Real-Time Diagrammatic Monte Carlo for
Non-Equilibrium Quantum Transport
Marco Schirò1 and Michele Fabrizio1,2
1
SISSA & Democritos
2 ICTP
Measuring Quantum Transport at Nanoscale
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Experiments can bridge small quantum objects to conducting leads
Current I is induced through the dot/molecule by an applied bias Vb
N. Roch et al., Nature (2008)
Why it can be interesting..
1. New-Tech: Molecular Electronics is coming!
2. New-Physics: Experimental realization of out-of-equilibrium quantum systems
Measuring Quantum Transport at Nanoscale
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Experiments measure I − V characteristics and conductance dI /dV as a
function of external parameters
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Very sensitive probes to local many body interactions (→ Meir-Wingreen)
Park et al., Nature (2000)
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e.g. Coulomb Blockade, Kondo Effect
Transport at finite bias requires a full out-of-equilibrium description
Out-of-Equilibrium Quantum Impurity Models
ΓL
Vg
ΓR
µL
L
†
†
H = ∑ ξkα fkσ
α fkσ α + Hloc [cσ , cσ ] +
k,α
R
QI
∑
Vb
µR
†
†
Vkα (fkσ
α cσ + cσ fkσ α )
kσ α
Relevant Energy/Time Scales
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Tunneling to the leads → Γα = π Vα2 ρ(εF )
Local energy scales:
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Molecular Level spacing, Vibrational frequency → ε0 , ω0
Local many-body interactions → Coulomb repulsion, electron-vibron coupling
External control parameters → Bias V , gate voltage Vg , Temperature T
How to solve Quantum Impurities out-of-equilibrium?
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Keldysh perturbation theory
L. Glazman et al. (2000), A. Rosch et al.(2000), J. Konig et al. (2000),
A.J. Millis et al.(2004), ..
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Bethe Ansatz for Open Quantum Systems
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Time-dependent NRG
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Iterative Summation of Real-Time Path Integrals
P. Metha and N. Andrei (2006)
F. Anders (2008)
S. Weiss, J. Eckel, M. Thorwart and R. Egger PRB(2008)
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Real-Time (Diagrammatic) MonteCarlo on the Keldysh Contour
L. Muhlbacher and E. Rabani, PRL (2008)
M. Schiro’ and M. Fabrizio, arXiv:0808.0589, Phys.Rev.B in press
P. Werner, T. Oka, A. Millis, Phys. Rev. B (2009)
Real-Time Dynamics for Quantum Impurity Models
I
t=0
µL
L
QI
Initial Condition, t = 0
H0 =
R
∑
†
†
ξkσ α fkσ
α fkσ α + Hloc [cσ , cσ ]
k,α =L,R
µR
ρ0 = ρL ⊗ ρloc ⊗ ρR
Real-Time Dynamics for Quantum Impurity Models
I
t>0
Real-Time Dynamics, t > 0
H(t > 0) = H0 +
µL
L
QI
R
∑
†
Vkα (fkσ
α cσ + h.c.)
kσ α
µR
ρ0 = ρL ⊗ ρloc ⊗ ρR
Real-Time Dynamics for Quantum Impurity Models
I
t>0
Real-Time Dynamics, t > 0
H(t > 0) = H0 +
µL
L
QI
†
Vkα (fkσ
α cσ + h.c.)
∑
kσ α
R
µR
h O(t) i = Tr ρ0 U † (t) O U(t)
ρ0 = ρL ⊗ ρloc ⊗ ρR
U(t) = T e −i
Rt
0
dτH (τ )
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Following the dynamics from the initial condition we may access both to
transient and to steady-state features.
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Due to the bias in the long-time limit the system will reach a non-equilibrium
steady-state → I (V ) 6= 0, dissipation!
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How do we compute real-time averages?
Real-Time Diagrammatic MC on the Keldysh Contour
Real-time dynamics can be formulated along the Keldyshcontour CK
R
i
dτ H0 +Hhyb
hO(t)i = Tr ρ0 TCK e CK
O
0
I
t
Real-Time Diagrammatic MC on the Keldysh Contour
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Real-time dynamics can be formulated along the Keldyshcontour CK
R
i
dτ H0 +Hhyb
hO(t)i = Tr ρ0 TCK e CK
O
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We formally expand Keldysh evolution operator in power of Hhyb
0
t
e
R
i C dτ H0 +Hhyb
K
=e
R
i C dτ H0
K
∞
∑ (i)n
n=0
Z
CK
Hhyb (τ1 ) · · ·
Z
CK
Hhyb (τn )
Real-Time Diagrammatic MC on the Keldysh Contour
I
Real-time dynamics can be formulated along the Keldyshcontour CK
R
i
dτ H0 +Hhyb
hO(t)i = Tr ρ0 TCK e CK
O
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We formally expand Keldysh evolution operator in power of Hhyb
0
t
e
R
i C dτ H0 +Hhyb
K
=e
R
i C dτ H0
K
∞
∑ (i)n
n=0
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Z
CK
Hhyb (τ1 ) · · ·
Z
CK
Hhyb (τn )
At any order n in Htun we can integrate-out exactly electrons in the leads
†
∆ τ, τ 0 = ∑ |Vkα |2 hTCK fkα (τ) fkα
τ0 i .
