1. How do you recognize the non-abelian quantum Hall effect when you see it
Ady Stern
(Weizmann)
Papers: Stern & Halperin , PRL
Grosfeld & Stern, Rap. Comm.
Grosfeld, Simon & Stern, PRL
Feldman, Gefen, Kitaev, Law & Stern, Cond-mat
Grosfeld, Cooper, Stern & Ilan, Cond-mat

2. The goal:
Reasonably realistic measurements that will show signatures of particles satisfying non-abelian statistics.
The list:
0. Pattern formation
Observing the zero energy Majorana modes
Fabry-Perot interferometry
Mach-Zehnder interferometry

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

4. For non-abelian states:
With N quasi-particles at fixed positions, the ground state is
-degenerate.
Interchange of quasi-particles shifts between ground states.
For n=5/2 (Moore-Read, Pfaffian), where l=
position of
quasi-particles
degenerate ground states
…..
Permutations between quasi-particles positions
topological unitary transformations in the
subspace of ground states test ground for TQC
(Kitaev, 1997)

5. What does it take to have non-abelian statistics?
1. Degeneracy of the ground state in the presence of localized quasi-particles
2. Topological interaction between the quasi-particles
How do you see them experimentally??

7. We
are
non-abelian
quasi-particles
Call
your leader
a patent lawyer
Read and Moore
“If only life was so simple” (Allen, Ann. Ha. 1977)

8. From electrons at n=5/2 to non-abelian quasi-particles in four steps:
Read and Green (2000)
Step I:
A half filled Landau level on top of
two filled Landau levels
Step II:
the Chern-Simons transformation to
Spin polarized composite fermions at zero (average)
magnetic field

9. Step III: fermions at zero magnetic field pair into Cooper pairs
Spin polarization requires pairing of odd angular momentum
a p-wave super-conductor of composite fermions
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 and thus
accommodates vortices. The vortices, which are charged, are the
non-abelian quasi-particles.

10. The quadratic BCS mean field Hamiltonian is diagonalized by solving
the Bogolubov-deGennes equations

11. For a single vortex – there is a zero energy mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
Ground state degeneracy
Skip steps I and II: Cold atoms forming a p-wave superfluid
Gurarie et al.

12. A p-wave superfluid of fermionic cold atoms
Fermionic atoms with two internal states, “” and “”
Initially, all atoms are in the “” state and form a p-wave superfluid.
How can one detect the different phases of the superfluid using absorption measurements?
Let me start right away with the model we use. We consider then fermionic atoms with two internal hyperfine states, to which we shall refer for convenience as “down” and “up”. The lower energy state interacts resonantly via a p-wave molecular state and forms a p-wave superfluid, which has a very rich and interesting phase diagram. The upper energy state is initially free. We then induce transitions between the two levels by shining RF radiation. The question we ask ourselves, is whether the absorption spectrum will contain information about the different superfluid states.
see also: Tewari, Das Sarma, Nayak, Zhang and Zoller (2006)

13. Free atoms – a delta function absorption spectrumEg
Eg- 
We first consider the simple case of free atoms in the two states. We assume that the system is translational invariant and that momentum is conserved. We expect a delta-function peak in the absorption spectrum when the external frequency exactly matches the frequency of the internal atomic transition.
-  Eg

14. -atoms form a p-wave superfluid
Rate of excitations between two states
Cooper pairs are broken by absorbing light, generating two quasi-particles with momenta k,-k.
One quasi-particle occupies a -state
Other quasi-particle occupies a -state

15. The absorption spectrum when the -atoms form a p-wave superfluid
weak-pairing
phase (>0)
Eg- 
We assume then that the Feshbach detuning and temperature are tuned so that the down-atoms form a superfluid; the energy required for the internal transition E_g now becomes a threshold for absorption. The absorption weight shows a discontinuity at E_g for weak pairing, but no discontinuity for strong pairing. In this way, one can distinguish these two states from each other. Eg
Strong pairing
phase (<0)
Eg+2||
- 

16. Vortices appear in the superfluid, forming a lattice.
Now, rotate the system (an analog to a magnetic field)
Vortices appear in the superfluid, forming a lattice.
Each vortex carries a Majorana zero mode at its core.
Due to tunneling between core states, a band is formed near zero energy.
We now rotate the system. Vortices are introduced into the superfluid, and form a lattice. The Majorana core states can now tunnel between the vortices, and a band is formed near zero energy.

