How do you recognize the non-

abelian quantum Hall effect when

you see itAdy 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

The goal:

Reasonably realistic measurements that will show signatures

of particles satisfying non-abelian statistics.

The list:

0. Pattern formation

1. Observing the zero energy Majorana modes

2. Fabry-Perot interferometry

3. Mach-Zehnder interferometry

Extending the notion of quantum statistics

),..,;............,.........( 411 RRrr NA ground state:

Adiabatically interchange the position of two excitations

Energy gap

ie

Laughlin quasi-

particlesElectrons

For abelian states:

With N quasi-particles at fixed positions, the ground state is

-degenerate.

Interchange of quasi-particles shifts between ground states.

For =5/2 (Moore-Read, Pfaffian), where =

N

For non-abelian states:

2

...,2..

...,2..

...,1..

212/

21

21

RRsg

RRsg

RRsg

N

…..

..., 21 RR

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)

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??

We are non-abelian quasi-particles

Callyour leader

a patent lawyer

Read and Moore

“If only life was so simple” (Allen, Ann. Ha. 1977)

From electrons at =5/2 to non-abelian quasi-particles in four steps:

A half filled Landau level on top of two filled Landau levels

Step II:

the Chern-Simons transformation to

Step I:Read and Green (2000)

2

12

2

5

Spin polarized composite fermions at zero (average) magnetic field

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.

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

..)(0 chrdrHH

)()()()( rrvrrudrE

..)(0 chrdrHH

The quadratic BCS mean field Hamiltonian is diagonalized by solving

the Bogolubov-deGennes equations

E

EEgs EEH

For a single vortex – there is a zero energy mode at the vortex’ core

Kopnin, Salomaa (1991), Volovik (1999)

E

EEgs EEH

Ground state degeneracy

Skip steps I and II: Cold atoms forming a p-wave superfluid

Gurarie et al.

• 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?

see also: Tewari, Das Sarma, Nayak, Zhang and Zoller (2006)

A p-wave superfluid of fermionic cold atoms

Eg

Free atoms – a delta function absorption spectrum

Eg

0

-

Eg-

-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

Eg

Eg+2||

The absorption spectrum when the -atoms form a p-wave superfluid

Strong pairingphase (<0)

weak-pairingphase (>0)

0

-

Eg-

• 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.

Now, rotate the system (an analog to a magnetic field)

Eg- Egc

t

c

0

-

Eg-Eg-

The absorption spectrum of a rotated system

And now back to the quantum Hall effect

Landau levels are the spectrum of the

-atoms

band formed by Majoranafermions near zero energy

The =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.

g(r) is a localized function in the vortex core

)()()()( * rRrgrRrgdr iii

ii

A zero energy solution is a spinor

A localized Majorana operator .

A subspace of degenerate ground states, with the ’s operating in that subspace.

In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator ij

Nayak and Wilczek (’96)Ivanov (’01)

All ’s anti-commute, and 2=1.

.... sgsg ji

backscattering = |tleft+tright|2

An experimental manifestation through interference:

Stern and Halperin (2005)Bonderson, Shtengel, Kitaev (2005)Following Das Sarma et al (2005)

interference pattern is observed by varying the cell’s area

Gate

Volt

age, V

MG

(mV

)

Magnetic Field

cell area

0 50 100 150 2000

5

10

15

-9.0 -7.5 -6.0 -4.5 -3.0

cell area

Curr

ent

(a.u

.)

Integer quantum Hall effect (adapted from Neder et al., 2006)The prediction for the =5/2 non-abelian state (weak backscattering limit)

Followed by an extension to a closed dot

vortex a around vortex 1 - a

vortex a around vortex 1 and vortex 2 - aa

The effect of the core states on the interference of backscattering amplitudes depends crucially on the parity of the number of localizedstates.

statescorerightleft Before encircling

1a left right2

After encircling

statescorestatescore arightleft 1

for an even number of localized vorticesonly the localized vortices are affected(a limited subspace)

for an odd number of localized vorticesevery passing vortex acts on a different subspace

statescorestatescore rightleft 12

Interference term:

for an even number of localized vorticesonly the localized vortices are affectedInterference is seen

for an odd number of localized vorticesevery passing vortex acts on a different subspaceinterference is dephased

statescorestatescorerightleft 12*

statescorestatescore arightleft 1*

|tleft + tright|2 |tright|2 + |tleft|2

Gate

Volt

age, V

MG

(mV

)

Magnetic Field (or voltage on anti-dot)

cell area

The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.

When interference is seen:

statescorestatescore nrightleft 12...

Interference term is proportional to

statescorestatescore n 12...

Two possible eigenvalues that differ by a minus sign.Cannot be changed by braiding of vortices

Closing the island into a quantum dot – Coulomb blockade:

Coulomb blockade !

Transport thermodynamics

For a conventional super-conductor, spacing alternates between

charging energy Ec (add an even electron)

charging energy Ec + superconductor gap

(add an odd electron)

The spacing between conductance peaks translates to the energy cost of adding an electron.

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)

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, nisa gapless (E=0) edge mode if nis is odd corresponds

to=0

a gapfull (E≠0) edge mode if nis is even corresponds

to ≠0

The gap diminishes with the size of the dot ∝ 1/L

Cell area

Magnetic field

(number of electrons in the dot)

(number of q.p.sin the dot)

Even Odd

What destroys the even-odd effect:

1. Fluctuating number of vortices on the island, nis

2. Fluctuations in the state of the nis vortices

3. Thermal fluctuations of the edges

All these fluctuations smear the interference picture,

but signatures of non-abelian statistics may still be

seen.

For example, what if nis is time dependent?

A simple way to probe exotic statistics:

tItnGtn isis )()(

For weak backscattering - a new source of current noise.

For Abelian states ():

q

nGnG is

is

2cos1)( 0

For the state:

Chamon et al. (1997)

G = G0 (nis odd) G0[1 ± cos( + nis/4)] (nis even)

time

G

G

0)0()( tt

isis entn

022

0

2 tVGI

compared to shot noise GVe*

bigger when t0 is long enough

close in spirit to 1/f noise, but unique to FQHE states.

(Kane PRL, 2003)

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.