proposed experimental probes of non-abelian anyons ady stern (weizmann) with: n.r. cooper, d.e....

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 ?

Introductory

pedagogical

Comprehensive

More precise and relaxed presentations:

The quantized Hall effect and unconventional

quantum statistics

• zero longitudinal resistivity - no dissipation, bulk

energy gap current flows mostly along the edges of the

sample

• quantized Hall resistivity2

1

e

hxy

The quantum Hall effect

is an integer,

or q even

or a fraction with q odd,qp

IB

Extending the notion of quantum statistics

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

Adiabatically interchange the position of two excitations

Energy gap

ie

Laughlin quasi-

particlesElectrons

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

-degenerate.

Interchange of quasi-particles shifts between ground states.

N2

More interestingly, non-abelian statistics (Moore and Read, 91)

ground statesN2

...,2..

...,2..

...,1..

21

21

21

RRsg

RRsg

RRsg

N

…..

..., 21 RR

position of quasi-particles

Permutations between quasi-particles positions unitary transformations in the ground state subspace

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

Up to a global phase, the unitary transformation depends only on the topology of the trajectory

1

1

3

2

2

3

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, 2008Yang & Halperin, 2008)

Measuring the entropy of quasi-particles in the bulk

The density of quasi-particles is

Zero temperature entropy is then

02/5 2

544

Bnnn

2log2

54

0

Bn

To isolate the electronic contribution from other contributions:

T

m

B

s

Tn

s

;

Leading to

2/5sgn2log5

2/5sgn2log2

0

T

m

T (~1.4)

(~12pA/mK)

Probing quasi-particle braiding - interferometers

When a vortex i encircles a vortex j, the ground state is multiplied by the operator ij

Nayak and WilczekIvanov

ii A localized Majorana operator .All ’s anti-commute, and 2=1.

.... sgsg ji

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

The interference term depends on the number and quantum state of the quasi-particles in the loop.

even odd evenNumber of q.p.’s in the interference loop,

Interference term

Brattelli diagram

Interferometers:

Odd number of localized vortices:vortex a around vortex 1 - a

1a left right

statescorestatescore arightleft 1

The interference term vanishes:

statescorestatescore arightleft 1*

Even number of localized vortices:vortex a around vortex 1 and vortex 2 - aa

1a left right2

statescorestatescore rightleft 12

The interference term is multiplied by a phase:

statescorestatescorerightleft 12*

Two possible values, mutually shifted by

Interference in the =5/2 non-abelian quantum Hall state: The Fabry-Perot interferometer

S1 D1

D2

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.

Coulomb blockade vs interference

(Stern, Halperin 2006,Stern, Rosenow, Ilan, Halperin, 2009Bonderson, 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 NA

2ANNEE bgc

N

bgc

T

NNEZ

2)(exp

Poisson summation

The simplest case, =1.

Energy is determined by the number of electrons

Partition function

sbg

cs

bgc sAiNsE

TiNs

T

ANNEdNZ )(2)(exp~2

))((exp~ 2

2

sbg

cNbg

c sAiNsE

TNN

T

E)(2)(exp~)(exp 22

• Sum over electron number. • Thermal suppression of high energy configurations

• 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

The energy of the dot is made of • A charging energy

• An energy of the neutral mode. The spectrum is

determined by the number and state of the bulk

quasi-particles.

2ANNE bgc

)())((exp 2 TANNT

EZ N

Nbg

c

The neutral mode partition function χ depends on nqp and their state.

Poisson summation is modular invariance

And now for the Moore-Read state

Lv

TS

T

Lv

n

n

/

/

(Cappelli et al, 2009)

The components of the vector correspond to

the different possible states of the bulk quasi-

particles, one state for an odd nqp (“”), and

two states for an even nqp (“1” and “ψ”).

A different thermal suppression factor for

each component.

