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. Outline:
Ady Stern (Weizmann)
with: D.E. Feldman, Eytan Grosfeld, Y. Gefen,
B.I. Halperin, Roni Ilan, A. Kitaev, K.T. Law, K. Schoutens
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
ground states states
Permutations between quasi-particles positions
unitary transformations in the ground state subspace
determined by the topology of the trajectories
Flux attachment (roughly): states
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 subjectedto flux quanta per electron
The statistics of the new particles is determined by the value ofa.
Even a: fermionic Odd a: bosonic
Fractional a: anyonic
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
p filled Landau levels
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.
n=1/3 – attaching three flux quanta to each electron turns it into a boson and cancels the magnetic field – good
n=1/2 – attaching two flux quanta to each electron cancels the magnetic field, but turns the electron into a fermion.
n=2/3 – attaching 1.5 flux quanta to each electron cancels the magnetic field, but turns the electron into an anyon.
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
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)
q.p.’s in the interference loop,
Brattelli diagram (for k=2)
The interference term depends on the number and quantum state of the bulk quasi-particles.
D states 1
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)
G states 4
q.p.’s in the interference loop
I states 1
a well-designed coin
The other extreme states – 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.
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
Coulomb blockade in non-abelian quantum Hall states dependence of the effective charge on flux is a consequence of unconventional statistics.
Two pinched-off point contacts define a quantum dot dependence of the effective charge on flux is a consequence of unconventional statistics.
Coulomb blockade !
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?
The energies involved: dependence of the effective charge on flux is a consequence of unconventional statistics.
Chiral Luttinger liquid mode –
leads invariably to an equal area separation between consecutive peaks
The energy of the parafermion edge mode needs to be added.
Chiral Luttinger liquid energy alone dependence of the effective charge on flux is a consequence of unconventional statistics.
Equal area spacing of charging peaks
When the energy of the parafermionic mode is added, the peaks move and bunch. The bunching depends on the number of localized quasi-particles.
The picture obtained (k=4): peaks move and bunch. The bunching depends on the number of localized quasi-particles.
Bunching of the Coulomb peaks to groups of n and k-n –
A signature of the Zk states