Transcript Ady Stern

Proposed experimental probes of non-abelian anyons
Ady Stern (Weizmann)
with: N.R. Cooper, D.E. Feldman, Eytan Grosfeld, Y. Gefen,
B.I. Halperin, Roni Ilan, A. Kitaev, K.T. Law, B. Rosenow, S. Simon
Outline:
1. Non-abelian anyons in quantum Hall states – what they are,
why they are interesting, how they may be useful for topological
quantum computation.
2. How do you identify a non-abelian quantum Hall state when you
see one ?
More precise and relaxed presentations:
Introductory
pedagogical
Comprehensive
The quantized Hall effect and unconventional quantum
statistics
The quantum Hall effect
• zero longitudinal resistivity - no dissipation, bulk energy gap
current flows mostly along the edges of the sample
B
I
• quantized Hall resistivity
 xy 
1 h
 e2
 is an integer,
or a fraction
p
q
or q even
with q odd,
Extending the notion of quantum statistics
Electrons
A ground state:
Laughlin quasiparticles
 ( r1,.....................rN ; R1,.., R4 )
Energy gap
Adiabatically interchange the position of two excitations

 ei
More interestingly, non-abelian statistics
(Moore and Read, 91)
In a non-abelian quantum Hall state, quasi-particles obey
non-abelian statistics, meaning that (for example)
with 2N quasi-particles at fixed positions, the ground state is
2 N -degenerate.
Interchange of quasi-particles shifts between ground states.
2
N
ground states




 
g .s. 1 R1 , R2 ...
 
g .s. 2 R1 , R2 ...
…..

 
R1 , R2 ...position
of
quasi-particles

 
g .s. 2 R1 , R2 ...
N
Permutations between quasi-particles positions
unitary transformations in the ground state subspace
Up to a global phase, the unitary transformation depends only on
the topology of the trajectory
1
2
2
3
3
1
Topological quantum computation
(Kitaev 1997-2003)
• Subspace of dimension 2N, separated by an energy gap from the
continuum of excited states.
• Unitary transformations within this subspace are defined by the
topology of braiding trajectories
• All local operators do not couple between ground states
– immunity to errors
The goal:
experimentally identifying non-abelian quantum Hall states
The way: the defining characteristics of the most prominent
candidate, the =5/2 Moore-Read state, are
1. Energy gap.
2. Ground state degeneracy exponential in the number N of
quasi-particles, 2 N/2.
3. Edge structure – a charged mode and a Majorana fermion
mode
4. Unitary transformation applied within the ground state
subspace when quasi-particles are braided.
In this talk:
1. Proposed experiments to probe ground state degeneracy –
thermodynamics
2. Proposed experiments to probe edge and bulk braiding by
electronic transport–
Interferometry, linear and non-linear Coulomb blockade,
Noise
Probing the degeneracy of the ground state
(Cooper & Stern, 2008
Yang & Halperin, 2008)
Measuring the entropy of quasi-particles in the bulk
The density of quasi-particles is
4 n  n5 / 2  4 n 
Zero temperature entropy is then
4n
5 B
2 0
5 B
log 2
2 0
To isolate the electronic contribution from other contributions:
s


n
T
;
s m

B T
Leading to

 2 log 2 sgn   5 / 2 
T
(~1.4)
m
5

log 2 sgn   5 / 2 
T
0
(~12pA/mK)
Probing quasi-particle braiding - interferometers
Essential information on the Moore-Read state:
• Each quasi-particle carries a single Majorana mode
• The application of the Majorana operators takes
one ground state to another within the subspace of
degenerate ground states