The modular S matrix. Sab encodes the outcome of a quasi-particle of type a going around one of type b

Lv

TS

T

Lv

n

n

/

/

)(T

S

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 (Ilan, Rosenow, AS 2010)

Source-drain voltage

Magnetic field

nE

nE

n

n

)2/1( Even number

Odd number

Interference in the =5/2 non-abelian quantum Hall state:

Mach-Zehnder interferometer

The Mach-Zehnder interferometer: (Feldman, Gefen, Kitaev, Law, Stern, PRB2007)

S D1D2

Compare:

S D1D2

S1 D1

D2

M-Z

F-P

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)

Number of q.p.’s in the interference loop

Interference term

• 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)

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.

If all rates are equal, current flows in “bunches” of one quasi-particle each – Fano factor of 1/4.

Summary:

Interference magnitude depends on the parity of the number of quasi-particlesPhase depends on the eigenvalue of

12

Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers

Mach-Zehnder:

Temperature dependence of the chemical potential and the magnetization reflect the ground state entropy

2/5sgn2log5

2/5sgn2log2

0

T

m

T

Coulomb blockade I-V characteristics may measure the spectrum of the edge Majorana mode

even odd evenNumber of q.p.’s in the interference loop,

Interference term

S1 D1

D2For a Fabry-Perot interferometer, the state of the bulk determines the interference term.

Das-Sarma-Freedman-Nayak qubit

The interference phases are mutually shifted by

0

2/2)(m

immegg

even odd evenNumber of q.p.’s in the interference loop,

Interference term

The sum of two interference phases, mutually shifted by

The area period goes down by a factor of two.

S1 D1

D2

0

4)(m

immegg

Gate

Volt

age, V

MG

(mV

)

Magnetic Field (or voltage on anti-dot)

cell area

Ideally,

The magnetic field Quasi-particles

number

The gate voltage Area

Are we getting there?

(Willett et al. 2008)

From electrons at n=5/2 to non-abelian quasi-particles:

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

Step II:

the Chern-Simons transformation

from: electrons at a half filled Landau level

Step I:Read and Green (2000)

2

12

2

5

to: spin polarized composite fermions at zero (average) magnetic field

GM87R89ZHK89LF90HLR93KZ93

Be-

B1/2 = 2ns0

B)c(

20

CF

)b(B

Electrons in a magnetic field B

Composite particles in a magnetic field )(2 0 rnB

Mean field (Hartree) approximation

02 0 nBB

H = E

ji ji rri

ii err)arg(2

})({})({

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

-2 -1 0 1 2-1

0

1

2

3

4

5x 10

-29

Impinging current (nA)

Sho

t no

ise

(A2 /H

z) e/2

e/4

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

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

Kopnin, Salomaa (1991), Volovik (1999)

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 (1996)Ivanov (2001)

All ’s anti-commute, and 2=1.

.... sgsg ji

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:

Interference involving multiple scatterings, Coulomb blockade

0 50 100 150 2000

5

10

15

-9.0 -7.5 -6.0 -4.5 -3.0

cell area

Curr

ent

(a.u

.)

12 1ais very different from

But, 121

212 a

so, interference of even number of windings always survives.

Equal spacing between peaks for odd number of localized vorticesAlternate spacing between peaks for even number of localized vortices

nis – a crucial quantity. How do we know it’s time independent?

What is ? )0()( isis ntn

isis ne

Q4

By the fluctuation-dissipation theorem,

02)0()( tt

isis CTeQtQ

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:

tItnGtn isis )()(

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.

When multiple reflections are taken into account, the average conductance and the noise, satisfy

4cos102 I

and

311 , BBB

III

I

A signature of the =5/2 state

(For abelian Laughlin states – the power is ) 11

A “cousin” of a similar scaling law for the Mach-Zehndercase (Law, Feldman and Gefen, 2005)

4cos1III BB

For a lattice, expect a tight-binding Hamiltonian

..,

chtHji

jiij

Analogy to the Hofstadter problem.

j

iijdlAii

ij etett

The phases of the tij’s determine the flux in each plaquette

Finally, a lattice of vortices

When vortices get close to one another, degeneracy is lifted by tunneling.

