Transcript tqc2007 6564

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
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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
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Proposed Wavefunction for 5/2
Moore, Read (1991)
Greiter, Wen, Wilczek (1992)
“Paired” Hall state
Pfaffian:
Moore/Read = Laughlin x BCS
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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
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
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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”
Complex fermion:
Vortex plus edge makes one q-bit
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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 fermion
from 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
charge e/4 signature of pairing
• Double vortex: charge e/2, Abelian semion
(Laughlin quasiparticle)
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5/2 Edge
Charged edge plasmon as in Laughlin
Neutral Majorana as in p+ip
Edge Operators
• Majorana fermion:
• vortex:
• double vortex:
Electron:
Pair:
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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
?
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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:
Bosonize edge Lagrangian and vortex tunneling term:
5/2 point contact is identical
to two-channel Kondo model !!
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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:
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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 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=1/2
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
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Vortex fusion
Fuse two vortices:
2 zero modes
split: 2 states
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Kane/MPAF PRL (1994)
Glattli et. al. PRL (1997)
Heiblum et. al. Nature (1997)
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