Transcript tqc2007 6542

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
1. Observing the zero energy Majorana modes
2. Fabry-Perot interferometry
3. Mach-Zehnder interferometry
Extending the notion of quantum statistics
Electrons
A ground state:
Laughlin quasiparticles
 ( r1,.....................rN ; R1,.., R4 )
Energy gap
For abelian states:
Adiabatically interchange the position of two excitations
i

 e 
For non-abelian states:
With N quasi-particles at fixed positions, the ground state is
lN -degenerate.
Interchange of quasi-particles shifts between ground states.
For n=5/2 (Moore-Read, Pfaffian), where l= 2


…..
degenerate
ground
states


 
g.s. 1 R1 , R2 ...
 
g.s. 2 R1 , R2 ...
g.s. 2
N /2

 
R1 , R2 ...
 
R1 , R2 ...position
of
quasi-particles

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
a patent lawyer
“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
5
1
2
2
2
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
H  H 0   dr( r )   
 h.c.
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
H  H 0   dr( r )   
 h.c.
   dr u(r)(r)  v(r) (r) 

E
H  Egs   E E E
E
H  Egs   E E E
E
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?
see also: Tewari, Das Sarma, Nayak, Zhang and Zoller (2006)
Free atoms – a delta function absorption spectrum


Eg
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
The absorption spectrum when the -atoms form a p-wave
superfluid
weak-pairing
phase (>0)
Eg-

0
-
Strong pairing
phase (<0)
Eg
Eg+2||
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.
The absorption spectrum of a rotated system
Landau levels are
the spectrum of the
-atoms
c
Eg-

t
0
-
Eg-
Eg
band formed by 
Majorana
c
fermions near zero energy
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
 i   dr g ( r  Ri ) ( r ) 
g * ( r  Ri )  ( r )

g(r) is a localized function in the vortex core

A localized Majorana operator  i   i . All ’s anti-commute, and 2=1.
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 ij
g.s. 
  i j g.s.
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
Gate Voltage, VMG (mV)
quantum
Hall
effect
(adapted fromstate
Neder (weak
et al., 2006)
TheInteger
prediction
for the
n=5/2
non-abelian
backscattering limit)
-9.0
-7.5
-6.0
0
50
100
-4.5
-3.0
Current (a.u.)
15
cell
area
10
5
0
150
cell area
Magnetic Field
Followed by an extension to a closed dot
n5/2
200
vortex a around vortex 1
-
1a
vortex a around vortex 1 and vortex 2
a
 left
1
1a2a ~ 21
-
2
 right
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

left
 right   core states
After encircling
 left  core states  right   2 1 core states
for an even number of localized vortices
only the localized vortices are affected
(a limited subspace)
 left  core states  right   a 1 core states
for an odd number of localized vortices
every passing vortex acts on a different subspace
Interference term:
 left* right core states  2 1 core states
for an even number of localized vortices
only the localized vortices are affected
Interference is seen
  right core states  a 1 core states
*
left
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
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)
n5/2
cell
area
Magnetic Field
(or voltage on anti-dot)
When interference is seen:
 left  core states  right   n ... 2 1 core states
Interference term is proportional to
core states  n ... 2 1 core states
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:
Coulomb blockade !
n5/2
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 
(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
a gapfull (E≠0) edge mode if nis is even
corresponds to =0
corresponds to ≠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)
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:
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:
nis (t )  Gnis (t )   I t 
For weak backscattering - a new source of current noise.
For Abelian states (n1/3):


2nis  

G(nis )  G0 1   cos  
 
q 


Chamon et al. (1997)
For the n5/2 state:
G = G0
(nis odd)
G0[1 ±  cos( + nis/4)] (nis even)
G
dG
time
nis (t )nis (0)  e
t
t0
compared to shot noise
dI 2
 0
 dG 2V 2t0
e*GV
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.