FQHE: Only known topological phases in nature, 5/2 state is the best non-Abelian candidate Chiral edge states are easiest to probe in experiment Can use edges to measure non-abelian statistics with multiple point contacts So: Let’s first try to understand the 5/2 edge and then the physics of a Single Point Contact

2. 2 FQHE: Filling nu=p/q Odd q is the “rule” - Fermi statistics All (but one?) odd denominator states believed to have quasiparticles with Abelian statistics Even denominator plateau: nu=5/2 Willett et. al. (1987), Eisenstein et. al.(2002), Stormer et. al.(2004) Well formed plateau

3. 3 Proposed Wavefunction for 5/2 Moore, Read (1991) Greiter, Wen, Wilczek (1992) “Paired” Hall state Moore/Read = Laughlin x BCS Pfaffian:

4. 4 Physics of p+ip superconductor Bogoliubov deGennes Hamiltonian: Eigenstates in +/- E pairs Spectrum with a gap Excitations: Fermionic quasiparticles above the gap

5. 5 p+ip Edge y x p+ip superconductor Edge state Edge Majorana fermion Chiral fermion propagates along edge 2-component spinor tangent to edge Edge state encircling a droplet Antiperiodic b.c. Spinor rotates by 2 pi encircling sample

6. 6 Vortex in p+ip superconductor Single vortex Fermion picks up pi phase around vortex: Changes to periodic b.c.!! E=0 Majorana fermion encircling sample, AND encircling vortex - a “vortex zero mode” Vortex plus edge makes one q-bit Complex fermion:

7. 7 Vortices have Non-Abelian Statistics N v vortices vortex: Majorana zero mode: Ground state degeneracy: N v /2 Qbits Massive degeneracy of E=0 Hilbert space Braid two vortices (eg. i and i+1): Unitary transformation - U i

8. 8 “Edge Vortices” Majorana fermion: Pass vortex thru edge: Changes b.c. for Majorana fermion from periodic to antiperiodic Can define “edge vortex” operator:

9. 9 nu=5/2: Add in charge Excitations: Majorana Fermion: charge Q=0 Vortex: charge e/4, non-Abelian Double vortex: charge e/2, Abelian semion (Laughlin quasiparticle) charge e/4 signature of pairing

10. 10 5/2 Edge Edge Operators Charged edge plasmon as in Laughlin Neutral Majorana as in p+ip Majorana fermion: vortex: double vortex: Electron: Pair:

11. 11 Probing the edge Electron tunneling into edge from “metal” Edge electron “charge”“neutral” Shot noise for hc/2e vortex backscattering at point contact Crossover from weak to strong (vortex) backscattering thru point contact??? ? Fendley/MPAF/Nayak PRL (2006) + PRB

12. 12 Weak constriction in p+ip Inter-edge Vortex tunneling: Perturbation expansion and Chiral decomposition: “Fusion channels”: Determine fusion channels using: together with braiding rules: Formal (!) perturbation expansion:

13. 13 Need clever bookkeeping! Define complex coordinate: 4th order in perturbation theory: 6th order in perturbation theory:

14. 14 p+ip Bosonization Flip direction of left mover: Define complex fermion and bosonize: Lagrangian for boson: Bosonize vortex tunneling Hamiltonian: Emergent spin 1/2 p+ip point contact is identical to (anisotropic) Kondo model

15. 15 5/2 Bosonization Reinstate the charge edge modes: Flip direction of leftmover, again: Define “odd” charge boson: 5/2 point contact is identical to two-channel Kondo model !! Bosonize edge Lagrangian and vortex tunneling term:

16. 16 Kondo Crossovers for Point Contact Weak vortex backscattering (UV)Two drops weakly coupled (IR) Upon cooling Thermodynamic Entropy Drop: UV: Unscreened spin 1/2 IR: Fully screened spin p+ip, Kondo: nu=5/2, two-channel Kondo: (“Boundary” entropy change - Ludwig and Affleck)

17. 17 Entanglement Entropy D is quantum dimension of the topological phase “Entanglement entropy” between two regions in an infinite sample: Thermodynamic Entropy Drop = Entropy of “Disentanglement” Thermodynamic (“Boundary”) Entropy drop under point contact crossovers:

18. 18 Conclusions: 5/2 (hopefully!) has non-Abelian quasiparticles A point contact is complicated due to the particle’s non-trivial braiding statistics. Dynamically breaking a drop into two is described by the two-channel Kondo model Open issues… Theory: Non-equilibrium transport thru point contact (noise and I-V, Keldysh etc) Multiple point contacts, for topological QC gates Point contacts in other non-Abelian states, ie Read-Rezayi Experiment: Measure e/4 charge, signature of pairing Detect presence of “neutral” edge modes (e-tunneling into edge?) Measure properties of a point contact Multiple junctions to detect non-Abelian statistics and build quantum computer!

19. 19 “Interpretation” of emergent s=1/2 Bosonized representation: Vortex tunneling event, pi/2 phase shift: Subsequent vortex tunneling event, -pi/2 phase shift s=1/2 keeps track of sign changes, spin flip during each tunneling event

20. 20 Vortex fusion Fuse two vortices: 2 zero modes split: 2 states

21. 21 Kane/MPAF PRL (1994) Glattli et. al. PRL (1997) Heiblum et. al. Nature (1997)

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