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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 experimentCan 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

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FQHE: Filling nu=p/q

Odd q is the “rule” - Fermi statistics

All (but one?) odd denominator states believedto have quasiparticles with Abelian statistics

Even denominator plateau: nu=5/2Willett et. al. (1987), Eisenstein et. al.(2002), Stormer et. al.(2004)

Well formed plateau

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Proposed Wavefunction for 5/2Moore, Read (1991)Greiter, Wen, Wilczek (1992)

“Paired” Hall state

Moore/Read = Laughlin x BCS

Pfaffian:

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Physics of p+ip superconductor

Bogoliubov deGennes Hamiltonian:

Eigenstates in +/- E pairs

Spectrum with a gap

Excitations: Fermionic quasiparticles above the gap

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p+ip Edge y

xp+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

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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:

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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

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“Edge Vortices”

Majorana fermion:

Pass vortex thru edge:Changes b.c. for Majorana fermionfrom periodic to antiperiodic

Can define “edge vortex” operator:

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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

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5/2 Edge

Edge Operators

Charged edge plasmon as in Laughlin

Neutral Majorana as in p+ip

• Majorana fermion:

• vortex:

• double vortex:

Electron:

Pair:

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Probing the edge • Electron tunneling into edge from “metal”

Edge electron

“charge” “neutral”

• Shot noise for hc/2e vortexbackscattering at point contact

• Crossover from weak to strong (vortex) backscattering thru point contact??? ?

Fendley/MPAF/Nayak PRL (2006) + PRB

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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:

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Need clever bookkeeping!

Define complex coordinate:

4th order in perturbation theory:

6th order in perturbation theory:

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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

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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:

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Kondo Crossovers for Point Contact

Weak vortex backscattering (UV) Two drops weakly coupled (IR)

Upon cooling

Thermodynamic Entropy Drop:

UV: Unscreened spin 1/2IR: Fully screened spin

p+ip , Kondo:

nu=5/2, two-channel Kondo:

(“Boundary” entropy change - Ludwig and Affleck)

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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:

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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!

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“Interpretation” of emergent s=1/2Bosonized 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

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Vortex fusion

Fuse two vortices:

2 zero modessplit: 2 states

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

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