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### 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

- Non-abelian anyons in quantum Hall states – what they are, why they are interesting, how they may be useful for topological quantum computation.
- How do you identify a non-abelian quantum Hall state when you see one ?

The quantum Hall effect

- zero longitudinal resistivity - no dissipation, bulk energy gap current flows mostly along the edges of the sample
- quantized Hall resistivity

B

I

n is an integer,

or q even

Extending the notion of quantum statistics

Laughlin quasi-particles

Electrons

A ground state:

Energy gap

Adiabatically interchange the position of two excitations

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

-degenerate.

Interchange of quasi-particles shifts between ground states.

position of

quasi-particles

…..

Permutations between quasi-particles positions

unitary transformations in the ground state subspace

2

3

2

1

3

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

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

- experimentally identifying non-abelian quantum Hall states
- The way: the defining characteristics of the most prominent candidate, the n=5/2 Moore-Read state, are
- Energy gap.
- Ground state degeneracy exponential in the number N of quasi-particles, 2 N/2.
- Edge structure – a charged mode and a Majorana fermion mode
- Unitary transformation applied within the ground state subspace when quasi-particles are braided.

- Proposed experiments to probe ground state degeneracy – thermodynamics
- Proposed experiments to probe edge and bulk braiding by electronic transport–
- Interferometry, linear and non-linear Coulomb blockade, Noise

Measuring the entropy of quasi-particles in the bulk

The density of quasi-particles is

Zero temperature entropy is then

To isolate the electronic contribution from other contributions:

All g’s anti-commute, and g2=1.

A localized Majorana operator .

- 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

When a vortex i encircles a vortex j, the ground state is

multiplied by the operator gigj

Nayak and Wilczek

Ivanov

term

Number of

q.p.’s in the interference loop,

even

odd

even

Brattelli diagram

Interferometers:

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

Even number of localized vortices:

vortex a around vortex 1 and vortex 2 - g1gag2ga~ g2g1

2

1

a

The interference term is multiplied by a phase:

Two possible values, mutually shifted by p

Gate Voltage, VMG (mV)

Magnetic Field

(or voltage on anti-dot)

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

cell

area

Coulomb blockade vs interference

(Stern, Halperin 2006,

Stern, Rosenow, Ilan, Halperin, 2009

Bonderson, Shtengel, Nayak 2009)

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

The simplest case, n=1.

Energy is determined by the number of electrons

Partition function

Poisson summation

- Thermal suppression of high winding number.
- An Aharonov-Bohm phase proportional to the winding number.
- At high T, only zero and one windings remain

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

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 by the number and state of the bulk quasi-particles.

The neutral mode partition function χdepends on nqp and their state.

Poisson summation is modular invariance

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

A different thermal suppression factor for each component.

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

High T

Probing excited states at the edge –

non linear transport in the Coulomb blockade regime

(Ilan, Rosenow, Stern, 2010)

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.

Odd number

Current-voltage characteristics (Ilan, Rosenow, AS 2010)

Source-drain voltage

Magnetic field

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

Mach-Zehnder interferometer

S

D2

D2

D1

S1

Compare:

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)

G2

G3/2

G1/2

G1

G4/2

G2/2

G3

Interference

term

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.

Interference magnitude depends on the parity of the number of quasi-particles

Phase depends on the eigenvalue of

Summary:

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

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

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

Mach-Zehnder:

odd

even

Das-Sarma-Freedman-Nayak qubit

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

D2

D1

S1

Interference

term

Number of

q.p.’s in the interference loop,

The interference phases are mutually shifted by p.

odd

even

D2

D1

S1

Interference

term

Number of

q.p.’s in the interference loop,

The sum of two interference phases, mutually shifted by p.

The area period goes down by a factor of two.

Magnetic Field

(or voltage on anti-dot)

Ideally,

The magnetic field Quasi-particles number

The gate voltage Area

cell

area

(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

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

20

(b)

CF

B

B1/2 = 2ns0

B

Electrons in a magnetic fieldB

e-

H y = E y

Composite particles in a

magnetic field

Mean field (Hartree) approximation

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.

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.

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(r) is a localized function in the vortex core

All g’s anti-commute, and g2=1.

A localized Majorana operator .

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 (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 n=5/2

cell area

Closing the island into a quantum dot:

n=5/2

Interference involving multiple scatterings, Coulomb blockade

is very different from

But,

so, interference of even number of windings always survives.

Equal spacing between peaks for odd number of localized vortices

Alternate spacing between peaks for even number of localized vortices

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

What is ?

By the fluctuation-dissipation theorem,

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:

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 + nis/4)] (nis even)

dG

time

compared to shot noise

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

and

A signature of the n=5/2 state

(For abelian Laughlin states – the power is )

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

Analogy to the Hofstadter problem.

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

Since the tunneling matrix elements must be imaginary.

The question – the distribution of + and -

For a square lattice:

Corresponds to half a flux quantum per

plaquette.

A unique case in the Hofstadter problem –

no breaking of time reversal symmetry.

is varied by varying

density

E

k

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

Exponential dependence on density

- 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)

- A proposed interference experiment to address the non-abelian
- nature of the quasi-particles, insensitive to localized quasi-particles.
- 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 !

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 D

(add an odd electron)

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

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

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

even

even

odd

odd

B

No interference

even

even

cell area

No interference

cell area

So what about peak spacings?

When nis is odd, peak spacing is “unaware” of D

peaks are equally spaced

When nis is even, peak spacing is “aware” of D

periodicity is doubled

Interference pattern

Coulomb peaks

From electrons at n=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

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:

- Schroedinger eq. H y = E y
- Define a new wave function:

describes electrons (fermions)

describes composite fermions

The effect on the Hamiltonian:

|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.

