Non Abelian Quantum Hall States:

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Non- Abelian Quantum Hall States An overview
experimental consequences Ady Stern (Weizmann)
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• Outline
• 1. What are non-abelian quantum Hall states? Why?
  Where?
• 2. Understanding them by
• Trial wave functions
• Chern-Simons theories
• Composite fermion theory
• 3. Experimental implications

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The quantum Hall effect

• zero longitudinal resistivity - no dissipation,
  bulk energy gap current flows mostly along the
  edges of the sample
• quantized Hall resistivity

n is an integer,
or q even
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Extending the notion of quantum statistics
Laughlin quasi-particles
Electrons
A ground state
Energy gap
Adiabatically interchange the position of two
excitations
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More interestingly, non-abelian statistics
(Moore and Read, 91)
In a non-abelian quantum Hall state,
quasi-particles obey non-abelian statistics,
meaning that with 2N quasi-particles at fixed
positions, the ground state is
-degenerate. Interchange of quasi-particles
shifts between ground states.
For n5/2 (main example)
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ground states
position of quasi-particles
..
Permutations between quasi-particles positions
unitary transformations in the
ground state subspace
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Up to a global phase, the unitary transformation
depends only on the topology of the trajectory
Topological quantum computation
(Kitaev 1997-2003)

• Subspace of dimension 2N, separated by an energy
  gap from the continuum of excited states.
• Unitary transformations within this subspace are
  defined by the topology of braiding trajectories
• All local operators do not couple between ground
  states
• immunity to errors

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Immunity to perturbations

• Diagonalized Hamiltonian
• Energy of all ground states is set to zero
• E is a general name for a positive energy of the
  excites states
• Lowest value of E is the energy gap

Perturbation No matrix element that connects
ground states Shaded part is protected. Virtual
transitions introduce exponentially small
splitting
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Non-abelian states through wave functions

• General comments on wave functions in the quantum
  Hall effect
• Surprisingly, first-quantized wave functions are
  a useful concept to construct.
• The wave function for one full Landau level may
  be constructed exactly, and it inspires an
  approximate wave function for the n1/3, 1/5
  fractions, the Laughlin wave function.

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Non-abelian states through wave functions
We consider many electrons in a magnetic field,
with filling fraction lt 1, and weak interaction
between electrons. So, we look for a wave
function thats made solely of lowest Landau
level single particle w.f.s. A single electron
in the lowest Landau level (symmetric gauge)
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• From now on, we do not talk about the Gaussian
  factor. The rest is
• a polynomial in the zis purely lowest Landau
  level
• the maximal power for z1 is (N-1). It determines
  the area, and through the area the filling
  factor.
• the polynomial in z1 has (N-1) zeroes, which are
  on the other (N-1) electrons. Each zero is of
  order one.
• anti-symmetric to the exchange of any two
  electrons

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Laughlin wave function for other filling factors

• a polynomial in the zis purely lowest Landau
  level
• maximal power for z1 is 3(N-1). It determines
  the filling factor to be 1/3.
• The polynomial in z1 has 3(N-1) zeroes, which
  are all on the other (N-1) electrons. Each zero
  is of order three. This efficient use of the
  zeroes is the reason to the great success of
  Laughlins wave function in minimizing the
  interaction energy.
• anti-symmetric to the exchange of any two
  electrons

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If it is so successful, why not using it for
other values of n (say, 1/2)?

• a polynomial in the zis purely lowest Landau
  level - GOOD
• maximal power for z1 is 2(N-1). Fixes filling
  factor to 1/2 - GOOD
• Making efficient use of the zeroes - GOOD
• The problem
• For n1/2 the wave function is not anti-symmetric
  to the exchange of any two electrons
• For other ns it is not even single valued.

Can the problem be fixed?
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The role of the function F to fix the symmetry
(or single valuedness) of the wave function,
without changing the filling factor. We need
For example, for n1/2 we need odd a and any
finite b.
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How shall we find a function that satisfies this?
With a little help from my friends (L-M, 1967).
Parafermionic Conformal Field Theories (CFTs)
(Fateev and Zamolochikov, the
Yellow Book, etc.)
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A fusion rule determines the singular behavior of
a correlator when two of its arguments get close
together
The function F will be a correlator of fields y1,
that represents the electrons. We know what
behavior we need for the wave function, so we
need to find a CFT that has this behavior.
Example n1/2
We need
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We ask around and find the Ising CFT. The fields
it has are 1, y, s. The first fusion rule is
The function F will be
The fusion rule gives us the short distance
behavior that we need. In fact, the correlator
may be exactly evaluated and shown to be
identical to the BCS wave function for a p-wave
super-conductor (the Moore-Read Pfaffian wave
function).
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But there are two more fusion rules, which
involve the s. If y represents the electron, what
is s? It is the quasi-particle (the vortex in
the p-wave superconductor).
Its non-abelian nature is revealed in the fusion
of two ss
The F functions for two quasi-holes will be

