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Real-Time Diagrammatic Monte Carlo for Non-Equilibrium Quantum Transport

Marco Schir`

• 1 and Michele Fabrizio1,2

1SISSA & Democritos

2ICTP

Quantum Transport from Macro to Nano

◮ How do we measure the current I flowing through a macroscopic sample? ◮ Conductance is defined by Ohm’s law

I = GV G = σ S L

◮ What is the conductance of a single molecule?

Measuring Quantum Transport at Nanoscale

◮ 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

◮ Experiments measure I −V characteristics and conductance dI/dV as a

function of external parameters

Park et al., Nature (2000) ◮ Very sensitive probes to local many body interactions (→ Meir-Wingreen)

◮ e.g. Coulomb Blockade, Kondo Effect

◮ Transport at finite bias requires a full out-of-equilibrium description

Modeling Non Equilibrium Transport through Nanodevices

◮ Electrons in the leads are non-interacting (Landau-Fermi-Liquid)

Hleads = ∑

α=L,R ∑ k,σ

ξkα f †

kσαfkσα

ξkα = ξk − µα

◮ Dot/Molecule has a small set of discrete levels

Hloc = Hloc

• c†

σ,cσ

Modeling Non Equilibrium Transport through Nanodevices

◮ Electrons in the leads are non-interacting (Landau-Fermi-Liquid)

Hleads = ∑

α=L,R ∑ k,σ

ξkα f †

kσαfkσα

ξkα = ξk − µα

◮ Dot/Molecule has a small set of discrete levels

Hloc = Hloc

• c†

σ,cσ

• ◮ Coupling between leads and molecule is due to Hhyb

N.J.Tao, Nature Nanotechnology (2006)

Hhyb = ∑

kσα

Vkα (f †

kσαcσ +c† σfkσα)

Out-of-Equilibrium Quantum Impurity Models

Γ

L

µR g V

L R

QI

µL Γ

R

V b H = ∑

k,α

ξkα f †

kσαfkσα +Hloc[c† σ,cσ]+ ∑ kσα

Vkα (f †

kσαcσ +c† σfkσα)

Relevant Energy/Time Scales

◮ Tunneling to the leads → Γα = π V 2 α ρ(εF) ◮ Local energy scales:

◮ 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

− → We are mainly interested in the non-perturbative regime!

Quantum Transport in the Linear-Response Regime

◮ Zero-bias conductance is related to the equilibrium spectral-function

dI dV |V =0 = 2e2

dω

2π ΓLΓR ΓL +ΓR ∂ nF(ω) ∂ ω

• ImGR(ω)

van der Wiel et al., Science(2002)

Kondo Effect in Quantum Dots!

see e.g. → Pustilnik, Glazman (2004)

Theoretical approaches to Equilibrium QIM

◮ Numerical Renormalization Group ◮ Bethe-Ansatz, Integrability, Boundary-CFT ◮ Diagrammatic MonteCarlo (→ see e.g. P. Werner, A. Millis (2006))

Quantum Transport in the non-equilibrium regime

A formula for the current →

Meir, Wingreen PRL (1992)

I(V ) =

dω

2π [(ΓL −ΓR)G <(ω;V )+(ΓL fL(ω)−ΓR fR(ω))ImGret(ω;V )]

◮ Averages taken over the non-equilibrium steady-state (= the ground state!) ◮ Out-of-Equilibrium both the spectrum and the statistics are needed ◮ Real-Time Dynamics is a possible route

How to solve Quantum Impurities out-of-equilibrium?

◮ Keldysh perturbation theory

• L. Glazman et al. (2000), A. Rosch et al.(2000), J. Konig et al. (2000),

A.J. Millis et al.(2004), ..

