The 52 Edge

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 Lets first try to understand the
5/2 edge and then the physics of a Single Point
Contact
2
FQHE Filling nup/q
Odd q is the rule - Fermi statistics
All (but one?) odd denominator states believed to
have quasiparticles with Abelian statistics
Even denominator plateau nu5/2
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 pip superconductor
Bogoliubov deGennes Hamiltonian
Eigenstates in /- E pairs
Spectrum with a gap
Excitations Fermionic quasiparticles above the
gap
5
pip Edge
y
Edge state
x
pip 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 pip superconductor
Single vortex
Fermion picks up pi phase around vortex Changes
to periodic b.c.!!
E0 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 E0 Hilbert space
Braid two vortices (eg. i and i1)
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
nu5/2 Add in charge

• Excitations
• Majorana Fermion charge Q0
• 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 pip
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 pip
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
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pip Bosonization
Flip direction of left mover
Define complex fermion and bosonize
Lagrangian for boson
Bosonize vortex tunneling Hamiltonian
pip 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)
UV Unscreened spin 1/2 IR Fully screened spin
pip , Kondo
nu5/2, two-channel Kondo
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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
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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!

19
Interpretation of emergent s1/2
Bosonized representation
Vortex tunneling event, pi/2 phase shift
Subsequent vortex tunneling event, -pi/2 phase
shift
s1/2 keeps track of sign changes, spin flip
during each tunneling event
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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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