kα =L,R
Real-Time Diagrammatic MC on the Keldysh Contour
I
Real-time dynamics can be formulated along the Keldyshcontour CK
R
i
dτ H0 +Hhyb
hO(t)i = Tr ρ0 TCK e CK
O
I
We formally expand Keldysh evolution operator in power of Hhyb
0
t
e
R
i C dτ H0 +Hhyb
K
=e
R
i C dτ H0
K
∞
∑ (i)n
n=0
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Z
CK
Hhyb (τ1 ) · · ·
Z
CK
Hhyb (τn )
At any order n in Htun we can integrate-out exactly electrons in the leads
†
∆ τ, τ 0 = ∑ |Vkα |2 hTCK fkα (τ) fkα
τ0 i .
kα =L,R
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Example: Second Order
O(t)
0
0
O(t)
det
O(t)
0
∆1 10
∆20 1
∆1 20
∆2 20
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
∑
Z
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
∑
Z
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
∑
Z
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
∑
Z
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
∑
Z
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
Z
∑
n=0 CK
I
ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
+
+
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
Z
∑
n=0 CK
I
ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
+
+
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
Z
∑
n=0 CK
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ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
I
+
+
Basic diag-MC updates
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Adding/Removing/Shifting vertex on the contour
Accept/Reject by standard Metropolis
W (Cnew )
A (Cnew ← Cold ) = min 1,
W (Cold )
+
Real-Time Diagrammatic MC on the Keldysh Contour
∞
h O(t) i =
Z
∑
n=0 CK
I
ˆ 1e , . . . , tne |t1s , . . . tns )hTK c(t1e )c † (t1s ) · · · c(tne )c † (tns )O(t) iloc
det∆(t
Real-time quantum average written as a sum over diagrams along the contour
< n (t) > =
I
+
+
+
Basic diag-MC updates
I
I
Adding/Removing/Shifting vertex on the contour
Accept/Reject by standard Metropolis
W (Cnew )
A (Cnew ← Cold ) = min 1,
W (Cold )
Similar structure as the hybridization expansion in imaginary time but
1. Real-time evolution means the weight is a complex number
2. Unitarity of quantum evolution −→ huge cancellations!
Z = Tr [ρ(t)] = 1
Benchmark: Biased Resonant Level Model
H=
∑
k,α =L,R
†
†
ξkα fkα
fkα + ε0 c † c + ∑ Vkα (fkα
c + c † fkα )
kα
I (t)
n (t)
1.0
eV = 0
eV = 4 Γ
eV = 8 Γ
0.9
1.5
eV = 1.5 Γ
eV = 1.0 Γ
eV = 0.75 Γ
eV =0.5 Γ
0.8
0.7
1
0.6
0.5
0.5
0.4
0.3
0
0.5
1.0
tΓ
1.5
2
0
0 0.25 0.5 0.75 1.0 1.25 1.5
tΓ
Lesson from a simple case:
1. All the perturbation theory in Hhyb is summed-up → cfr exact result!
2. Dissipation occurs entirely within the fermionic reservoirs
Non-Equilibrium Transport through an Anderson Impurity
Hloc = εd n̂ + U (n̂ − 1)2
D (t)
dI(t)/dV
1.0
0.9
0.8
1.5
U = 4Γ, eV = 0.1Γ
U = 8Γ, eV = 2Γ
U = 8Γ, eV = 2.0 Γ
U = 4Γ, eV = 0.1 Γ
0.7
1
0.6
0.5
0.4
0.5
0.3
0.2
0
0.5
tΓ
1.0
1.5
0
0 0.25 0.5 0.75 1.0 1.25 1.5
tΓ
Lesson from an hard case:
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Charge degrees of freedom relax on short-time scales
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A large bias eV TK cuts-off Keldysh evolution →
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An exp-long-time controls the low-bias conductance
dI
dV
∼ 1/log 2 (eV /TK )
Hystogram of Perturbative Order
0,4
0,2
Γ t = 0.5
Γt = 1.0
Γt=1.5
Γ t=2.0
proba(k)
0,3
0,15
0,2
0,1
0,1
0,05
0
0
10
5
k
Γ t=2.0 U=0
Γt = 2.0 U=4Γ
15
0
0
10
5
15
k
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hki ∼ Γt , independently of U ←→ imaginary time: hki ∼ TK /T
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For a finite quantum system Trloc [....] it’s a pure phase!
Keldysh diag-MC: conclusions and outlooks
What we like
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Numerically exact method (no truncations): infinite size limit, continuous-time
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Treat exactly local physic: strong correlations, phonons
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Direct access to Current, Conductance, Impurity Green’s Function
What we still don’t like
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Lack of renormalization in the hystogram severely limits max time
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Starting point is too far from strong-coupling steady-state
Thank you!
Non-Equilibrium Transport through a single molecule
Tal, Krieger et al. PRL (08)
µL
L
QI
R
µR
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A simple model of molecular-conductor
Hloc (n) =
ω0 2
1
1
(x + p 2 ) + gx(n − ) + εd (n − )
2
2
2
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Electron-Vibron Coupling affects dI /dV spectrum
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Lowest order e-ph perturbation theory predicts sharp jump in dI /dV at
eV ' h̄ω0
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Point Contact Spectroscopy reveals a smoother behaviour and a tiny jump
Differential Conductance dI /dV from K-diagMC
0.75
εd = 1.0
0.32
εd = 3.0
0.745
dI/dV
0.315
0.74
0.31
0.735
ω0
0.305
ω0
0.73
1.8
1.9
2.0
V
2.1
2.2
0.3
1.7 1.8 1.9 2.0 2.1 2.2 2.3
V
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Step-Down to Step-Up crossover when εd is tuned across G (V = 0) = 1/2
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Non-Perturbative effects act to broaden the feature