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Nonequilibrium current and relaxation dynamics
of a charge-fluctuating quantum dot
Volker Meden
Institut fur Theorie der Statistischen Physik
with Christoph Karrasch, Sabine Andergassen, Dirk Schuricht,
Mikhail Pletyukhov, Laszlo Borda, Herbert Schoeller
Finite-bias current through quantum dots—limiting cases
spin fluctuations: nonequilibrium physics of the Kondo model
charge fluctuations: the interacting resonant level model (lattice version)
1 2 3t t t t t t
εL L R,t’R
µL µR
u u,t’
.H = εn2 + uL
(
n1 −12
)(
n2 −12
)
+ uR
(
n2 −12
)(
n3 −12
)
− t′L(
d†1d2 + H.c.)
.−t′R(
d†2d3 + H.c.)
− t∑
α=L,R
∞∑
j=1
(
c†j,αcj+1,α + H.c.)
− t(
d†1c1,L + d†3c1,R + H.c.)
model parameters:
• level position: ε (ε = 0: p-h symmetry)
• local Coulomb interaction: uα (= Uα/ρα)
• local tunneling: t′α2 (= Γ∞α /(2πρα))
• band width: B = 4t
• bias voltage: µL/R = ±V/2
In equilibrium with L-R symmetry: scaling limit
consider field theoretical limit with |ε|, |u|, t′� B: ρ(ω)→ ρ(0)• relation to x-ray edge problem (Noziere & de Dominicis ’69)
• single lead: mapping to Kondo model (Wiegmann & Finkel’shtein ’78)
• bosonization and RG methods (Schlottmann ’80-82, . . . )
• numerical RG (Borda et al. ’07; Kashcheyevs et al. ’09)
here: functional RG and real-time RG in frequency space, small U(Karrasch, Enss & VM ’06; Schoeller ‘09)
example: susceptibility χ = − d〈n2〉dε
∣
∣
∣
ε=0→ power law
0 0.1 0.2U
0
0.1
0.2
0.3
α χ
NRGFRGRTRG-FS2U
χ−1 ∼ (Γ∞)1−αχ
αχ = 2U +O(U 2)
RG: Γ∞ acts as cutoff
define scale: TK = 2πχ
In equilibrium with L-R symmetry: lattice model
use numerical DMRG and Kubo formula (Bohr & Schmitteckert ’07)
here: functional RG for small u (Karrasch, Pletyukhov, Borda & VM ’10)
0 0.04 0.08ε / t
0
0.5
1
G /
G0
u/t=0u/t=0.3u/t=1u/t=5
t’ / t = 0.1
symbols: DMRG
lines: FRG
Finite bias: steady state at particle-hole and L-R symmetry
scaling limit and V � TK: I ∼ V −αV , αV = 2U +O(U2)(Doyon ’07; Boulat, Saleur & Schmitteckert ’08)
does V only act as (trivial) IR cutoff? (Borda & Zawadowski ’10)
lattice model, td-DMRG: (Boulat, Saleur & Schmitteckert ’08)
hopping parameter: t′/t = 0.5
⇒ not in scaling limit
⇒ no power law!
questions:
• compare nonequ. FRG to td-DMRG
• confirm power law in scaling limit• go away from p-h and L-R symmetry
Steady-state FRG on the Keldysh contour: ε = 0, L-R symmetric(Gezzi, Pruschke & VM ’07; Jakobs, VM & Schoeller ’07; Jakobs, Pletyukhov & Schoeller ’10)
cutoff: auxiliary leads to sites 1,2,3; coupling Λ from ∞→ 0
flow equation (first order in u):
∂Λt′Λ = i
u
4π
∫
dω{
GΛ∂Λ[GΛ0 ]−1GΛ
}off-diag.
Keldysh, t′Λ=∞ = t′
Green functions: (Γlead(ω) = πρ(ω)t2)
[GΛret]−1 =
ω + iΓlead + iΛ −t′Λ 0−t′Λ ω + iΛ −t′Λ
0 −t′Λ ω + iΓlead + iΛ
, GΛadv = [GΛ
ret]†
GΛKeldysh = −2iGΛ
ret
iΛ[1− 2fC] + Γlead
[1− 2fL]0
[1− 2fR]
GΛadv
occupation functions:
fL/R(ω) = f(ω ∓ V/2) physical leads
fC(ω) = f(ω) auxiliary leads
current (at Λ = 0): I = 4∫
Γ2lead(ω) [fL(ω)− fR(ω)]
∣
∣G13ret(ω)
∣
∣
2dω
Comparison to td-DMRG
0 1 2 3V / t
0
0.5
1
1.5
I / e
/h
u/t=0u/t=0.3u/t=-1u/t=-1
t’ / t = 0.5
negative diff.
cond.
agreement
FRG-DMRG
bla
bla
bla
bla
bla
bla
(Boulat, Saleur & Schmitteckert ’08; Karrasch, Borda, Pletyukhov & VM ’10)
FRG and RTRG-FS in scaling limit: ε = 0, L-R symmetric
flow equation for renormalized rate ΓΛ:
∂ΛΓΛ = −2UΓΛ Λ + ΓΛ
(V/2)2 + (Λ + ΓΛ)2, ΓΛ=∞ = Γ∞
⇒ Γ ≈ Γ∞(
Λ0
max{V/2,Γ}
)2U
, Λ0 ∼ B
⇒ V and Γ act as IR cutoffs
observables:
• susceptibility at V = 0:
χ−1 = πΓ/2 = πTK/2 ∼ (Γ∞)1−αχ , αχ = 2U +O(U2)
• current for V � Γ = TK: (agrees with Doyon ’07)
I =2Γπ
arctanV
Γ∼ Γ ∼ V −αV , αV = 2U +O(U2)
no interesting nonequilibrium physics??