A localized Majorana operator g i  g i . All g’s anti-commute, and g2=1.
When a vortex i encircles a vortex j, the ground state is
multiplied by the operator gigj
g.s. 
 g ig j g.s.
Nayak and Wilczek
Ivanov
Interferometers:
The interference term depends on the number and quantum
state of the quasi-particles in the loop.
Interference
term
even
Brattelli diagram
odd
even
Number of
q.p.’s in the
interference loop,
Odd number of localized vortices:
vortex a around vortex 1
- g1ga
a
 left
1
 left  core states  right  g ag 1 core states
The interference term vanishes:
  right core states g ag 1 core states
*
left
 right
Even number of localized vortices:
vortex a around vortex 1 and vortex 2
a
 left
g1gag2ga ~ g2g1
-
1
2
 left  core states  right  g 2g 1 core states
The interference term is multiplied by a phase:
  right core states g 2g 1 core states
*
left
Two possible values, mutually shifted by p
 right
Interference in the =5/2 non-abelian quantum Hall state:
The Fabry-Perot interferometer
D2
S1
D1
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)
5/2
cell
area
Magnetic Field
(or voltage on anti-dot)
Coulomb blockade vs interference
(Stern, Halperin 2006,
Stern, Rosenow, Ilan, Halperin, 2009
Bonderson, Shtengel, Nayak 2009)
Interferometer (lowest order)
Quantum dot
For non-interacting electrons – transition from one limit to another
via Bohr-Sommerfeld interference of multiply reflected trajectories.
Can we think in a Bohr-Sommerfeld way on the transition when
anyons, abelian or not, are involved?
Yes, we can (BO, 2008)
(One) difficulty – several types of quasi-particles may tunnel
Thermodynamics is easier than transport. Calculate the
thermally averaged number of electrons on a closed dot. Better
still, look at

N
A
The simplest case, =1.
Energy is determined by the number of electrons E  Ec N  Nbg  A 2
Partition function
 Ec ( N  N bg ) 2 
Z   exp 
N

T


Poisson summation
 Ec ( N  N bg ( A)) 2

 T

Z ~   dN exp 
 2piNs  ~  exp  (ps ) 2  2piN bg ( A) s 
T
s

 s
 Ec

 Ec
2
exp

(
N

N
)

bg
 T



N
• Sum over electron
number.
• Thermal suppression of
high energy configurations
~
 T

2
s exp  E (ps)  2piN bg ( A)s
 c

• Sum over windings.
• Thermal suppression of high winding
number.
• An Aharonov-Bohm phase
proportional to the winding number.
• At high T, only zero and one windings
remain
And now for the Moore-Read state
The energy of the dot is made of
•
A charging energy
•
An energy of the neutral mode. The spectrum is determined
Ec N  N bg  A
2
by the number and state of the bulk quasi-particles.
 Ec

Z   exp  ( N  N bg ( A)) 2   N (T )
 T

N
The neutral mode partition function χ depends on nqp and their state.
Poisson summation is modular invariance
 v / L 
 n 
 T 
  T 

S  
 vn / L 
(Cappelli et al, 2009)
  T 
  vn / L 


  S  
 T 
 vn / L 

 (T )
The components of the vector correspond to the
different possible states of the bulk quasi-particles,
one state for an odd nqp (“s”), and two states for an
even nqp (“1” and “ψ”).
A different thermal suppression factor for each
component.

S
The modular S matrix. Sab encodes the outcome of a
quasi-particle of type a going around one of type b
Low T
High T
Probing excited states at the edge –
non linear transport in the Coulomb blockade regime
(Ilan, Rosenow, Stern, 2010)
A nu=5/2 quantum Hall system
=2
Goldman’s group, 80’s
Non-linear transport in the Coulomb blockade regime:
dI/dV at finite voltage – a resonance for each many-body
state that may be excited by the tunneling event.
dI/dV
Vsd
Energy spectrum of the neutral mode on the edge
Single fermion:
For an odd number of q.p.’s
En=0,1,2,3,….
For an even number of q.p.’s
En= ½, 3/2, 5/2, …
Many fermions:
For an odd number of q.p.’s
Integers only
For an even number of q.p.’s
Both integers and half integers
(except 1!)
The number of peaks in the differential conductance varies with
the number of quasi-particles on the edge.
Current-voltage characteristics
En  (n  1 / 2)
Even number
En  n
Odd number
(Ilan, Rosenow, AS 2010)
Sourcedrain
voltage
Magnetic
field
Interference in the =5/2 non-abelian quantum Hall state:
Mach-Zehnder interferometer
The Mach-Zehnder interferometer:
(Feldman, Gefen, Kitaev, Law, Stern, PRB2007)
S
D2
D1
Compare:
M-Z
F-P
S
D2
D1
D2
S1
D1
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)
Interference
term
G2
G1/2
G1
G2/2
G4
G3
G3/2
G4/2
Number of
q.p.’s in the
interference loop
•
The system propagates along the diagram, with transition rates
assigned to each bond.
•
The rates have an interference term that
•
depends on the flux
•
depends on the bond (with periodicity of four)
If all rates are equal, current flows in “bunches” of one
quasi-particle each – Fano factor of 1/4.
The other extreme – 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.
Summary:
Temperature dependence of the chemical
potential and the magnetization reflect the
ground state entropy
Interference magnitude depends on the
parity of the number of quasi-particles
Phase depends on the eigenvalue of

