The 5/2 Edge IPAM meeting on Topological Quantum

The 5/2 Edge IPAM meeting on Topological Quantum

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

Recently Viewed Presentations

  • Welcome to myBlueprint…

    Welcome to myBlueprint…

    VISIT: WWW.MYBLUEPRINT.CA ENTER ACTIVATION KEY 2 1 EXAMPLE FROM A POPULAR INTEREST INVENTORY The exact same content designed for Grade 5 Students all the way through to Adults DISCOVERY AND EXPLORATIONS DISCOVERY designed with content for Grade 7 - 10...
  • Selfishness beats altruism within groups. Altruistic groups beat

    Selfishness beats altruism within groups. Altruistic groups beat

    Fletcher et al. spend most of their time advocating the second (a form of kin selection theory) but then conclude that group selection is best. In reality, all three models are important and useful tools for investigating and modeling social...
  • Energy Efficiency and Zero Energy Ready Housing Amy

    Energy Efficiency and Zero Energy Ready Housing Amy

    (Dave PM (Pelunis-Messier)was project lead) 300 person village, home to Alaska's only inland Inupiat population, the Nunamiut. The Nunamiut historically hunted caribou, following the herds through the mountains. They were the last documented hunter-gather population in North America, and finally...
  • Emergence of Semantic Knowledge from Experience Jay McClelland

    Emergence of Semantic Knowledge from Experience Jay McClelland

    Emergent domain-specificity. Basic level, expertise & frequency effects. Conceptual reorganization. Tim Rogers. Phenomena in Development(Rogers & McClelland, 2004) Progressive differentiation. U-shaped over-generalization of. Typical properties. Frequent names.
  • GSM GSM (Global System for Mobile Communications, originally

    GSM GSM (Global System for Mobile Communications, originally

    The use of the SIM card allows GSM network users to quickly move their phone number from one GSM phone to another by simply moving the SIM card. Currently GSM networks operate on the 850MHz, 900MHz, 1800MHz, and 1900MHz frequency...
  • It is a report card UNTRUTHS Places doctors

    It is a report card UNTRUTHS Places doctors

    Clinical Practice Guidelines,Continuing Education credits. From this point forward, the report will be refreshed quarterly and delivered almost exclusively through our portal. Our physician portal was initially developed as Registration tool so physicians could complete their annual practice permit renewal
  • Applied Economics for Business Management

    Applied Economics for Business Management

    Since the tangencies represent points where slopes of isoquant and isocost lines are equal, we can solve for the equation of the expansion path: Slope of isoquant: Slope of isocost line: Using our previous production example: equation of the expansion...


    85 YEARS OF US RURAL WOMEN'S TIME USE Teresa Harms and Jonathan GershunyCentre for Time Use Research University of Oxford. ACKNOWLEDGEMENTS: We gratefully acknowledge the financial support of the UK Economic and Social Research Council (grant references ES/L011662/1; ES-060-25-0037, ES-000-23-TO704...