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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)
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
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?
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