• Two functions per each pair of ss
• wave functions are single valued w.r.t. the
  electronic coordinates, but not w.r.t. the
  quasi-particle coordinates.

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• Other CFTs describe clustering of the electrons
  to condensates of groups of 3,4,5 with
  quasi-particles being sort-of vortices in these
  condensates.
• Quasi-particles satisfy non-abelian statistics
  whose details are presently partially worked out.
  
• So far, we looked at the CFTs as working in the
  two dimensional world of (x,y) we looked only
  at ground state wave functions.
• In fact, conclusions may be drawn also about
  dynamics, at the place where it exists the
  edge.

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From electrons to non-abelian quasi-particles
the QFT way
x,y
11D WZW model on the edge
edge, t
The spectrum and Fock space of the edge
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One non-abelian quantum Hall state, that of
nu5/2, may be understood by Composite Fermion
theory, following four steps
Step I
A half filled Landau level on top of two filled
Landau levels
Step II the Chern-Simons transformation
from electrons at a half filled Landau level
to spin polarized composite fermions at zero
(average) magnetic field
GM87 R89 ZHK89 LF90 HLR93 KZ93
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B
Electrons in a magnetic field B
e-
H y E y
Composite particles in a magnetic field
Mean field (Hartree) approximation
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Spin polarized composite fermions at zero
(average) magnetic field
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Dealing with vortices in a p-wave super-conductor
First, a single vortex focus on the mode at the
vortex core
Kopnin, Salomaa (1991), Volovik (1999)
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The functions are solutions of the
Bogolubov de-Gennes eqs.
Ground state should be annihilated by all s
For uniform super-conductors
For a single vortex there is a zero energy mode
at the vortex core
Kopnin, Salomaa (1991), Volovik (1999)
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A zero energy solution is a spinor
g(r) is a localized function in the vortex core
A localized Majorana operator .
All gs anti-commute, and g21.
To connect between ground states one needs an
even number of different gs, which are located
far away from one another. A Perturbation of the
form will never translate into
two different gs and therefore will never
connect one ground state to another.
Immunity to perturbation
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Furthermore, the zero energy solutions are
protected the BDG spectrum is even with
respect to zero energy. A perturbation must
connect two zero energy states for their energy
to be shifted.
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Experimental implications
Bunching of the Coulomb peaks to groups of n and
k-n A signature of the Zk states
Fano factor changing between 1/4 and about three
a signature of non-abelian statistics in
Mach-Zehnder interferometers
Mach-Zehnder
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A Fabry-Perot interferometer
Stern and Halperin (2005) Bonderson, Shtengel,
Kitaev (2005) Following Das Sarma et al (2005)
Chamon et al (1996)
n5/2
backscattering tlefttright2
interference pattern is observed by varying the
cells area
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Integer quantum Hall effect (adapted from Neder
et al., 2006)
The prediction for the n5/2 non-abelian state
(weak backscattering limit)
cell area
Gate Voltage, VMG (mV)
Magnetic Field
(the number of quasi-particles in the bulk)
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vortex a around vortex 1 - g1ga vortex a
around vortex 1 and vortex 2 -
g1gag2ga g2g1
1
a
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After encircling
for an even number of localized vortices only the
localized vortices are affected (a limited
subspace)
for an odd number of localized vortices every
passing vortex acts on a different subspace
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Interference term
for an even number of localized vortices only the
localized vortices are affected Interference is
seen
for an odd number of localized vortices every
passing vortex acts on a different
subspace interference is dephased
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The number of quasi-particles on the island may
be tuned by charging an anti-dot, or more simply,
by varying the magnetic field.
cell area
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When interference is seen
Interference term
Interference magnitude depends on the parity of
the number of quasi-particles Phase depends on
the eigenvalue of
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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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Summary Non-abelian quantum Hall states are
a very exciting theoretical possibility, which
requires much more research for a complete
understanding. Even more so, it requires
experimental testing.