Charging and noise as probes of nonabelian quantum Hall states

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Charging and noise as probes of non-abelian
quantum Hall states Ady Stern (Weizmann)
with D.E. Feldman, Eytan Grosfeld, Y.
Gefen, B.I. Halperin, Roni Ilan, A. Kitaev, K.T.
Law, K. Schoutens
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• Outline
• Non-abelian quantum Hall states what they are
  and where they are on the map of the fractional
  quantum Hall effect
• Bulk and edge in non-abelian quantum Hall states
• Experimental consequences of non-abelian quantum
  Hall states
• Interferometers
• Coulomb blockade

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More precise, and relaxed, presentations of the
subject 1. Anyons in the QHE a pedagogical
introduction AS, Annals of physics, 2008 2.
Review paper Non Abelian Anyons and Topological
Quantum Computation by Nayak, Simon,
Stern, Freedman and Das Sarma on the arxiv/soon
on RMP 3. Next week, in a talk I will give here
at the GGI
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Non-abelian quantum Hall states introduction
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• Defining properties
• A quantum Hall state vanishing longitudinal
  resistivity, quantized Hall resistivity. Gapped
  bulk.
• Current flows with no dissipation, along the
  gapless edge.
• 2. In the presence of localized quasi-particles,
  the ground state is degenerate, and the
  degeneracy is exponential in the number of
  quasi-particles
• 3. Local perturbations (phonons, photons, etc.)
  do not couple ground states. The (almost) only
  way to shift the system from one ground state to
  another is by having quasi-particles braid one
  another.

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2
2
3
1
3
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ground states
position of quasi-particles
..
Permutations between quasi-particles positions
unitary transformations in the
ground state subspace determined by the
topology of the trajectories
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• Non-abelian quantum Hall states location on the
  map
• 1. The fractional quantum Hall effect is a state
  of
• Dissipationless flow of current
• Quantized Hall resistivity
• 2. Understanding it by mapping onto another
  system where current flows with no dissipation.
• Two possibilities
• The integer quantum Hall effect composite
  fermion theory
• Bose Einstein condensate (Bosons at zero
  magnetic field, or nearly zero)
  Moore-Read-Rezayi non-abelian states
• Both are based on flux attachment

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Flux attachment (roughly)
H y(z1..zN) E y(z1..zN) Interacting
electrons at a partially filled Landau level,
with 1/n flux quanta per electron. Define a
new wave function y(z1..zN) Piltj
(zi-zj) a F (z1..zN) The new wave function
describes interacting particles subjected to
flux quanta per electron The statistics
of the new particles is determined by the value
of a. Even a fermionic
Odd a bosonic Fractional a anyonic
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Mapping the fractional onto the integer - Even a
Reducing the magnetic field and increasing the
filling factor from a fraction of a Landau level
to an integer number of Landau levels, keeping
the statistics fermionic
ne p/(ap1)
ncf p
ne p/(ap1)
p filled Landau levels
Abelian excitations
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Non-abelian quantum Hall states choosing
such that the composite particles feel no
magnetic field, with the goal of Bose condensing
these particles. Good news the composite
particles are at zero field Bad news they are
not necessarily bosons. Examples n1/3
attaching three flux quanta to each electron
turns it into a boson and cancels the magnetic
field good But n1/2 attaching two flux
quanta to each electron cancels the magnetic
field, but turns the electron into a fermion.
Moreover n2/3 attaching 1.5 flux quanta to
each electron cancels the magnetic field, but
turns the electron into an anyon.
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Question How does one Bose-condense particles
which are not bosons? Answer (Bardeen, Cooper,
Schrieffer) pairs of fermions may condense like
bosons similarly, clusters of k-anyons with
statistics of p/k may condense like
bosons Examples n1/2 (the Moore-Read state of
the n5/2 state) Bose condensate of pairs of
composite fermions n2/5 (the Fibonacci anyon
state) a condensate of clusters of three anyons
a bosonic phase is accumulated upon encircling
Other ways of looking at these states exist
(Cappelli, Georgiev, Todorov)
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Non-abelian quantum Hall states bulk and edge
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Properties of such condensates

• Bulk excitations
• A Bose condensate has topological excitations
  vortices.
• If the boson is a cluster of k particles, the
  vortex carries a flux of 1/k, and a charge of
  e/(k2).
• On a compact geometry,
• of vortices minus of anti-vortices
• must be a multiple of k
• 4. Clusters may disintegrate and populate inner
  core states.
• If these inner core states are modes of zero
  energy
• The ground state becomes degenerate in the
  presence of (anti)vortices, and a non-abelian
  quantum Hall state is formed.

