Eytan Grosfeld

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Experimental signatures of non-abelian statistics
in clustered quantum Hall states
Eytan Grosfeld
Roni Ilan, EG, Ady Stern, Phys. Rev. Lett. 100,
086803 (2008)
Roni Ilan, EG, Kareljan Schoutens, Ady Stern,
arXiv0803.1542 (March, 2008)
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Outline of the talk
I. Introduction to non-abelian quantum hall
effect II. Coulomb blockade peaks in Read-Rezayi
states III. Relaxation processes involving
bulk-edge coupling IV. Signatures of
spin-singlet (Ardonne-Schoutens) state
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The Quantum Hall Effect
Zero longitudinal resistivity - Bulk is
gapped - Current flows mainly along the
edges Quantized Hall resistivity
integer, Integer QHE
p/q, Fractional QHE
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Ground state trial wave-functions
- Wavefunctions are correlators of certain fields
in a conformal field theory
- The same CFT describes the dynamics of the edge
- For ?1/q (q odd) Chiral bosonic field theory
Laughlin wavefunction
Moore Read (1991)
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Quasi-particles
Shifting the density away from the center of the
plateau, localized quasi- particles appear in the
bulk.
Quasi-particles positions are parameters in the
wavefunction
electrons
localized quasi-particles
Quasi-particles may carry fractional charge and
obey non-conventional statistics
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Non-abelian statistics
Ground state is degenerate in the presence of
quasi-particles
A set of ground states
time
Two matrices U do not necessarily commute
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Clustered Read-Rezayi non-abelian states
States in the 2nd Landau level upper LL is
spin-polarized.
I. Chern-Simons transformation
attach 12/k flux quanta to each electron
II. Spin-polarized composite-anyons
k2 composite-fermions form a p-wave
superconductor. kgt2 composite-anyons form
clusters of size k.
III. Excitations
fractional vortices, broken clusters, and
combinations Carry Zk topological charge
IV. Wavefunctions and edge dynamics
described by Zk parafermionic theories
Moore Read (1991), Read Green (2000), Read
Rezayi
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Quantum Hall effect in the second LL
Are these states well-described by the
Read-Rezayi states?
will be determined by an experiment
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How to probe in experiment?

• Can quasi-particles carry information on
  non-abelian theories?

Yes! Through interference of quasi-particle
trajectories.
E. Fradkin, C. Nayak, A. Tsvelik, F. Wilczek A.
Stern, B. Halperin P. Bonderson, A. Kitaev, K.
Shtengel EG, S. Simon, A. Stern D.E. Feldman, A.
Kitaev E. Ardonne, E.A. Kim R. Ilan, EG, K.
Schoutens, A. Stern

• Can electrons?

Yes! quasi-particles in the bulk determine the
energy levels of the edge.
A. Stern B. Halperin R. Ilan, EG, A. Stern R.
Ilan, EG, K. Schoutens, A. Stern
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Fabry-Perot interferometer
Localized quasi- particles in the bulk
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Weak-backscattering limit
Interference of quasi-particle trajectories.
Conductivity carries information about qp
braiding statistics
k2 odd number of qps in bulk no interference
term! kgt2 suppression of interference term.
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Strong back-scattering limit
A finite quantum Hall system number of electrons
quantized to an integer.
electrons are allowed to tunnel into and out of
the dot
Measure conductivity as function of the area of
the dot
Condition for non-zero conductivity
Claim for non-abelian states, energy depends on
number of quasi-particles in the bulk.
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Coulomb blockade peaks 2d conductor
G
positive background density
Energy cost for adding an electron
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Coulomb blockade peaks superconductor
G
positive background density
Energy cost for adding an odd electron
Energy cost for adding an even electron
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Coulomb blockade peaks Read-Rezayi
G
k4
0 qp
S
1 qp
2 qp
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Spectrum of the edge
Two independent 11 field theories
Free chiral boson
Zk parafermionic field theory

• bulk is a condensate of clusters of k anyons.
• clustering reflected in edge spectrum

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Zk parafermions
The electrons are represented by the field
The quasi-particles are represented by
Example Z2 parafermions - A free Majorana
(neutral) fermion
a completely paired condensate
unpaired electron on the edge
associated with an odd number of quasi-particles
in the bulk changes the boundary conditions for
the fermion
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Spectrum of Z2 parafermions 5/2
even number of electrons all electrons are
paired.
condensate of pairs
Stern and Halperin
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Spectrum of Z2 parafermions 5/2
odd number of electrons extra electron goes to
the edge.
Stern and Halperin
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Spectrum of Z2 parafermions 5/2

• Adding a quasi-particle in the bulk
• bulk edge should fuse to identity,
• boundary conditions on the edge change from odd
  to even.

Stern and Halperin
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Energy spectrum 5/2 state (k2)
E
?
S
N
N1
N2
N3
N4
G
S
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Coulomb blockade peaks Read Rezayi k4
0 qp
1 qp
2 qp
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Coulomb blockade peaks Read Rezayi
S
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Summary so far
Bunching pattern of coulomb blockade peaks in the
Read-Rezayi states into groups of size Nqp and
k-Nqp
Nqp number of quasi-particles in the area
between the two quantum point contacts
k 2,3,4 the number appearing in the filling
factor
Next
1. Edge-bulk coupling
2. Ardonne-Schoutens state
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Relaxation 5/2 state (k2)
energy levels - edge
energy levels - bulk
Make use of available states in the bulk!
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Relaxation 5/2 state (k2)
Bunching effect disappears
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Relaxation 12/5 state (k3)
2/3
2/5
1/15
S
S1
S2
Bunching effect survives
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Spin-singlet Ardonne-Schoutens state
Can we relax the assumption that the electrons
are spin-polarized?
Non-abelian state at
Three independent 11 field theories
Gepner parafermions, su2(3)/u(1)2
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SSAS state experimental signatures
Strong backscattering limit
Conductance peaks show modulations from average
spacing whose size depend on the two ratios
The bunching pattern shows a periodicity of 2 or
4.
Weak backscattering limit
Suppression of interference term, similar to k3
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Summary
Clustered quantum Hall states show a
bunching pattern of coulomb blockade peaks.
The pattern depends on the number of
quasi-particles localized in the bulk a
signature of clustered states.
Bulk-edge coupling may change the pattern.
G
S