◮ Bethe Ansatz for Open Quantum Systems

• N. Andrei PRL(2006)

◮ Time-dependent NRG

• F. Anders PRL(2008)

◮ Iterative/Stochastic Summation of Real-Time Path Integrals

• R. Egger PRB(2008),E. Rabani PRL(2008)

◮ Real-Time Diagrammatic MonteCarlo on the Keldysh Contour

• 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

L R

QI

µ µ

L R

t = 0

◮ Initial Condition, t = 0

H0 =

∑

k,α=L,R

ξkσα f †

kσαfkσα +Hloc[c† σ,cσ]

ρ0 = ρL ⊗ρloc ⊗ρR

Real-Time Dynamics for Quantum Impurity Models

L R

QI

µ µ

L R

t > 0

◮ Real-Time Dynamics, t > 0

H(t > 0) = H0 + ∑

kσα

Vkα (f †

kσαcσ +h.c.)

ρ0 = ρL ⊗ρloc ⊗ρR

Real-Time Dynamics for Quantum Impurity Models

L R

QI

µ µ

L R

t > 0

◮ Real-Time Dynamics, t > 0

H(t > 0) = H0 + ∑

kσα

Vkα (f †

kσαcσ +h.c.)

ρ0 = ρL ⊗ρloc ⊗ρR O(t) = Tr

• ρ0 U†(t)O U(t)
• U(t) = T e−i

t

0 dτH(τ)

◮ Following the dynamics from the initial condition we may access both to

transient and to steady-state features.

◮ Due to the bias in the long-time limit the system will reach a non-equilibrium

steady-state → I(V ) = 0, dissipation!

◮ How do we compute real-time averages?

Real-Time Diagrammatic MC on the Keldysh Contour

◮ Real-time dynamics can be formulated along the Keldysh contour CK

t

O(t) = Tr ρ0 TCK

• ei
• CK dτ H0+Hhyb O

Real-Time Diagrammatic MC on the Keldysh Contour

◮ Real-time dynamics can be formulated along the Keldysh contour CK

t

O(t) = Tr ρ0 TCK

• ei
• CK dτ H0+Hhyb O
• ◮ We formally expand Keldysh evolution operator in power of Hhyb

ei

• CK dτ H0+Hhyb = ei
• CK dτ H0

∞

∑

n=0

(i)n

• CK

Hhyb(τ1)···

• CK

Hhyb(τn)

Real-Time Diagrammatic MC on the Keldysh Contour

◮ Real-time dynamics can be formulated along the Keldysh contour CK

t

O(t) = Tr ρ0 TCK

• ei
• CK dτ H0+Hhyb O
• ◮ We formally expand Keldysh evolution operator in power of Hhyb

ei

• CK dτ H0+Hhyb = ei
• CK dτ H0

∞

∑

n=0

(i)n

• CK

Hhyb(τ1)···

• CK

Hhyb(τn)

◮ At any order n in Htun we can integrate-out exactly electrons in the leads

∆

• τ,τ′

= ∑

kα=L,R

|Vkα|2TCK

• fkα (τ)f †

kα

• τ′

.

Real-Time Diagrammatic MC on the Keldysh Contour

◮ Real-time dynamics can be formulated along the Keldysh contour CK

t

O(t) = Tr ρ0 TCK

• ei
• CK dτ H0+Hhyb O
• ◮ We formally expand Keldysh evolution operator in power of Hhyb

ei

• CK dτ H0+Hhyb = ei
• CK dτ H0

∞

∑

n=0

(i)n

• CK

Hhyb(τ1)···

• CK

Hhyb(τn)

◮ At any order n in Htun we can integrate-out exactly electrons in the leads

∆

• τ,τ′

= ∑

kα=L,R

|Vkα|2TCK

• fkα (τ)f †

kα

• τ′

.