Since the tunneling matrix elements must be imaginary. ii

ji

jiijitH,

The question – the distribution of + and -

For a square lattice:

ii i

i

Corresponds to half a flux quantum per

plaquette.

A unique case in the Hofstadter problem –

no breaking of time reversal symmetry.

Spectrum – Dirac: kvkE 0)( tav 0

E

k

A mechanism for dissipation, without a motion of the charged vortices

0v is varied by varyingdensity

22~0,0 qqe

Exponential dependence on density

Protection from decoherence:

• 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!

(Kitaev, 1997-2003)

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 !

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.

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

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

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

cell area

odd

even

even

No interference

No interference

B

cell area

odd

even

even

Interference patternCoulomb peaks

From electrons at =5/2 to a lattice of non-abelian quasi-particlesin four steps:

A half filled Landau levelon top of Two filled Landau levels

Step II:

From a half filled Landau level of electrons to composite fermions

at zero magnetic field - the Chern-Simons transformation

Step I:

Read and Green (2000)

2

12

2

5

• The original Hamiltonian:

• Schroedinger eq. H • Define a new wave function:

Hm

P A re

r rii i ji j

1

2

1

2

22

( )| |,

ji ji rri

ii err)arg(2

})({})({

)()()( rarAPrAP ii

The Chern-Simons transformation

})({ ir

})({ ir

describes electrons (fermions)

describes composite fermions

The effect on the Hamiltonian:

r

r

arg

|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:)(

~)( 0 rra

B

ns

e-)a(

B1/2 = 2ns0

B)c(

20

CF

)b(B

Electrons in a magneticfield

BA

Composite particles ina magnetic field

)(2 0 rnBaA

Mean field (Hartree) approximation

02 0

nBaA

B

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-conductorRead and Green (2000)

Step IV: introducing quasi-particles into the super-conductor

- shifting the filling factor away from 5/2

022 10 nBaAB

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

..)2

()2

(),('0 chr

Rr

RrRdrdrHH

)(),( rfRrR

Dealing with Abrikosov lattice of vortices in a p-wave super-conductor

First, a single vortex – focus on the mode at the vortex’ core

A quadratic Hamiltonian – may be diagonalized

(Bogolubov transformation)

E

EEEEH 0

)()()()( rrvrrudrE BCS-quasi-particle annihilation operator

Ground state degeneracy requires zero energy modes

Kopnin, Salomaa (1991), Volovik (1999)

The functions are solutions of the Bogolubov de-Gennes eqs.)(),( rvru

Ground state should be annihilated by all ‘sE

For uniform super-conductorsrkiervru )()(

kkkk vacccgsg 1..

.const

)()()()( rrvrrudrE

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

Kopnin, Salomaa (1991), Volovik (1999)

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 WilczekIvanov

All ’s anti-commute, and 2=1.

.... sgsg ji

backscattering = |tleft+tright|2

Interference experiment: Stern and Halperin (2005)Following Das Sarma et al (2005)

interference pattern is observed by varying the cell’s area

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 subspaceinterference is dephased

statescorestatescore rightleft 12

|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

cell area

even

odd

even

occupation ofanti-dot

interference

no interference

interference

Localized quasi-particles shift the red lines up/down

Be-

B1/2 = 2ns0

B)c(

20

CF

)b(B

Electrons in a magnetic field B

Composite particles in a magnetic field )(2 0 rnB

Mean field (Hartree) approximation

02 0 nBB

H = E

ji ji rri

ii err)arg(2

})({})({

A yet simpler version:

B

cell area

equi-phase lines

odd

even

even

No interference

No interference

For a lattice, expect a tight-binding Hamiltonian

..,

chtH jji

iij

Analogy to the Hofstadter problem.

j

iijdlAii

ij etett

The phases of the tij’s determine the flux in each plaquette

And now to a lattice of quasi-particles.

When vortices get close to one another, degeneracy is lifted by tunneling.