 2 log 2 sgn   5 / 2 
T
m
5

log 2 sgn   5 / 2 
T
0
g 2g 1
Coulomb blockade I-V characteristics may
measure the spectrum of the edge Majorana
mode
Mach-Zehnder:
Fano factor changing between 1/4 and about
three – a signature of non-abelian statistics in
Mach-Zehnder interferometers
D2
For a Fabry-Perot interferometer, the
state of the bulk determines the
interference term.
S1
D1
Interference
term
even
odd
even
Das-Sarma-Freedman-Nayak qubit
The interference phases are mutually shifted by p.

g ()   g m e
m 0
2pim  p / 2 
Number of
q.p.’s in the
interference loop,
Interference
term
even
odd
Number of
q.p.’s in the
interference loop,
even
The sum of two interference phases, mutually shifted by p.

g ()   g m e
m 0
4pim
D2
S1
The area period goes down by a factor of two.
D1
Ideally,
Quasi-particles number
The gate voltage
Area
Gate Voltage, VMG (mV)
The magnetic field
cell
area
Magnetic Field
(or voltage on anti-dot)
Are we getting there?
(Willett et al. 2008)
From electrons at n=5/2 to non-abelian quasi-particles:
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
from: electrons at a half filled Landau level
to: spin polarized composite fermions at zero (average)
magnetic field
GM87
R89
ZHK89
LF90
HLR93
KZ93
B
Electrons in a magnetic field B
e-
H=E
(b)
20
B
Composite particles in a
magnetic field B  2 0n( r )
CF
 ({ri }) ({ri })e
i2
i  j arg( ri  rj )
Mean field (Hartree) approximation
(c)
B
B1/2 = 2ns0
B  B  20 n  0
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.
Shot noise (A2/Hz)
5
x 10
-29
4
e/2
3
2
1
0
-1
-2
e/4
-1
0
1
Impinging current (nA)
2
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.
For a single vortex – there is a zero energy mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
A zero energy solution is a spinor
g i   dr g ( r  Ri ) ( r ) 
g * ( r  Ri )  ( r )

g(r) is a localized function in the vortex core

A localized Majorana operator g i  g i . 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
g.s. 
 g ig j g.s.
Nayak and Wilczek (1996)
Ivanov (2001)
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 =5/2
Closing the island into a quantum dot:
5/2
-9.0
-7.5
-6.0
-4.5
-3.0
Current (a.u.)
15
Interference involving
multiple scatterings,
Coulomb blockade
10
5
0
0
50
100
150
cell area
200
5/2
g 2g 1
But,
is very different from
g 2g 1 
2
g ag 1
 g ag 1   1
2
so, interference of even number of windings always survives.
Equal spacing
between
peaks
forfor
oddeven
number
of localized
vortices
Alternate
spacing
between
peaks
number
of localized
vortices
nis – a crucial quantity. How do we know it’s time independent?
What is
5/2
nis (t )nis (0) ?
e
Qis  nis
4
By the fluctuation-dissipation theorem,
t
Qis (t )Qis (0)  2CTe
t0
C – capacitance
t0 – relaxation time = C/G
G – longitudinal conductance
Best route – make sure charging energy >> Temperature
A subtle question – the charging energy of what ??
And what if nis is time dependent?
A simple way to probe exotic statistics:
nis (t )  Gnis (t )   I t 
A new source of current noise.
For Abelian states (1/3):


2pnis  

G(nis )  G0 1   cos  
 
q 


Chamon et al. (1997)
For the 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.
When multiple reflections are taken into account, the average
conductance and the noise, satisfy
I 2   0  1 cos 4
I B  I B  I1 cos 4
and
A signature of the =5/2 state
I1
IB
,
1
IB

