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: Lets first try to understand the 5/2 edge
and then the physics of a Single Point Contact 1 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 2 Proposed Wavefunction for 5/2 Moore, Read (1991) Greiter, Wen, Wilczek (1992) Paired Hall state Pfaffian: Moore/Read = Laughlin x BCS
3 Physics of p+ip superconductor Bogoliubov deGennes Hamiltonian: Eigenstates in +/- E pairs Spectrum with a gap Excitations: Fermionic quasiparticles above the gap 4
p+ip Edge y Edge state x p+ip superconductor Edge Majorana fermion 2-component spinor tangent to edge Chiral fermion propagates along edge
Edge state encircling a droplet Antiperiodic b.c. Spinor rotates by 2 pi encircling sample 5 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 6 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 7 Edge Vortices Majorana fermion: Pass vortex thru edge: Changes b.c. for Majorana fermion
from periodic to antiperiodic Can define edge vortex operator: 8 nu=5/2: Add in charge Excitations: Majorana Fermion: charge Q=0 Vortex: charge e/4, non-Abelian charge e/4 signature of pairing Double vortex: charge e/2, Abelian semion
(Laughlin quasiparticle) 9 5/2 Edge Charged edge plasmon as in Laughlin Neutral Majorana as in p+ip Edge Operators Majorana fermion: vortex: double vortex:
Electron: Pair: 10 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 ? 11 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: 12
Need clever bookkeeping! Define complex coordinate: 4th order in perturbation theory: 6th order in perturbation theory: 13 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 14 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 !! 15 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:
16 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 17 Conclusions:
5/2 (hopefully!) has non-Abelian quasiparticles A point contact is complicated due to the particles 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! 18 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 19 Vortex fusion Fuse two vortices: 2 zero modes split: 2 states 20
Kane/MPAF PRL (1994) Glattli et. al. PRL (1997) Heiblum et. al. Nature (1997) 21 22