Since the tunneling matrix elements must be imaginary. ii

ji

jiijitH,

The question – the distribution of + and -

For a square lattice:

ii i

i

Corresponds to half a flux quantum per

plaquette.

A unique case in the Hofstadter problem –

no breaking of time reversal symmetry.

Spectrum – Dirac: kvkE 0)( tav 0

E

k

What happens when an electric field E(q,) is applied?

0v is varied by varyingdensity

E

k

Given a perturbation

tAJ cos

the rate of energy absorption is

f

iffAJi

22

Distinguish between two different problems –

1. Hofstadter problem – electrons on a lattice

2. Present problem – Majorana modes on a lattice

f

iffAJi

22

E

k

For both problems the rate of energy absorption

is

20

~)(

~ field electric

vdos

AiE

2Re E

The difference between the two problems is in the matrix elements

tev

ev

fJi

if0

0 for the electrons

for the Majorana modes

So the real part of the conductivity is

h

e

8

2

for the electrons

22

8

th

e for the Majorana modes

The reason – due the particle-hole symmetry of the Majorana mode,it does not carry any current at q=0.

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

022

2

0

22

18

0

0

18

,

qvqae

m

ni

qv

qae

m

ni

qCF

From composite fermions to electrons

02

202

11

e

hCFe

2~,0 qqe 0, 0 qvqe

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.

i

i

e

e localized function in the direction perpendicular to the =0 line

When vortex i encircles vortex i+1, the unitary transformation operating on the ground state is

)'()'()'()'(

)()()()('

11 *11

*1

rerwrerw

rerwrerwdrdr

ii

ii

ii

ii

ii

iiii

• How does the zero energy state at the i’s vortex “know” that it is encircled by another vortex?

• No tunneling takes place?

Unitary transformations:

A more physical picture?

• 2N localized intra-vortex states, each may be filled (“1”) or empty (“0”)

• Notation: means 1st, 3rd, 5th vortices filled, 2nd, 4th vortices empty....11001

1111100101010110

1010110000110000

1111100101010110

1010110000110000

iiii

iiii

eeee

eeee

and all possible combinations with odd numbers of filled states

1110110110110111

1000010000100001

1110110110110111

1000010000100001

iiii

iiii

eeee

eeee

Product states are not ground states:

0110*0110 2222

4321 iiii

ieeiee

The emerging picture – two essential ingredients:

•Full entanglement: Ground states are fully entangled super-positions of

all possible combinations with even numbers of filled states

• Phase accumulation depend on occupation

When a vortex traverses a closed

trajectory, the system’s wave-function

accumulates a phase that is N2

N – the number of fluid particles encircled by the trajectory

(|0000 + |1100 ) (|0000 - |1100 )

HalperinArovas, Schrieffer, Wilczek

Vortex 2 encircling vortex 3

1111100101010110

1010110000110000

1111100101010110

1010110000110000

iiii

iiii

eeee

eeee

A “+” changing into a “”

Vortex 2 and vortex 3 interchanging positions

1111100101010110

1010110000110000

1111100101010110

1010110000110000

iiii

iiii

eeee

eeee

Permutations of vortices change relative phases in the superposition

Four vortices:

A vortex going around a loop generates a unitary transformation in the ground state subspace

1

2

34

A vortex going around the same loop twice does not generate any transformation

1

2

34

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 rule.

3. In transitions between different plateaus, is non-monotonous12

p

p8

19,

2

7,

2

5

xy

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 )4

More systematically: what are the ground states? N2

The goal: ground statesN2

...,2..

...,2..

...,1..

21

21

21

RRsg

RRsg

RRsg

N

…..

that, as the vortices move, evolve without being mixed.

..., 21 RR

position of vortices

The condition: nkfornsgR

ksgi

0....

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

*

0

0

w

wis a purely zero energy state

*

0

0

w

wis a purely non-zero energy state =

EEEEE CC *

1*

0*

10

iww

iwwCY

EEE

defines a localized function correlates its occupation with that of

1w

0w

There is an operator Y for each vortex.