IB
3
(For abelian Laughlin states – the power is
1

1 )
A “cousin” of a similar scaling law for the Mach-Zehnder
case (Law, Feldman and Gefen, 2005)
Finally, a lattice of vortices
When vortices get close to one another, degeneracy is lifted by
tunneling.
For a lattice, expect a tight-binding Hamiltonian
H   tijg ig j  h.c.
i, j
Analogy to the Hofstadter problem.
tij  t e
i ij
 t e i
The phases of the tij’s determine the flux in each plaquette
i
j
Adl
Since g i  g i
the tunneling matrix elements must be imaginary.
H    it ij g ig j
i, j
The question – the distribution of + and For a square lattice:
Corresponds to half a flux quantum per
i
i
i
i
plaquette.
A unique case in the Hofstadter problem –
no breaking of time reversal symmetry.
Spectrum – Dirac:
E (k )  v0k
v0  ta

v0 is varied by varying
E
density
k
A mechanism for dissipation, without a motion of the charged vortices
s e q  0,   0 ~ q 2 2
Exponential dependence on density
Protection from decoherence:
(Kitaev, 1997-2003)
• The ground state subspace is separated from the rest of the
spectrum by an energy gap
• Operations within this subspace are topological
But:
• In present schemes, the read-out involves interference of two
quasi-particle trajectories (subject to decoherence).
• In real life, disorder introduces unintentional quasi-particles. The
ground state subspace is then not fully accounted for.
A theoretical challenge!
Summary
1. A proposed interference experiment to address the non-abelian
nature of the quasi-particles, insensitive to localized quasi-particles.
2. A proposed “thermodynamic” experiment to address the
non-abelian nature of the quasi-particles, insensitive to localized
quasi-particles.
3. Current noise probes unconventional quantum statistics.
Closing the island into a quantum dot:
Coulomb blockade !
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
Reason: consider a compact geometry (sphere). By Dirac’s
quantization, the number of flux quanta (h/e) is quantized to an integer,
the number of vortices (h/2e) is quantized to an even
integer
In a non-compact closed geomtry, the edge “completes” the pairing
So what about peak spacings?
When nis is odd, peak spacing is “unaware” of 
peaks are equally spaced
When nis is even, peak spacing is “aware” of 
periodicity is doubled
B
Interference
pattern
Coulomb peaks
even
even
No interference
odd
odd
even
even
No interference
cell area
From electrons at =5/2 to a lattice of 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:
From a half filled Landau level of electrons to composite fermions
at zero magnetic field - the Chern-Simons transformation
The Chern-Simons transformation
•
The original Hamiltonian:
1
H 
2m
P
i
 A( r ) 
i
2
1

2

i, j
e2
| ri  r j |
• Schroedinger eq. H   E 
• Define a new wave function:
 ({ri }) ({ri })e
i2
i  j arg( ri  rj )
({ri }) describes electrons (fermions)
({ri })
describes composite fermions
The effect on the Hamiltonian:
Pi  A( r )


Pi  A(r )a(r )

r

arg r
5/2
|tleft + tright|2
for an even number of localized quasi-particles
|tright|2 + |tleft|2
for an odd number of localized quasi-particles
The number of quasi-particles on the island may be tuned by
charging an anti-dot, or more simply, by varying the magnetic
field.
The new magnetic field:
~
 a( r )  0 ( r )
(a)
e-
B
ns
 A  B
20
(b)
B
Electrons in a magnetic
field
CF
Composite particles in
a magnetic field
   A  a   B  2 0n( r )
Mean field (Hartree) approximation
(c)
B
B1/2 = 2ns0
B 
   A  a   B  2 0 n  0
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
Read and Green (2000)
Step IV: introducing quasi-particles into the super-conductor
- shifting the filling factor away from 5/2
B     A  a   B  20 n   1  2  0
The super-conductor is subject to a magnetic field
an Abrikosov lattice of vortices in a p-wave super-conductor
Look for a ground state degeneracy in this lattice
Dealing with Abrikosov lattice of vortices in a p-wave super-conductor
First, a single vortex – focus on the mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
H  H0
r 
r
  drdr ' ( R, r ) ( R  ) ( R  )  h.c.
2
2

( R, r )  R  f (r )
A quadratic Hamiltonian – may be diagonalized
(Bogolubov transformation)
H  E0   E g E g E

E
g E   dr  u (r ) (r )  v(r )  (r )  BCS-quasi-particle annihilation operator
Ground state degeneracy requires zero energy modes
The functions u( r ), v ( r ) are solutions of the Bogolubov de-Gennes eqs.
g   dr u(r)(r)  v(r) (r) 

E
Ground state should be annihilated by all
For uniform super-conductors

ik r
u(r)  v(r)  e
g E ‘s
  const.

