Fractional quantum Hall spectroscopy investigated by a resonant - - PowerPoint PPT Presentation
Fractional quantum Hall spectroscopy investigated by a resonant detector Alessandro Braggio CNR-SPIN, Genoa https://sites.google.com/site/alessandrobraggio/ M. Sassetti M.Carrega D.Ferraro Genoa Genoa Geneve-Marseille New. J. Phys. 16
2012
Alessandro Braggio CNR-SPIN, Genoa
Genoa M.Carrega Genoa D.Ferraro Geneve-Marseille
https://sites.google.com/site/alessandrobraggio/
Laughlin PRL’83
Wen PRB90,Halperin PRB 82, Buttiker PRB 88, Beenakker PRL 90
Jain, PRL’89Jain, PRL’89
σxy = ν e2 h σxx = 0 ν = 1 2np + 1 = 1, 1 3, 1 5, .. ν = p 2np + 1 = 2 5, 2 3, ...
Jain PRL’89, Wen & Zee PRB’92, Kane & Fisher PRB’95
Kane & Fisher PRB’95
Hierarchical models ν = N NΦ
m = 1 m > 1
Single-qp m-agglomerate e∗ = 1 2np + 1 me∗
Laughlin PRL 83, Arovas, Schrieffer & Wilczek PRL 84
Ψ(m)(x)Ψ(m)(y) = Ψ(m)(y)Ψ(m)(x)e−iθmsgn(x−y) G(m)(τ) = hΨ(m)(τ)Ψ(m)†(0)i G(m)(τ) / |τ|−∆m Abelian + = ∆m Scaling dimension Ψl(x) ∝ elT ·K·φ
scaling dimension I = ν e2 h V IB IB ⌧ I
S(ω = 0) = Z +∞
−∞
h{δIB(t), δIB(0)}+i δIB = IB hIBi m-qps G(m)
B
∝ T 2∆m−2 I(m)
B
∝ V 2∆m−1 ∆m S(m) = I(m)
B
coth ✓me∗V 2kBT ◆ S(m) ≈ me∗I(m)
B
S(m) ≈ 2kBTGB
k
B
T
e
∗
V kBT ⌧ me∗V
Theory:Kane & Fisher PRL 94, Fendley, Ludwig & Saleur PRL 95 Exp:De-Picciotto… Nature 97,Saminadayar.… PRL’97,Reznikov… Nature’99
Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations"
B u n c h i n g B u n c h i n g S i n g l e q p S i n g l e q p
eeff/e T(mK)
Chung…PRL03, Bid PRL03,Dolev….
ν = 5/2
ν = 2 5
ωn ⌧ ωc ∆m =gc∆c
m + gn∆n m
Coupling other degrees: Rosenow & Halperin PRL 02, Papa & MacDonald PRL 05 1/f noise + dissipation: A. B., D. Ferraro, M. Carrega, N. Magnoli, M. Sassetti NJP12
Bid et al., Nature 10, Gross et al. PRL12, Gurman et al. Nature 12, Shtanko et al PRB14, Takei et al. PRB11, Dolev et al. PRL11 Kou et al. PRL12,D. T. McClure et al. PRL12, Safi & Sukhorukov EPL10 Altimiras et al PRL12, Aita et al PRB13, Inoue et al. Nature14
Meier et al. 1406.4517
Blanter&Buettiker Phys.Rep.00, Rogovin&Scalapino Ann. Phys 74
Chamon..PRB95; Chamon..PRB96; Dolcini..PRB05; Bena..PRB06; Bena..PRB07; Sukhorukov..PRB01; Sukhorukov..EPL10; Schoelkopf…03; Deblock…Science ’03; Engel…’04;Hekking….06;……
Lesovik..JETP97;Gavish U..PRB00;Gavish U.. arXiv:0211646; Bednorz& Belzig PRL13; Aguado..PRL00;
Aguado PRL00, Blanter 05, Martin&Crepieux 04-05-06,…..
