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

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

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

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)

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 Call your leader a patent lawyer Read and Moore “If only life was so simple” (Allen, Ann. Ha. 1977)

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

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.

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

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.

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)

Free atoms – a delta function absorption spectrum Eg 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

-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

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|| - 

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.

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

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.

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)

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

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

vortex a around vortex 1 - g1ga vortex a around vortex 1 and vortex 2 - 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

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

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

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

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

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)

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)

(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)

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.

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 = G0 (nis odd) G0[1 ± b cos(f + pnis/4)] (nis even)

bigger when t0 is long enough dG 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.

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.