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Properties of such condensates

• Bulk excitations
• A Bose condensate has topological excitations
  vortices.
• If the boson is a cluster of k particles, the
  vortex carries a flux of 1/k, and a charge of
  e/(k2).
• 3. Clusters may disintegrate and populate inner
  core states.
• If these inner core states are modes of zero
  energy
• The ground state becomes degenerate in the
  presence of (anti)vortices, and a non-abelian
  quantum Hall state is formed. Also, a vortex and
  an anti-vortex have two ways to annihilate one
  another.

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• Edges
• The mere existence of the quantum Hall effect
  forces the edge to have a charged chiral gapless
  mode a Luttinger liquid (Wen)
• In non-abelian states, the edge has another
  gapless mode, which is neutral.
• Both the quasi-particle operator and the electron
  operator affect the state of the two edge modes
  the charged and the neutral.

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Interferometers in non-abelian quantum Hall states
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Interferometers
The interference term depends on the number and
quantum state of the bulk quasi-particles.
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Interferometers
Main difference the interior edge is/is not part
of interference loop
For the M-Z geometry every tunnelling
quasi-particle advances the system along the
Brattelli diagram
(Feldman, Gefen, Law PRB2006)
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Interference term
Number of q.p.s in the interference loop

• The system propagates along the diagram, with
  transition rates assigned to each bond.
• The rates have an interference term that
• depends on the flux
• depends on the bond (with periodicity of 4)

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• The probability p, always ltlt1, varies according
  to the outcome of the tossing. It depends on flux
  and on the number of quasi-particles that have
  already tunneled.
• Consider two extremes (two different values of
  the flux)
• If all rates are equal, there is just one value
  of p, and the usual binomial story applies Fano
  factor of 1/4.
• But

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The other extreme some of the bonds are broken
Charge flows in bursts of many quasi-particles.
The maximum expectation value is around 12
quasi-particles per burst Fano factor of about
three.
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Effective charge span the range from 1/4 to about
three. The dependence of the effective charge on
flux is a consequence of unconventional
statistics. Charge larger than one is due to the
Brattelli diagram having more than one floor,
which is due to the non-abelian statistics
In summary, flux dependence of the effective
charge in a Mach-Zehnder interferometer may
demonstrate non-abelian statistics at n5/2
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Coulomb blockade in non-abelian quantum Hall
states
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Two pinched-off point contacts define a quantum
dot
Coulomb blockade !
current
n5/2
area S, B
A Coulomb blockade peak appears in the
conductance through the dot whenever the energy
cost for adding an electron is zero
For a fixed magnetic field B, what is the area
separation between consecutive peaks?
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The energies involved
Chiral Luttinger liquid mode charging
energy leads invariably to an equal area
separation between consecutive peaks
The energy of the parafermion edge mode needs to
be added.
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Chiral Luttinger liquid energy alone
E
N
N1
current
Equal area spacing of charging peaks
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• The second (parafermionic) edge mode
• accommodates the un-clustered Nmod(k) electrons
  (for an energy cost) energy is periodic with k
• the energy cost is determined in a
  Bohr-Sommerfeld manner, but in a way that depends
  both on Nmod(k) and on the number of
  quasi-particles localized in the bulk. This
  number varies with the magnetic field.

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When the energy of the parafermionic mode is
added, the peaks move and bunch. The bunching
depends on the number of localized
quasi-particles.
E
1
3/4
N
N1
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The picture obtained (k4)
Bunching of the Coulomb peaks to groups of n and
k-n A signature of the Zk states
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