FRG and RTRG-FS in scaling limit: general case
flow of level positions is of order U2 and can be neglected
flow equation for renormalized rates ΓΛα: (ΓΛ =
∑
α ΓΛα)
∂ΛΓΛα = −2UαΓΛ
α
Λ + ΓΛ/2(µα − ε)2 + (Λ + ΓΛ/2)
, ΓΛ=∞α = Γ∞α
⇒ Γα ≈ Γ∞α
(
Λ0
max{|µα − ε|,Γ/2}
)2Uα
, Λ0 ∼ B
current (at Λ = 0):
I =1π
ΓLΓRΓ
[
arctanV − 2ε
Γ+ arctan
V + 2εΓ
]
introduce TαK: TαK = Γ∞α (2Λ0/TK)2Uα , TK =∑
α TαK ( 6= Γ)
introduce asymmetry parameter: c2 = TLK/TRK
Away from p-h symmetry, still L-R symmetric
10-2 10-1 11 101 102
ε /TK
0
0.2
0.4
0.6
0.8
1
G/G
0
V/TK =0
V/TK =0.6
V/TK =1
V/TK =2
V/TK =5
V/TK=50
L-R symmetric
U = 0.1/π
lines: RTRG-FS
symbols: FRG
⇒ maximum of G = dI/dV at ε = V/2: resonance!
On-resonance current
11 102 104
V/TK
10-2
10-1
I/T K
-0.06
0
log. deriv.
ε=V/2
ε=0 L-R symmetric
U = 0.1/π
lines: RTRG-FS
symbols: FRG
current for V � TK: I(V ) ≈ TK2
(
TKΓ
)2UL(
TK2V
)2UR
c(
TKΓ
)2UL+1c
(
TK2V
)2UR
c1+c2
.
⇒ V is not a simple IR cutoff
power law only for
• V ≫ TK
• c� 1 ⇒ I ∼ V −2UR
(does not agree with Doyon ’07)
Left-right asymmetry, off-resonance
10-2 100 102 104 106
V/TK
0
0.1
0.2
(1+c
2 )2I/
4c2 T
K
γ = 0.75, c2=21.4
γ = 0.5, c2=7.9
γ = 0.25, c2=2.8
γ = 0, c2=1
L-R asymmetric
UL/R = (1± γ) 0.1/π
ε = 0
lines: RTRG-FS
current for V � TK, V � ε: I(V ) ≈ TK(
TKV
)2UL(
TKV
)2UR
c(
TKV
)2UL+1c
(
TKV
)2UR
c1+c2
.
⇒ V is not a simple IR cutoff for generic cases
power law only for• UL = UR ⇒ I ∼ V −2U
• UL 6= UR, c� 1 ⇒ I ∼ V −2UL
• UL 6= UR, c� 1 ⇒ I ∼ V −2UR
Relaxation dynamics into the steady state: RTRG-FS
long-time behavior, off-resonance (ε, V, |ε− V/2| � TK, 1/t)
〈n2(t)〉 ≈(
1− e−Γ1t)
〈n2〉 −1
2πe−Γ2t (TKt)
1+2U
×[
sin(
(ε+V2 )t)
(ε+V2 )2 t2
−πU
4
cos(
(ε+V2 )t)
(ε+V2 )2 t2
+ (V →−V )
]
0
0.2
0.4
<n2(t)
>
0 1 2 3 4 5TK t
0
0.2
0.4
I(t)/
T K
V/TK =10V/TK =20V/TK =50V/TK =200
0 1 20
0.04
L-R symmetric
〈n2(t = 0)〉 = 0
U = 0.1/π, ε = 10TK
Γ1 ≈ Γ (charge relaxation)
Γ2 ≈ Γ/2 (level broadening)
oscillation freqs. ε± V/2
⇒ exponential and power-law decay, two decay rates
Summary
• finite bias IRLM contains interesting nonequilibrium physics . . .
• which doesn’t show up for left-right symmetry and off-resonance
• interesting relaxation dynamics
• FRG and RTRG-FS are suitable tools to study the nonequ. IRLM
• td-DMRG results (not in scaling limit) confirmed
Refs.:
• Karrasch, Andergassen, Pletyukhov, Schuricht, Borda, VM & Schoeller, EPL90, 30003 (2010)
• Karrasch, Pletyukhov, Borda & VM, PRB 81, 125122 (2010)
