Charge and Statistics of Dilute Laughlin Quasiparticles

C. L. Kane University of Pennsylvania

I. Introduction

II. Transmission of Dilute Quasiparticlesthrough a point contact

• Laughlin Quasiparticle and Shot Noise

• Corbino Geometry (Moty Heiblum et al. ’03)• Telegraph Noise and fractional statistics• Luttinger Liquid Theory

• Three Terminal Geometry for preparing“dilute quasiparticle beam”

(Moty Heiblum et al. ’02)• Shot Noise Puzzle• Luttinger Liquid Theory

PRL 90, 226802 (03)

III. Proposed measurement of fractional statistics: Telegraph Noise

PRB 67, 45307 (03) w/ Matthew Fisher (ITP)

Laughlin Quasiparticle for ν = 1/m

• Energy Gap• Quantized Hall Conductance

Fractionally Charged Excitations

xy xydQ dE ddt dt

σ σ Φ= ⋅ =∫

( / ) / *xyQ h e e m eσ= = ≡

ν = 1/m

σxx = 0σxy = (1/m)e2/h

Fractional Statistics

• Phase Θ = π/m under interchange:

• Phase when one circles another:

e*

e*

Halperin 1984;Arovas, Schrieffer, Wilczek 1984

ie ΘΨ → Ψ

2ie ΘΨ → Ψ

• Signature of topological orderThouless 1989;Wen, Niu 1990

m-fold topologicaldegeneracy on torus

Measurement of Fractional Charge1. Capacitive Measurement

Resonant Tunneling Through Anti-dotGoldman & Su (Science, 1995)

∆VBG = e*/C

2. Shot Noise MeasurementBackscattering at Quantum Point Contact

Kane, Fisher (PRL 1994);De Picciotto et al. (Nature 1997); Glattli et al. (PRL 1997)

ItV

0<I2>ω~0 = e* Ib<I2>ω~0 = e It

Ibe

-e-e*

e* 0

V

Strong Pinch Off Weak Pinch Off

Electron Tunneling Quasiparticle Tunneling

Three Terminal ConfigurationComforti et al. (Nature 2002)

• When QPC2 is open, a “Dilute Beam” of quasiparticles is isolated in lead 3.

• Transport of Dilute beam through QPC2 probestransmission of individual quasiparticles.

Transmission of Dilute QuasiparticlesThrough a Point Contact

Dilute : q ~ .45 e

Not Dilute : q ~ e

Can Dilute Charge e/3 Quasiparticles Traverse a Nearly Opaque Barrier ?

q

t1

Comforti et al. (Nature 2002)

e over three ?

How can that be ?

E THREEe over three.

Luttinger Liquid Model : ν = 1/m

[ ]3

2

1

1 1 2 2 2 3

( )4

v cos( ) v cos( )

Fi x i i

i

mvH dx x

Vt

φπ

φ φ φ φ=

= ∂ +

− − + −

∑∫

Analysis:

QPC1:• Perturbative for small v1 (Dilute Limit)

QPC2:• Perturbative for small v2

• Perturbative for large v2 (small t2)• Exact Solution for m=2 via fermionization

Perturbative Analysis1. Small t2 (T<<V)

2. Small v2 (T<<V)

2 2 2 2/ 31 2( ) v t m mI V c V + −=

2 2 2 2/ 31 2( ) v t m mS V ec V + −=

2 21 2

1 21 2/ 2 2 /

v v( , ) 1m mI V T c cV T− −

⎡ ⎤= −⎢ ⎥

⎣ ⎦2 21 2

1 31 2/ 2 2 /

v v( , ) * 1m mS V T e c cV T− −

⎡ ⎤= −⎢ ⎥

⎣ ⎦

22

2-2/m

v*T

Q e c= +

Q e=

Perturbation theory diverges for T=0 even for fixed finite V.

Exact Solution

1. Zero Temperature:

For m=2 map problem to free fermions

2

in 42

2( ) ( ) 1 ( )16vVI V I V K

π⎛ ⎞

= − −⎜ ⎟⎝ ⎠

( ) ( )S V eI V=

Transmission through QPC2

V/v22

Q = e , independent of v2, even when transmission through QPC2 is nearly perfect.