g.s.   1  g k ck ck


 vac
k
For a single vortex – there is a zero energy mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
A zero energy solution is a spinor
g i   dr g ( r  Ri ) ( r ) 
g * ( r  Ri )  ( r )

g(r) is a localized function in the vortex core

A localized Majorana operator g i  g i . 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
g.s. 
 g ig j g.s.
Nayak and Wilczek
Ivanov
Interference experiment:
Stern and Halperin (2005)
Following Das Sarma et al (2005)
5/2
backscattering = |tleft+tright|2
interference pattern is observed by varying the cell’s area
vortex a around vortex 1
-
g1ga
vortex a around vortex 1 and vortex 2
a
 left
1
g1gag2ga ~ g2g1
-
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  g 2g 1 core states
for an even number of localized vortices
only the localized vortices are affected
(a limited subspace)
 left  core states  right  g ag 1 core states
for an odd number of localized vortices
every passing vortex acts on a different subspace
interference is dephased
5/2
|tleft + tright|2
for an even number of localized quasi-particles
|tright|2 + |tleft|2
for an odd number of localized quasi-particles
• the number of quasi-particles on the dot may be tuned by a gate
• insensitive to localized pinned charges
occupation of
anti-dot
interference
no interference
interference
cell area
Localized quasi-particles shift the red lines up/down
B
Electrons in a magnetic field B
e-
H=E
(b)
20
B
Composite particles in a
magnetic field B  2 0n( r )
CF
 ({ri }) ({ri })e
i2
i  j arg( ri  rj )
Mean field (Hartree) approximation
(c)
B
B1/2 = 2ns0
B  B  20 n  0
A yet simpler version:
equi-phase
lines
5/2
B
even
odd
No interference
even
No interference
cell area
And now to a lattice of quasi-particles.
When vortices get close to one another, degeneracy is lifted by
tunneling.
For a lattice, expect a tight-binding Hamiltonian
H   tij g i g j  h.c.
i, j
Analogy to the Hofstadter problem.
tij  t e
i ij
 t e i
The phases of the tij’s determine the flux in each plaquette
i
j
Adl
Since g i  g i
the tunneling matrix elements must be imaginary.
H    it ij g ig j
i, j
The question – the distribution of + and For a square lattice:
Corresponds to half a flux quantum per
i
i
i
i
plaquette.
A unique case in the Hofstadter problem –
no breaking of time reversal symmetry.
Spectrum – Dirac:
E (k )  v0k
v0  ta

v0 is varied by varying
E
density
k
What happens when an electric field E(q,) is applied?
Given a perturbation
E
J  Acost
the rate of energy absorption is
2p
 i J  A f

f
2
d  f   i   
Distinguish between two different problems –
1. Hofstadter problem – electrons on a lattice
2. Present problem – Majorana modes on a lattice
k
For both problems the rate of energy absorption
E
2
is Re sE

2p
 i J  A f

f
2
d  f   i   
electric field E ~ iA
dos( ) ~
k

v0
2
The difference between the two problems is in the matrix elements
i J
f

 ev0

 

if
ev0
t

for the electrons
for the Majorana modes
The reason – due the particle-hole symmetry of the Majorana mode,
it does not carry any current at q=0.
So the real part of the conductivity is
e2
8h
e2   
 
8h  t 
for the electrons
2
for the Majorana modes
From the conductivity of the Majorana modes to the electronic
response
The conductivity of the p-wave super-conductor of composite
fermions, in the presence of the lattice of vortices
2


n
e2
qa 
 i

2
 m 8
 v0 q 
1 

s CF q,    
  


0

0
n
e2 qa 
i

m
8
From composite fermions to electrons
s e q  0,   ~ q 2
2





2 
v q
1   0  
   
s s
1
e
1
CF
h  0 2
 2 

e   2 0 
s e q, v0q   0
Summary
1. A proposed interference experiment to address the non-abelian
nature of the quasi-particles.
2. Transport properties of an array of non-abelian quasi-particles.
 ei  localized function in the
 i   direction perpendicular to the
 e  =0 line
Unitary transformations:
When vortex i encircles vortex i+1, the unitary transformation
operating on the ground state is