B

20

(b)

CF

B

B1/2 = 2ns0

The new magnetic field:

(a)

Electrons in a magnetic

field

e-

B

ns

Composite particles in

a magnetic field

Mean field (Hartree) approximation

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

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

Spin polarized composite fermions at zero (average)

magnetic field

A quadratic Hamiltonian – may be diagonalized

(Bogolubov transformation)

Ground state degeneracy requires zero energy modes

BCS-quasi-particle annihilation operator

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)

The functions are solutions of the Bogolubov de-Gennes eqs.

Ground state should be annihilated by all ‘s

For uniform super-conductors

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(r) is a localized function in the vortex core

All g’s anti-commute, and g2=1.

A localized Majorana operator .

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

Ivanov

Stern and Halperin (2005)

Following Das Sarma et al (2005)

n=5/2

backscattering = |tleft+tright|2

interference pattern is observed by varying the cell’s area

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

vortex a around vortex 1 - g1ga

vortex a around vortex 1 and vortex 2 - g1gag2ga~ g2g1

1

a

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

|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

odd

even

occupation of

anti-dot

interference

no interference

interference

cell area

Localized quasi-particles shift the red lines up/down

B

20

(b)

CF

B

B1/2 = 2ns0

B

Electrons in a magnetic fieldB

e-

H y = E y

Composite particles in a

magnetic field

Mean field (Hartree) approximation

For a lattice, expect a tight-binding Hamiltonian

Analogy to the Hofstadter problem.

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.

The question – the distribution of + and -

For a square lattice:

Corresponds to half a flux quantum per

plaquette.

A unique case in the Hofstadter problem –

no breaking of time reversal symmetry.

is varied by varying

density

E

k

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

k

Given a perturbation

the rate of energy absorption is

- Distinguish between two different problems –
- Hofstadter problem – electrons on a lattice
- Present problem – Majorana modes on a lattice

k

For both problems the rate of energy absorption

is

The difference between the two problems is in the matrix elements

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

for the electrons

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

From composite fermions to electrons

- 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.

When vortex iencircles vortex i+1, the unitary transformation

operating on the ground state is

- 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: 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

and all possible combinations with odd numbers of filled states

Product states are not ground states:

(|0000 - |1100)

- Phase accumulation depend on occupation

When a vortex traverses a closed trajectory, the system’s wave-function accumulates a phase that is

Halperin

Arovas, Schrieffer, Wilczek

N – the number of fluid particles encircled by the trajectory

Permutations of vortices change relative phases in the superposition

Four vortices:

Vortex 2 encircling vortex 3

Vortex 2 and vortex 3 interchanging positions

A “+” changing into a “-”

4

3

1

A vortex going around a loop generates a unitary transformation in the

ground state subspace

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

transformation

2

4

3

1

The Landau filling range of 2<n<4

- Unconventional fractional quantum Hall states:
- Even denominator states are observed
- Observed series does not follow the rule.
- In transitions between different plateaus, is non-monotonous

as opposed to

(Pan et al., PRL, 2004)

Focus on n=5/2

The effect of the zero energy states on interference

Dephasing, even at zero temperature

No dephasing (phase changes of )

More systematically: what are the ground states?

The goal: ground states

position of vortices

…..

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

The condition:

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

is a purely zero energy state

is a purely non-zero energy state =

defines a localized function

correlates its occupation with that of

There is an operator Y for each vortex.

defines a localized function

correlates its occupation with that of

This generates a set of orthogonal vortex states

near each vortex (the process must end when states from

different vortices start overlapping).

The requirements for j=1..k determine the occupations

of the states near each vortex.

The functions are solutions of the Bogolubov de-Gennes eqs.

Ground state should be annihilated by all ‘s

For uniform super-conductors

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

only

To get a localized mode of zero energy,

we need a localized region of m=0. A vortex is a closed curve of m=0

with a phase winding of 2p in the order parameter D.

The phase winding is turned into a boundary condition

A change of sign

localized function in the

direction perpendicular to the

m=0 line

Spinor is

- The phase a depends on the direction of the m=0 line. It changes by
- around the square.

A vortex is associated with a localized Majorana operator.

For a lattice, expect a tight-binding Hamiltonian

Analogy to the Hofstadter problem.

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

- Questions –
- What are the tij?
- How do we calculate electronic response functions from the
- spinors’ Hamiltonian?

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.

These requirements are satisfied for a given vortex by

either one of two wave functions:

or:

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

Still, two states per vortex, altogether and not

We took care of the operators

For the last state, we should take care of the operator

which creates and annihilates E0 quasi-particles.

Doing that, we get ground states that entangle states of different

vortices (example for two vortices):

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

1st vortex

2nd vortex

2N’th vortex

The operator creates a particle at the state near the

vortex. When the vortex is encircled by another

- Experimental tests of non-abelian states
- The expected QH series in the second Landau level
- The nature of the transition between QH states in the second
- Landau level
- 4. Linear response functions in the second Landau level
- Physical picture of the clustered parafermionic states
- 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

after measurement current has been flown through the system

A measurement of the interference pattern implies

The system is now either at the g1g2=i or at the g1g2=-i state.

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.

For both cases the interference pattern is shifted by p

by the transition of one quasi-particle through the gates.

Entanglement between

the occupation of states near

different vortices

The geometric phase

accumulated by

a moving vortex

Non-abelian statistics

(~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 quasi-particles – 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

localized states

qhe gap

Phonons of a quasi-particles Wigner crystal

temperature

non-abelians

ground state

Shot noise as a way to measure charge:

D1

1-p

S

coin tossing

p

I2

D2

Binomial distribution

For p<<1, current noise is

S=2eI2

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