17. t c -   Eg- Eg- Eg c The absorption spectrum of a rotated system
-  
Eg-
Landau levels are the spectrum of the -atoms
Eg- Eg c
band formed by Majorana
fermions near zero energy
This is the absorption spectrum for the rotating system. It now contains peaks, at frequencies lower than the energy required for the internal transition. These peaks appear due to transitions from the band near zero energy into the Landau levels. Consequently, their appearance in the spectrum would prove the existence of Majorana zero modes on the cores of the vortices. Let us draw the envelope function of the peak structure. Remarkably, we find that when we rotate the same system in the opposite direction, the envelope function changes its form, now starting linearly and going through a maximum. This is a manifestation of the time-reversal breaking by the pairing function.
And now back to the quantum Hall effect

18. The n=5/2 state is mapped onto a p-wave superfluid of
composite fermions, with a zero mode in the core of every
vortex (a 1/4 charge quasi-particle). We want to demonstrate
the topological interaction between the vortices.

19. 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.
A subspace of degenerate ground states, with the g’s operating
in that subspace.
In particular, when a vortex i encircles a vortex j, the ground state is
multiplied by the operator gigj
Nayak and Wilczek (’96)
Ivanov (’01)

20. An experimental manifestation through interference:
Stern and Halperin (2005)
Bonderson, Shtengel, Kitaev (2005)
Following Das Sarma et al (2005)
n=5/2
backscattering = |tleft+tright|2
interference pattern is observed by varying the cell’s area

21. 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)
Gate Voltage, VMG (mV)
Magnetic Field
cell area
Current (a.u.)
cell
area
n=5/2
Followed by an extension to a closed dot

22. 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

23. 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

24. 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
|tleft + tright|2
|tright|2 + |tleft|2

25. 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

26. When interference is seen:
Interference term is proportional to
n=5/2
Two possible eigenvalues that differ by a minus sign.
Cannot be changed by braiding of vortices

27. Closing the island into a quantum dot – Coulomb blockade:
Transport thermodynamics
The spacing between conductance peaks translates to the energy cost of adding an electron.
For a conventional super-conductor, spacing alternates between
charging energy Ec (add an even electron)
charging energy Ec + superconductor gap D
(add an odd electron)

28. But this super-conductor is anything but conventional…
For the p-wave super-conductor at hand, crucial dependence on
the number of bulk localized quasi-particles, nis
a gapless (E=0) edge mode if nis is odd corresponds to D=0
a gapfull (E≠0) edge mode if nis is even corresponds to D≠0
The gap diminishes with the size of the dot ∝ 1/L
The gap is with respect to the chemical potential, and not with respect to an absolute energy (similar to the gap in a super-conductor, unlike the gap in the quantum Hall effect)

29. (number of electrons in the dot)
Cell area
(number of electrons in
the dot)
Even Odd
Magnetic field
(number of q.p.s
in the dot)

30. What destroys the even-odd effect:
Fluctuating number of vortices on the island, nis
Fluctuations in the state of the nis vortices
Thermal fluctuations of the edges
All these fluctuations smear the interference picture, but signatures of non-abelian statistics may still be seen.

31. For example, what if nis is time dependent?
A simple way to probe exotic statistics:
For weak backscattering - a new source of current noise.
For Abelian states (n=1/3):
Chamon et al. (1997)
For the n=5/2 state:
G = G (nis odd)
G0[1 ± b cos(f + pnis/4)] (nis even)

32. bigger when t0 is long enoughdG
time
compared to shot noise
bigger when t0 is long enough
(Kane PRL, 2003)
close in spirit to 1/f noise, but unique to FQHE states.

33. Summary
1. Non-abelian quantum Hall states are theoretically exciting.
2. Experimental demonstration is highly desired
3. Needed for that – large experimental effort, new theoretical ideas for experiments.