ωm = me∗V/~ [I(t), I(t0)] 6= 0
S(m)
+/−(ω) =
Z +∞
−∞
e±iωthδI(m)
B
(t)δI(m)
B
(0)i S(m)(ω) = Z +∞
−∞
eiωth{δIB(t), δIB(0)}+i = X
i=±
S(m)
i
(ω)
Altimiras et al. APL13, PRL14
ω = p 1/LC
Cold detector Hot detector Resonant Tc ⌧ T Tc T S(m)
meas(ω) = K
n S(m)
+
(ω) + nB(ω) h S(m)
+
(ω) − S(m)
−
(ω) io δhx2i K = ⇣ α 2L ⌘2 1 2η ⌧ 1
S(m)
−
(ω) S(m)
+
(ω)Emission
Absorption
Lesovik G B and Loosen R JETP 65 295 (1997); Gavish U,….arXiv:0211646
nB(ω) = 1 eω/TC − 1
ω <e h G(m)
ac (ω)
i
25kΩ 50Ω ω ≈ 5GHz T ≈ 15mK
kBTc ⌧ ω
S(m)
meas(ω) ≈ KS(m) +
(ω) + O(e−~ω/kBTc)
S(m)
meas(ω) ⇡ K
n S(m)
+
(ω) kBTc<e h G(m)
ac (ω)
io kBTc ω
ω0 = e∗V/~
|tm|2
Ssym(ω) = X
m
S(m)
sym(ω)
Smeas(ω) = X
m
S(m)
meas(ω)
Γ(m)(E) Smeas ≡ Sex T = Tc
Ssym(ω, ω0)
a)
c)
Smeas(ω, ω0)/K
Electron
Smeas(ω, ω0) ⇡ KS+(ω, ω0) = K 2 ✓ Ssym(ω, ω0) 2 ˜ S0 ω ωc ◆ kBTc ⌧ ω
Ssym(ω, ω0) = 2 ˜ S0 ωc [θ(ω0 − ω)ω0 + θ(ω − ω0)ω] ˜ S0 = e2 2 |t1|2 2πα2 1 ωc
Tc = 15mK ω = 7.9GHz(60mK) ωc = 660GHz(5K) T = 0.1, 5, 15, 30[mK]
Γ(1)(E) ∝ θ(E)E
Lesovik G B and Loosen R JETP 65 295 (1997)
ν = 1 ∝ V ∝ V
b)
Ssym(ω, ω0)
d)
Smeas(ω, ω0)/K
Single-qp Tc = 15mK ω = 7.9GHz(60mK) ωc = 660GHz(5K) T = 0.1, 5, 15, 30[mK]
Ssym(ω, ω0) ≈ |ω − ω0|4∆(1)
1/3−1
Chamon, Freed & Wen PRB95,PRB96
kBTc ⌧ ω kBT ⌧ ω0
ν = 1/3 e∗ = e 3
S(m)
meas(ω, ω0) ≈ S(m) +
(ω) ≈ K (me∗)2 2 Γ(m) (−ω + mω0) ω ∼ ω0
returns directly the rates…….
S(m)
meas(ω, ω0)
∝ V ∝ V
a)
Smeas(ω, ω0)/K
b)
Sex(ω, ω0)/K
ν = 1/3, 1/5, 1/7 Tc = T T = 10mK Tc = 10, 30, 60, 90 mK Dashed lines theoretical rates It is possible to extract the scaling dimensions without requiring an extended window in frequency and bias simplifying the experimental requirements
ν
= ν 2 Note that Smeas ≡ Sex T = Tc
b)
Smeas(ω, ω0)/K
d)
∂Smeas(ω, ω0) K∂ω0
ν = 1/3 Tc = 5, 15, 30, 60 mK The QPC cannot excite detector modes only absorptive The QPC excites detector The combined effect is an enhancement of jump/peak Tc = 15mK ω = 7.9GHz(60mK) ωc = 660GHz(5K)
∝ V
Safi & Sukhorukov EPL10
c) d)
Smeas(ω, ω0)/K Smeas(ω, ω0)/K
ν = 2 5 ν = 2 3 T = 0.1, 5, 15, 30[mK] Note that Smeas ≡ Sex T = Tc Smeas(ω, ω0) ≈ α1Γ(1)(ω0 − ω) + α2Γ(2)(2ω0 − ω)
e∗ = e 5 e∗ = e 3
sensibility
(Kubo)
σxy = −ie2~ X
Eα<EF <Eβ
(vy)βα(vx)αβ − (vx)αβ(vx)βα (Eα − Eβ)2
Thouless, Kohmoto, Nightingale,den Nijs PRL’82; Kohmoto Ann. Phys. 160, 343 (1985)
σ(α)
xy = e2
2πi Z d2k Z d2r ✓∂uα∗
k1,k2
∂k2 ∂uα
k1,k2
∂k1 − ∂uα∗
k1,k2
∂k1 ∂uα
k1,k2
∂k2 ◆ σ(α)
xy = ne2
h Topological invariant
around
L = 1 4⇡ (Kij@xi@tj + Vij@xi@xj + 2✏µνtj@µjAν) Ψl(x) ∝ elT ·K·φ ν = tT · K−1 · t ql = 1 2π lT · K−1 · t = me∗ θl = 2π lT · K−1 · l qp 2-qps 3-qps 2π e− e−
Wen & Zee PRB 92, J. Fröhlich et al JSTAT 97 Wen, Kane & Fisher ,….
