Real-Time Diagrammatic Monte Carlo for Non-Equilibrium Quantum Transport o 1 and Michele Fabrizio 1 , 2 Marco Schir` 1 SISSA & Democritos 2 ICTP
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 G = σ S I = GV 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 V b 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) H leads = ∑ ξ k α f † α = L , R ∑ k σα f k σα k , σ ξ k α = ξ k − µ α ◮ Dot/Molecule has a small set of discrete levels � c † � H loc = H loc σ , c σ
Modeling Non Equilibrium Transport through Nanodevices ◮ Electrons in the leads are non-interacting (Landau-Fermi-Liquid) H leads = ∑ ξ k α f † α = L , R ∑ k σα f k σα k , σ ξ k α = ξ k − µ α ◮ Dot/Molecule has a small set of discrete levels � c † � H loc = H loc σ , c σ ◮ Coupling between leads and molecule is due to H hyb H hyb = ∑ V k α ( f † k σα c σ + c † σ f k σα ) k σα N.J.Tao, Nature Nanotechnology (2006)
Out-of-Equilibrium Quantum Impurity Models V g Γ Γ L R µ L V QI R L b µ R H = ∑ σ , c σ ]+ ∑ ξ k α f † V k α ( f † k σα f k σα + H loc [ c † k σα c σ + c † σ f k σα ) k , α k σα Relevant Energy/Time Scales ◮ Tunneling to the leads → Γ α = π V 2 α ρ ( ε F ) ◮ Local energy scales: ◮ Molecular Level spacing, Vibrational frequency → ε 0 , ω 0 ◮ Local many-body interactions → Coulomb repulsion, electron-vibron coupling ◮ External control parameters → Bias V , gate voltage V g , 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 � d ω � ∂ n F ( ω ) � dI Γ L Γ R dV | V = 0 = 2 e 2 Im G R ( ω ) 2 π Γ L +Γ R ∂ ω Kondo Effect in Quantum Dots! see e.g. → Pustilnik, Glazman (2004) van der Wiel et al. , Science(2002) 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) � d ω 2 π [(Γ L − Γ R ) G < ( ω ; V )+(Γ L f L ( ω ) − Γ R f R ( ω ))Im G ret ( ω ; V )] I ( 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 ◮ Initial Condition, t = 0 t = 0 ξ k σα f † µ ∑ k σα f k σα + H loc [ c † H 0 = σ , c σ ] L QI R L k , α = L , R µ R ρ 0 = ρ L ⊗ ρ loc ⊗ ρ R
Real-Time Dynamics for Quantum Impurity Models ◮ Real-Time Dynamics, t > 0 t > 0 H ( t > 0) = H 0 + ∑ V k α ( f † k σα c σ + h . c . ) µ L QI R L k σα µ R ρ 0 = ρ L ⊗ ρ loc ⊗ ρ R
Real-Time Dynamics for Quantum Impurity Models ◮ Real-Time Dynamics, t > 0 t > 0 H ( t > 0) = H 0 + ∑ V k α ( f † k σα c σ + h . c . ) µ L k σα QI R L µ R ρ 0 = ρ L ⊗ ρ loc ⊗ ρ R � t ρ 0 U † ( t ) O U ( t ) U ( t ) = T e − i 0 d τ H ( τ ) � � � O ( t ) � = Tr ◮ 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 C K � � CK d τ H 0 + H hyb O � e i � O ( t ) � = Tr ρ 0 T C K 0 t
Real-Time Diagrammatic MC on the Keldysh Contour ◮ Real-time dynamics can be formulated along the Keldysh contour C K � � CK d τ H 0 + H hyb O � e i � O ( t ) � = Tr ρ 0 T C K 0 t ◮ We formally expand Keldysh evolution operator in power of H hyb ∞ � � � CK d τ H 0 + H hyb = e i � CK d τ H 0 e i ∑ ( i ) n H hyb ( τ 1 ) ··· H hyb ( τ n ) C K C K n = 0
Real-Time Diagrammatic MC on the Keldysh Contour ◮ Real-time dynamics can be formulated along the Keldysh contour C K � � CK d τ H 0 + H hyb O � e i � O ( t ) � = Tr ρ 0 T C K 0 t ◮ We formally expand Keldysh evolution operator in power of H hyb ∞ � � � CK d τ H 0 + H hyb = e i � CK d τ H 0 e i ∑ ( i ) n H hyb ( τ 1 ) ··· H hyb ( τ n ) C K C K n = 0 ◮ At any order n in H tun we can integrate-out exactly electrons in the leads � τ ′ �� = ∑ τ , τ ′ � | V k α | 2 � T C K f k α ( τ ) f † � � ∆ � . k α k α = L , R
Real-Time Diagrammatic MC on the Keldysh Contour ◮ Real-time dynamics can be formulated along the Keldysh contour C K � � CK d τ H 0 + H hyb O � e i � O ( t ) � = Tr ρ 0 T C K 0 t ◮ We formally expand Keldysh evolution operator in power of H hyb ∞ � � � CK d τ H 0 + H hyb = e i � CK d τ H 0 e i ∑ ( i ) n H hyb ( τ 1 ) ··· H hyb ( τ n ) C K C K n = 0 ◮ At any order n in H tun we can integrate-out exactly electrons in the leads � τ ′ �� = ∑ τ , τ ′ � | V k α | 2 � T C K f k α ( τ ) f † � � ∆ � . k α k α = L , R ◮ Example: Second Order O(t) 0 O(t) 0 � � ∆ 11 ′ ∆ 12 ′ det ∆ 2 ′ 1 ∆ 22 ′ O(t) 0
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∑ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > =
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > = +
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > = +
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > = +
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > = + + +
Real-Time Diagrammatic MC on the Keldysh Contour ∞ � det ˆ ∆( t e 1 ,..., t e n | t s 1 ,... t s c ( t e 1 ) c † ( t s 1 ) ··· c ( t e n ) c † ( t s ∑ � � � O ( t ) � = n ) � T K n ) O ( t ) � loc C K n = 0 ◮ Real-time quantum average written as a sum over diagrams along the contour < n (t) > = + + +
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