1. 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

2. 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

3. 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

4. Non-abelian quantum Hall states – introduction

5. Current flows with no dissipation, along the gapless edge.
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. I B 1 2 2 3 3 1

6. Permutations between quasi-particles positions
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

7. 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

8. 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) =Pi<j (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

9. ne =p/(ap+1) ncf = p ne =p/(ap+1)
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/(ap+1)
ncf = p
ne =p/(ap+1)
p filled Landau levels
Abelian excitations

10. 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:
n=1/3 – attaching three flux quanta to each electron turns it into a boson and cancels the magnetic field – good
But:
n=1/2 – attaching two flux quanta to each electron cancels the magnetic field, but turns the electron into a fermion.
Moreover:
n=2/3 – attaching 1.5 flux quanta to each electron cancels the magnetic field, but turns the electron into an anyon.

11. How does one Bose-condense particles which are not bosons?
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:
n=1/2 (the Moore-Read state of the n=5/2 state) – Bose condensate of pairs of composite fermions
n=2/5 (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)

12. Non-abelian quantum Hall states – bulk and edge

13. 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/(k+2).
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.

15. 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/(k+2).
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.

16. 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.

17. Interferometers in non-abelian quantum Hall states

18. Brattelli diagram (for k=2)
Interferometers:
The interference term depends on the number and quantum state of the bulk quasi-particles.
even
odd
Number of
q.p.’s in the interference loop,
Interference
term
Brattelli diagram (for k=2)

19. 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)

20. 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)

21. 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:

22. 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.

23. 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

24. Coulomb blockade in non-abelian quantum Hall states

25. Two pinched-off point contacts define a quantum dot
Coulomb blockade !
current
n=5/2
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?

26. 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.

27. E Chiral Luttinger liquid energy alone current
Equal area spacing of charging peaks

28. 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.

29. 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+1 N

30. The picture obtained (k=4):
Bunching of the Coulomb peaks to groups of n and k-n –
A signature of the Zk states