1. Non- Abelian Quantum Hall States:
An overview + experimental consequences
Ady Stern (Weizmann)

2. 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

3. 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 a fraction with q odd,
or q even

4. Laughlin quasi-particles
Extending the notion of quantum statistics
Laughlin quasi-particles
Electrons
A ground state:
Energy gap
Adiabatically interchange the position of two excitations

5. 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 n=5/2 (main example)

6. Permutations between quasi-particles positions
ground states
position of
quasi-particles
…..
Permutations between quasi-particles positions
unitary transformations in the ground state subspace

7. Topological quantum computation (Kitaev 1997-2003)
Up to a global phase, the unitary transformation depends only on the topology of the trajectory 1 3 2
Topological quantum computation (Kitaev )
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

8. 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

9. 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 n=1/3, 1/5 fractions, the Laughlin wave function.

10. Non-abelian states through wave functions
We consider many electrons in a magnetic field, with filling fraction < 1, and weak interaction between electrons.
So, we look for a wave function that’s made solely of lowest Landau level single particle w.f.’s.
A single electron in the lowest Landau level (symmetric gauge):
Simplest many-electron problem – a full Landau level
The wave function is a Slater determinant,

11. From now on, we do not talk about the Gaussian factor. The rest is:
a polynomial in the zi’s – 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

12. Laughlin wave function for other filling factors
a polynomial in the zi’s – 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 Laughlin’s wave function in minimizing the interaction energy.
anti-symmetric to the exchange of any two electrons

13. If it is so successful, why not using it for other values of n (say, 1/2)?
a polynomial in the zi’s – 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 n=1/2 the wave function is not anti-symmetric to the exchange of any two electrons
For other n’s it is not even single valued.
Can the problem be fixed?

14. 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 n=1/2 we need odd a and any finite b.

15. 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.)
The essential minimum:
CFTs in this context are a device to generate functions:
A CFT has a set of fields:
and a set of fusion rules:
with conformal dimensions hi
where the d’s are determined by the conformal dimensions

16. 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: n=1/2
We need

17. 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).

18. But there are two more fusion rules, which involve the s
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 s’s
The F functions for two quasi-holes will be
Two functions per each pair of s’s
wave functions are single valued w.r.t. the electronic coordinates, but not w.r.t. the quasi-particle coordinates.

19. 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.

20. + From electrons to non-abelian quasi-particles: the QFT way
The quantum Hall effect a 2+1D Chern-Simons theory
gauge invariance
2+1D Chern-Simons theory +
x,y
1+1D WZW model on the edge
edge, t
The spectrum and Fock space of the edge

21. A half filled Landau level on top of two filled Landau levels
One non-abelian quantum Hall state, that of nu=5/2, 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

22. Electrons in a magnetic field B
H y = E y
20 CF
(b) B
Composite particles in a
magnetic field
Mean field (Hartree) approximation
B1/2 = 2ns0 B
(c)

23. Spin polarized composite fermions at zero (average) magnetic field
Step III: fermions at zero magnetic field pair into Cooper pairs
Spin polarization requires pairing of odd angular momentum
a p-wave super-conductor
Read and Green (2000)
Step IV: introducing quasi-particles into the super-conductor
- shifting the filling factor away from 5/2
The super-conductor is subject to a magnetic field
The super-conductor is subject to a magnetic field and thus
accommodates vortices. The vortices, which are charged, are the
non-abelian quasi-particles.

25. 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)
A quadratic Hamiltonian – may be diagonalized
(Bogolubov transformation)
BCS-quasi-particle annihilation operator
Ground state degeneracy requires zero energy modes

26. 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)

27. A zero energy solution is a spinor
g(r) is a localized function in the vortex core
A localized Majorana operator
All g’s anti-commute, and g2=1.
To connect between ground states one needs an even number of different g’s, which are located far away from one another. A Perturbation of the form will never translate into two different g’s and therefore will never connect one ground state to another.
Immunity to perturbation

28. 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.

29. Experimental implications:
Interference magnitude depends on the parity of the number of quasi-particles
Phase depends on the eigenvalue of
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:

30. A Fabry-Perot interferometer:
Stern and Halperin (2005)
Bonderson, Shtengel, Kitaev (2005)
Following Das Sarma et al (2005)
Chamon et al (1996)
n=5/2
backscattering = |tleft+tright|2
interference pattern is observed by varying the cell’s area

31. Integer quantum Hall effect (adapted from Neder et al., 2006)
The prediction for the n=5/2 non-abelian state (weak backscattering limit)
Integer quantum Hall effect (adapted from Neder et al., 2006)
cell area
Current (a.u.)
cell
area
Gate Voltage, VMG (mV)
Magnetic Field
(the number of quasi-particles in the bulk)
n=5/2
Followed by an extension to a closed dot

32. vortex a around vortex 1 - g1ga
vortex a around vortex 1 and vortex g1gag2ga ~ g2g1 2 1 a
The effect of the core states on the interference of backscattering
amplitudes depends crucially on the parity of the number of localized
states.
Before encircling

33. 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

34. interference is dephased
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

35. n=5/2
The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.
Gate Voltage, VMG (mV)
Magnetic Field
(or voltage on anti-dot)
cell
area

36. When interference is seen:
Interference term
n=5/2
Interference magnitude depends on the parity of the number of quasi-particles
Phase depends on the eigenvalue of

37. Main difference: the interior edge is/is not part of interference loop
Interferometers: S1 D1 D2
F-P S D1 D2
M-Z
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)

38. The rates have an interference term that depends on the flux G1
G1/2 G2
G2/2 G3 G4
G4/2
G3/2
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)

39. Consider two extremes (two different values of the flux): p
1-p I2 I1
a well-designed coin
The probability p, always <<1, 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:

40. 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.

41. Effective charge span the range from 1/4 to about three
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 n=5/2

42. 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.