◮ Example: Second Order

O(t) O(t) O(t)

det

• ∆11′

∆12′ ∆2′ 1 ∆22′

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > =

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = +

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = +

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = +

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = + + +

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = + + +

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = + + +

◮ Basic diag-MC updates

◮ Adding/Removing/Shifting vertex on the contour ◮ Accept/Reject by standard Metropolis

A (Cnew ← Cold) = min

• 1, W (Cnew)

W (Cold)

Real-Time Diagrammatic MC on the Keldysh Contour

O(t) =

∞

∑

n=0

• CK

det ˆ ∆(te

1,...,te n|ts 1,... ts n)TK

• c(te

1)c†(ts 1)···c(te n)c†(ts n)O(t)

• loc

◮ Real-time quantum average written as a sum over diagrams along the contour

< n (t) > = + + +

◮ Basic diag-MC updates

◮ Adding/Removing/Shifting vertex on the contour ◮ Accept/Reject by standard Metropolis

A (Cnew ← Cold) = min

• 1, W (Cnew)

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

ξkαf †

kαfkα +ε0c†c +∑ k α

Vkα (f †

kαc +c†fkα)

0.5 1.0 1.5 2

t Γ

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

n (t)

eV = 0 eV = 4 Γ eV = 8 Γ

0 0.25 0.5 0.75 1.0 1.25 1.5

t Γ

0.5 1 1.5

I (t)

eV = 1.5 Γ eV = 1.0 Γ eV = 0.75 Γ eV =0.5 Γ

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 = εdˆ n +U (ˆ n −1)2

0.5 1.0 1.5

t Γ

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

D (t)

U = 8Γ, eV = 2.0 Γ U = 4Γ, eV = 0.1 Γ

0 0.25 0.5 0.75 1.0 1.25 1.5

t Γ

0.5 1 1.5

dI(t)/dV

U = 4Γ, eV = 0.1Γ U = 8Γ, eV = 2Γ

Lesson from an hard case:

◮ Charge degrees of freedom relax on short-time scales ◮ A large bias eV ≫ TK cuts-off Keldysh evolution → dI dV ∼ 1/log2(eV /TK) ◮ An exp-long-time controls the low-bias conductance

Hystogram of Perturbative Order

5 10 15

k

0,1 0,2 0,3 0,4

proba(k)

Γ t = 0.5 Γt = 1.0 Γt=1.5 Γ t=2.0

5 10 15

k

0,05 0,1 0,15 0,2

Γ t=2.0 U=0 Γt = 2.0 U=4Γ

◮ k ∼ Γt , independently of U ←

→ imaginary time: k ∼ TK/T

◮ For a finite quantum system Trloc[....] it’s a pure phase!

Keldysh diag-MC: conclusions and outlooks

What we like

◮ Numerically exact method (no truncations), infinite size limit ◮ Direct access to Current, Conductance, Impurity Green’s Function ◮ Solve a Quantum Impurity Model for a given ∆(τ,τ′) → Noneq-DMFT

What we still don’t like

◮ Lack of renormalization in the hystogram severely limits max time ◮ Starting point is too far from strong-coupling steady-state

Thank you!

Non-Equilibrium Transport through a single molecule

L R

QI

µ µ

L R

Tal, Krieger et al. PRL (08)

◮ A simple model of molecular-conductor

Hloc(n) = ω0 2 (x2 +p2)+gx(n − 1 2)+εd(n − 1 2)

◮ Electron-Vibron Coupling affects dI/dV spectrum ◮ Lowest order e-ph perturbation theory predicts sharp jump in dI/dV at

eV ≃ ¯ hω0

◮ Point Contact Spectroscopy reveals a smoother behaviour and a tiny jump

Differential Conductance dI/dV from K-diagMC

1.8 1.9 2.0 2.1 2.2

V

0.73 0.735 0.74 0.745 0.75

dI/dV

εd = 1.0 1.7 1.8 1.9 2.0 2.1 2.2 2.3

V

0.3 0.305 0.31 0.315 0.32

εd = 3.0

ω0 ω0

◮ Step-Down to Step-Up crossover when εd is tuned across G(V = 0) = 1/2 ◮ Non-Perturbative effects act to broaden the feature