g ig i 1   dr  dr ' wi (r )ei (r )  wi * (r )e  i   (r )
i
i


 wi 1 (r ' )eii1  (r ' )  wi 1 (r ' )e ii1   (r ' )
*

• No tunneling takes place?
• How does the zero energy state at the i’s vortex “know” that it is
encircled by another vortex?
A more physical picture?
The emerging picture – two essential ingredients:
• 2N localized intra-vortex states, each may be filled (“1”) or empty (“0”)
• Notation: 1 0 0 1 1... means 1st, 3rd, 5th vortices filled, 2nd, 4th vortices empty.
•Full entanglement: Ground states are fully entangled super-positions of
all possible combinations with even numbers of filled states
ei 0000 0000  ei 0011 0011  ei1100 1100  ei1010 1010
 ei 0110 0110  ei 0101 0101  ei1001 1001  ei1111 1111
and all possible combinations with odd numbers of filled states
ei 0001 0001  ei 0010 0010  ei 0100 0100  ei1000 1000
 ei 0111 0111  ei1011 1011  ei1101 1101  ei1110 1110
Product states are not ground states:



i 2
i 3
i 4


 i 21

2
2
2

e
 * e
10

ie
01
10

ie
01

 


 

• Phase accumulation depend on occupation
When a vortex traverses a closed
trajectory, the system’s wave-function
accumulates a phase that is
2p N
Halperin
Arovas, Schrieffer, Wilczek
N – the number of fluid particles encircled by the trajectory
(|0000 + |1100 )
(|0000 - |1100 )
Permutations of vortices change relative phases in the superposition
Four vortices:
Vortex 2 encircling vortex 3
ei 0000 0000  ei 0011 0011  ei1100 1100  ei1010 1010
 ei 0110 0110  ei 0101 0101  ei1001 1001  ei1111 1111
Vortex 2 and vortex 3 interchanging positions
ei 0000 0000  ei 0011 0011  ei1100 1100  ei1010 1010
 ei 0110 0110  ei 0101 0101  ei1001 1001  ei1111 1111
A “+” changing into a “”
A vortex going around a loop generates a unitary transformation in the
ground state subspace
2
3
4
1
A vortex going around the same loop twice does not generate any
transformation
2
3
1
4
The Landau filling range of 2<<4
Unconventional fractional quantum Hall states:
1. Even denominator states are observed
2. Observed series does not follow the

p
2 p 1
5 7 19
 , ,
2 2 8
rule.
3. In transitions between different plateaus,  xy is non-monotonous
as opposed to
(Pan et al., PRL, 2004)
Focus on =5/2
The effect of the zero energy states on interference
Dephasing, even at zero temperature
No dephasing (phase changes of
4p )
More systematically: what are the 2 N ground states?




 
The goal: 2 ground states g.s. 1 R1 , R2 ...
 
g .s. 2 R1 , R2 ...
N
…..

 
R1 , R2 ...
position of vortices

 
g .s. 2 R1 , R2 ...
N
that, as the vortices move, evolve without being mixed.
The condition:
g.s. k

g.s. n  0
Ri
for
kn
How does the wave function near each vortex look?
To answer that, we need to
• define a (partial) single particle basis, near each vortex
• find the wave function describing the occupation of these states
 w0 
 *  is a purely zero energy state
 w0 
 w0 

*
 is a purely non-zero energy state =

w
 0 
 w0  iw1 


Y   CE GE  
 w*0  iw*1 
E


 C
E
GE  C * E GE


E
defines a localized function w1
correlates its occupation with that of w0
There is an operator Y for each vortex.
We may continue the process
 w1 
 * 
 w1 
 w1 

*

  w1 

 C
E
( 2)
E
GE  h.c.


 C
( 2)
E
GE  h.c.

E
 w  iw2 
( 2)
Y ( 2 )   CE GE   *1

* 
w

iw
1
2
E


defines a localized function w2
correlates its occupation with that of w0 , w1
This generates a set of orthogonal vortex states w0 , w1, w2 ,...wl
near each vortex (the process must end when states from
different vortices start overlapping).
The requirements Y  j  g.s.  0 for j=1..k determine the occupati
of the states w0 ..wk near each vortex.
The functions u( r ), v ( r ) are solutions of the Bogolubov de-Gennes eqs.
g   dr u(r)(r)  v(r) (r) 

E
Ground state should be annihilated by all
For uniform super-conductors

ik r
u(r)  v(r)  e
g E ‘s
  const.