φi
j0
qp = elδ(r − r0)
L = LCS + aµjµ
qp
J0 e = ν B φ0 + lνδ(r − r0) e∗
l = eνl
Single-qp 2-agglomerate l = 1 l = 2 ν = 1 3 electron ≡ l = 3 +
Laughlin PRL 83, Arovas, Schrieffer & Wilczek PRL 84
Ψ(m)(x)Ψ(m)(y) = Ψ(m)(y)Ψ(m)(x)e−iθmsgn(x−y)
Wen Adv. Phys. 95, Read&Rezayi PRB99
6= Fusion Rules eiθm → ei ˆ
Θmn
e− ν = 1 2np + 1 = 1, 1 3, 1 5, ...
Jµ = xy✏µνρ@νAρ J0 e = ν e 2π B = ν B φ0 ν = N NΦ σxy = ν e2 2π Ji = xy✏ijEj Jµ = e 2⇡ ✏µνρ@νaρ ∂µJµ = 0
Froehlich & Zee, NPB 91; Wen, Adv.Phys. 95,....
ν = 1 k Current cons. LCS = − k 4⇡ ✏µνρaµ@νaρ + e 2⇡ ✏µνρAµ@νaρ
LCS = −KIJ 4⇡ ✏µνρaI
µ@νaJ ρ + e
2⇡ tI✏µνρAµ@νaI
ρ
loop around ν = tT · K−1 · t lqp
I = (l1, .., ln)
ql = 1 2π lT · K−1 · t = me∗ θl = 2π lT · K−1 · l 2π e−
Wen & Zee PRB 92, J. Fröhlich et al JSTAT 97
e− ν = p 2np + 1 = 2 5, 2 3, ...
chiral currents Kač-Moody algebra WZW model
[a(z), a(z0)] = 2π k δ(z − z0) z = (x + t)/ √ 2 Stot = Sbulk + Sbd δStot = 0 aµ = ∂µφ Sedge = k 2π Z dtdx∂xφ(∂t − ∂x)φ [φ(x), φ(x0)] = iπνsgn(x − x0) χ
Wen Adv Phys. 95, Zee 95 Wen ; Fisher&Kane
Ψl(x) ∝ elT ·K·φ Bosonization
Qeff e ν = 2 5
b
20 40 60 1 2 3 4
80
ν =2/5
82mK q=e/5 9mK q=2e/5 Shot Noise, S (10
2/Hz)
Back Scattered Current, IB (pA)
Qeff(T) = " 3T G(tot)
B
✓d2Sexc dV 2 2 3T d3IB dV 3 ◆# 1
2
V →0
⇡ hIBi3 (e2IB) 1
2
V →0
Chung et al PRL 03
Sexc = Qeff(T) coth Qeff(T)V 2T
Sexc (10−30A2/Hz) V (µV)
eeff/e T(mK)
ν = 5/2
Non-Abelian theory + 6=
Data from M. Dolev and M.Heiblum
Valid also for Abelian Non-Abelian Ψ(2)
I
Ψ(1)
σ
Ψ(1)
σ
Ssym(ω, ω0) Ssym(ω, ω0)
a) b)
ν = 2 5 ν = 2 3
S(m)
sym(ω, ω0) ≈ |ω − ω0|4∆(m)
ν
−1
Chamon, Freed & Wen PRB95,PRB96
ω ≈ mω0
∆(m)
ν
> 1/4 ∆(m)
ν
< 1/4
T = 0.1, 5, 15, 30[mK] e∗ = e 5 e∗ = e 3
Sexc (10−30A2/Hz) V (µV)
∆m ˜ ∆m Good agreement with many observations, simple & coherent explanations Theorist Exp ?
S(m)
+
(ω) = e−~ω/kBT S(m)
+
(−ω)
S(m)
+
(ω) S(m)
−
(ω) = ω <e h G(m)
ac (ω)
i
G(m)
ac (ω) =
Z dteiωt 1 ω h[δI(m)
B
(t), δI(m)
B
(0)]i
kBTc ⌧ ω S(m)
meas(ω) ≈ KS(m) +
(ω) + O(e−~ω/kBTc)
S(m)
meas(ω) ⇡ K
n S(m)
+
(ω) kBTc<e h G(m)
ac (ω)
io
kBTc ω Sex(ω, ω0) = Smeas(ω, ω0) − Smeas(ω, ω0 = 0) ω0 = e∗V/~
|tm|2
Ssym(ω) = X
m
S(m)
sym(ω)
Smeas(ω) = X
m
S(m)
meas(ω)
Γ(m)(E) = |tm|2 Z +∞
−∞
dτ eiEtG<
m,−(−t)G> m,+(t)
G>
m,±(t) = hΨ(m) ν,±(t)Ψ(m)† ν,± (0)i
S(m)
+
(ω, ω0) = (me∗)2 2 h Γ(m) (−ω + mω0) + Γ(m) (−ω − mω0) i
S(m)
sym(ω, ω0) = (me∗)2
2 X
j,k=±
Γ(m) (jω + kmω0)
<e h G(m)
ac (ω)
i = (me∗)2 2ω X
j,k=±
jΓ(m) (jω + kmω0)