2. Finite Temperature:

Limits T 0 and v2 0 do not commute:

Q(T 0,v2=0) = e/m

Q(T=0,v2 0) = e

A nasty integral

= 1 (independent of X) !

with

Interpretation: Andreev ReflectionThree possible scattering products

for an incident quasiparticle

1. Transmission

Probability T

2. Reflection

Probability R

3. Andreev Reflection

Probability A

Theory predicts T=0 at zero temperature• Strong Pinch-off: R~1, 0<A<<1• Weak Pinch-off: R=1-1/m, A=1/m

Observe Andreev processes by measuring correlations between transmitted and reflected currents.

What about the experiments?

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

23mK 20% Dilution23mK 10% Dilution40mK 20% Dilution74mK 20% Dilution

Effe

ctiv

eC

harg

e, q

2/e

Transmission of QPC2, t2

Theoryfor T 0

The measured charge does approach e for small t2 and low temperature, but “high” temperaturedata remains unexplained.

• Role of irrelevant operators, eg. (dφ/dx)3

• Role of smooth edges near point contact.

Fractional Statistics

“h/e”

• Phase Θ = π/m under interchange:

• Phase when one circles another:

e*

e*

e*

• Model: Bosons + Statistical Flux

Halperin 1984;Arovas, Schrieffer, Wilczek 1984

*2 ( / )

2 /

e h e

mπ

Θ =

=

ie ΘΨ → Ψ

2ie ΘΨ → Ψ

Measurement of Fractional StatisticsQuantum Interference Necessary To

Measure Statistical Phase

Two Point Contact InterferometerChamon et al. (1997)

Φ

I = I0 + ∆I cos[e* Φ /h + 2Θ Nqpenc]

Aharonov Bohm Phase Statistical Phase

I

• Net phase changes by 2Θ = 2π/m when qp tunnels from outside ring to inside of ring.

• Equilibrium h/e* oscillations eliminated by qp tunneling.

Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu and H. Shtrikman, Nature (03)

A “Mach-Zehnder Interferometer”

An Ohmic contact in the middle of a ring

D1

S

BS1 M1

M2 BS2

D2

airbridge

QPC1

MG1

SD1

QPC2

mesa

MG2

D2

0 50 100 150 2000

5

10

15

-9.0 -7.5 -6.0 -4.5 -3.0

Time (minute) ~ Magnetic Field

Modulation Gate Voltage, VMG (mV)

Curr

ent

(a.u

.)

ν = 1

Telegraph Noise in ring with inner contact:A Direct Signature of Fractional Statistics

• Net phase changes by 2Θ = 2π/m when qp passes from lead 1 or 2 to lead 3.

• m-state telegraph noise for n = 1/m

• Related to m-fold “topological degeneracy”

• <I2>ω~0 = e* ∆I2/I3

Chiral Luttinger Liquid Model

( * ) ( * )*v vi e V t i e V tU e eα α α αφ φα α α α ακ κ∆ − − ∆ −+ −= +

Tunneling at QPCα :

Klein Factors

( )( )

232

2 2~03 3

coth * / 2*2 1 sin / sinh * / 2

e V Te III e V Tω

∆=+ Θ

Low Frequency Noise :

[ ]3 3

2

1 1( )

4F

i x i ii

mvH dx x Uαα

φπ = =

= ∂ +∑ ∑∫

2

3

*~2

e II∆

2 2

23

*~4 sin

e IG T

∆Θ

e*V3 >> kT

e*V3 << kT

Conclusion• Fractionally charged quasiparticles can not traverse

a nearly opaque barrier. At T=0 they can’t even getpast a nearly perfectly transmitting barrier!

• Telegraph Noise is predicted for a ring with an inner contact. It is a direct consequence of a fractional statistical phase, and has a unique signature in the low frequency noise.

Questions• Origin of observed suppression of transmitted

charge for moderately weak tunneling and moderately low temperature?

Smooth edges?Irrelevant operators?

• Observation of Andreev processes?

• Exact Solution for n=1/3?

• Telegraph Noise for Hierarchical Quantum Hall States?

• Effect of dephasing on edge state transport?