g.s.   1  g k ck ck
k


 vac
The simplest model – take a free Hamiltonian with a potential part
only
h( r )    ( r )
To get a localized mode of zero energy,
we need a localized region of 0. A vortex is a closed curve of =0
with a phase winding of 2p in the order parameter .
The phase winding is turned into a boundary condition
A change of sign
 ei  localized function in the
Spinor g i is  i   direction perpendicular to the
 e  =0 line


The phase  depends on the direction of the =0 line. It changes by
p around the square.
A vortex is associated with a localized Majorana operator.
For a lattice, expect a tight-binding Hamiltonian
H   tij g i g j  h.c.
i, j
Analogy to the Hofstadter problem.
tij  t e
i ij
 t e i
i
j
Adl
The phases of the tij’s determine the flux in each plaquette
Questions –
1. What are the tij?
2. How do we calculate electronic response functions from the
spinors’ Hamiltonian?
Two close vortices:
Solve along this line to
get the tunneling matrix
element
We find tij=i
A different case – the line going through
the tunneling region changes the sign of
the tunneling matrix element.
i
i
i
i
These requirements are satisfied for a given vortex by
either one of two wave functions:












p  1  c0 1  c1 1  c2 ... 1  c.. 1  c... vacuum
or:
m  1  c0 1  c1 1  c2 ... 1  c.. 1  c... vacuum
The occupation of all vortex states is particle-hole symmetric.
Still, two states per vortex, altogether 2 2 N
We took care of the operators
Y
( j)
and not 2 N
 w j 1  iw j 



 w* j 1  iw* j 


For the last state, we should take care of the operator
which creates and annihilates E0 quasi-particles.
 wk 




* 

w
k


Doing that, we get ground states that entangle states of different
vortices (example for two vortices):
g .s.1   p p  m m  env1   p m  m p  env2
g .s.2   p p  m m  env2   p m  m p  env1
For 2N vortices, ground states are super-positions of states of the form
  1 c
(1) 
i
i
1st vortex
   1 c
( 2)
i
i
2nd vortex
...  1  c
( 2 N )
i
 vacuum
i
2N’th vortex

The ci operator creates a particle at the state wi (r ) near the
vortex. When the vortex is encircled by another
ci 
  ci
Open questions:
1. Experimental tests of non-abelian states
2. The expected QH series in the second Landau level
3. The nature of the transition between QH states in the second
Landau level
4. Linear response functions in the second Landau level
5. Physical picture of the clustered parafermionic states
6. Exotic directions – quantum computing, BEC’s
Several comments on the Das Sarma, Freedman and Nayak proposed
experiment
One fermion mode, two possible states, two pi-shifted interference
patterns
n=0
n=1
Comment number 1: The measurement of the interference pattern
initializes the state of the fermion mode
initial core state
 g 1g 2  i   g 1g 2  i
after measurement current has been flown through the system
 g 1g 2  i  current 1   g 1g 2  i  current 2
A measurement of the interference pattern implies
current 1 current 2  0
The system is now either at the g1g2=i or at the g1g2=-i state.
At low temperature, as we saw

N
A
odd nqp (“s”)
even nqp (“1” and “ψ”).
At high temperature, the charge part will thermally suppress all
but zero and one windings, and the neutral part will thermally
suppress all but the “1” channel (uninteresting) and the “s”
channel. The latter is what one sees in lowest order
interference
(Stern et al, Bonderson et al, 2006).
High temperature Coulomb blockade gives the same
information as lowest order interference.
g1
g2
For both cases the interference pattern is shifted by p
by the transition of one quasi-particle through the
gates.
Summary
Entanglement between
the occupation of states near
different vortices
The geometric phase
accumulated by
a moving vortex
Non-abelian statistics

 2 log 2 sgn   5 / 2 
T
(~1.4)
m
5

log 2 sgn   5 / 2 
T
0
(~12pA/mK)
The positional entropy of the quasi-particles
If all positions are equivalent, other than hard core constraint –
positional entropy is ∝n log(n), but Interaction and disorder lead to the localization of the quasiparticles – essential for the observation of the QHE – and to
the suppression of their entropy.
The positional entropy of the quasi-particles depends on their
spectrum:
excitations
qhe gap
localized states
Phonons of a quasi-particles Wigner crystal
temperature
ground state
non-abelians
Shot noise as a way to measure charge:
D1
1-p
S
coin tossing
p
Binomial distribution
For p<<1, current noise is
S=2eI2
I22
D