How does the wave function near each vortex look?

We may continue the process

*

1

1

w

w

*

1

1

w

w E

EE chC ..)2(

E

EE chC ..)2(

2

*1

*21)2()2(

iww

iwwCY

EEE

defines a localized function correlates its occupation with that of

2w

10, ww

This generates a set of orthogonal vortex states near each vortex (the process must end when states from different vortices start overlapping).

lwwww ,...,, 210

The requirements for determine the occupations of the states near each vortex.

0.. sgY j

kww ..0

The functions are solutions of the Bogolubov de-Gennes eqs.)(),( rvru

Ground state should be annihilated by all ‘sE

For uniform super-conductorsrkiervru )()(

kkkk vacccgsg 1..

.const

)()()()( rrvrrudrE

The simplest model – take a free Hamiltonian with a potential part

only )()( rrh

To get a localized mode of zero energy,

we need a localized region of . A vortex is a closed curve of =0with a phase winding of 2in the order parameter

The phase winding is turned into a boundary condition

A change of sign

Spinor is

i

i

e

e localized function in the direction perpendicular to the =0 line

The phase depends on the direction of the =0 line. It changes by around the square. A vortex is associated with a localized Majorana operator.

i

For a lattice, expect a tight-binding Hamiltonian

..,

chtH jji

iij

Questions –

1. What are the tij?

2. How do we calculate electronic response functions from the

spinors’ Hamiltonian?

Analogy to the Hofstadter problem.

j

iijdlAii

ij etett

The phases of the tij’s determine the flux in each plaquette

Two close vortices:Solve along this line toget the tunneling matrixelement

We find tij=i

A different case – the line going through the tunneling region changes the sign of the tunneling matrix element.

ii i

i

These requirements are satisfied for a given vortex by either one of two wave functions:

vacuumcccccp .....210 11...111

or:

vacuumcccccm .....210 11...111

The occupation of all vortex states is particle-hole symmetric.

Still, two states per vortex, altogether and not N2N22

We took care of the operators

jj

jjj

iww

iwwY

*1

*

1)(

For the last state, we should take care of the operatorwhich creates and annihilates quasi-particles

k

k

w

w

*

Doing that, we get ground states that entangle states of different vortices (example for two vortices):

12

21

2..

1..

envpmmpenvmmppsg

envpmmpenvmmppsg

For 2N vortices, ground states are super-positions of states of the form

i

Ni

ii

ii vacuumccc )2()2()1( 1...11

1st vortex2nd vortex 2N’th vortex

The operator creates a particle at the state near the vortex. When the vortex is encircled by another

ic )(rwi

ii cc

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

One fermion mode, two possible states, two pi-shifted interference patterns

Several comments on the Das Sarma, Freedman and Nayak proposed experiment

n=0n=1

Comment number 1: The measurement of the interference patterninitializes the state of the fermion mode

initial core state ii 2121

after measurement current has been flown through the system

21 2121 currenticurrenti

A measurement of the interference pattern implies

021 currentcurrent

The system is now either at the =i or at the =-i state.

NA

At low temperature, as we saw

odd nqp (“”) 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 “” 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.

1 2

For both cases the interference pattern is shifted by by the transition of one quasi-particle through the gates.

Summary

The geometric phase accumulated by a moving vortex

Entanglement between the occupation of states near

different vortices

Non-abelian statistics

2/5sgn2log5

2/5sgn2log2

0

T

m

T

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 quasi-

particles – essential for the observation of the QHE – and to

the suppression of their entropy.

(~1.4)

(~12pA/mK)

The positional entropy of the quasi-particles depends on their

spectrum:

qhe gap

ground state

excitations

non-abelianstemperature

localized states

Phonons of a quasi-particles Wigner crystal

Shot noise as a way to measure charge:

p

1-p

I2Binomial distribution

For p<<1, current noise is

S=2eI2

coin tossingS

D1

D2