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The 5/2 Edge

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

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The 5/2 Edge

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

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

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

  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” Complex fermion: Vortex plus edge makes one q-bit

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

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

  11. Probing the edge • Electron tunneling into edge from “metal” “charge” “neutral” Edge electron • 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. 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. Need clever bookkeeping! Define complex coordinate: 4th order in perturbation theory: 6th order in perturbation theory:

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

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

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

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

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

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

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