Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach

Transcription

1 Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach Frithjof B Anders Institut für theoretische Physik, Universität Bremen Concepts in Electron Correlation, Hvar, 30. September 2005 Collaborator: A. Schiller, Hebrew University, Jerusalem, Israel R. Bulla, S. Tornow, University of Augsburg, Germany M. Vojta, University of Karlsruhe, Germany Concepts in Electron Correlation, Hvar, 30/9/2005 p.1/18

2 What is a Quantum-Impurity System (QIS)? Quantum Impurity α> γ> (metallic) host bosonic bath quantum-impurity: Problem: embedded in a (metallic) host interacting with the environment of non-interacting particles (Bosons/Fermions) infrared divergence due to local degeneracy Concepts in Electron Correlation, Hvar, 30/9/2005 p.2/18

3 What is a Quantum-Impurity System (QIS)? Quantum Impurity α> γ> (metallic) host bosonic bath Examples: transition metal ion Cu, Mn, Ce in a metal two-level system (Qubit) in a bosonic bath Quantum dot coupled to leads donor-acceptor centers of a large bio-molecule Concepts in Electron Correlation, Hvar, 30/9/2005 p.2/18

4 Goal of the Talk Our new Approach to Non-Equilibrium of QIS: based on the non-perturbative NRG uses the complete basis of the many body Fock space takes into account all energy scales describes short and long time scales does not accummulate an error t as the TD-DMRG breakthrough in the description of real time dynamics of non-equilibrium quantum systems: Concepts in Electron Correlation, Hvar, 30/9/2005 p.3/18

5 Contents 1. Introduction Modelling of quantum dots Charge transfer in molecules (spin-boson model) 2. Non-equilibrium dynamics Time evolution of quantum systems New approach to quantum impurity problems 3. Results Dissipation and decoherence in a two level system Spin- and charge dynamics in ulta-small quantum dots AF-Kondo model spin precession 4. Summary and outlook Concepts in Electron Correlation, Hvar, 30/9/2005 p.4/18

6 Modelling of a Quantum Dot H = kσ ǫ kσ c kσ c kσ + σ [E d σh]d σd σ + Un d n d + kσ ) V (c kσ d σ + d σc kσ Single Impurity Anderson Model (SIAM) charge fluctuation scale: Γ i = V 2 i πρ F infrared problem low temperature scale: T K exp( πu/8γ) Concepts in Electron Correlation, Hvar, 30/9/2005 p.5/18

7 Spin-Boson Model qubit plus environment (Unruh) electron transfer in (bio)-molecules (Marcus, Schulten) = A, = D H = ǫσ z 2 σ x + q ω q b ( ) qb q + σ z M q b q + b q q J(ω) = q M q 2 δ(ω ω q ) 2παωc 1 s ω s Leggett et. al. (RMP 1987), Xu and Schulten 1994, Bulla et. al. (2003) Questions: influence of the bosonic spectrum J(ω) on the real time dynamics critical slowdown of the charge transfer process for large coupling Concepts in Electron Correlation, Hvar, 30/9/2005 p.6/18

8 Where do we stand in the description of non-equilibrium, dissipation and decoherence in quantum systems? Concepts in Electron Correlation, Hvar, 30/9/2005 p.7/18

9 Non-Equilibrium Dynamics of Quantum Systems quantum dynamics single quantum state: Schrödinger equation i t ψ >= H ψ > ensemble: density operator i tˆρ = [H, ˆρ] ; ρ = e iht/ ρ 0 e iht/ finite size quantum system: only unitary dynamics, no dissipation dissipation and decoherence: infinitly large environment needed Size of Subsystem Size of environment Subsystem 0 Environment Concepts in Electron Correlation, Hvar, 30/9/2005 p.8/18

10 NRG Approach to Quantum Impurity Problems H = H imp + H bath + H imp bath 1. discretizing the bath Hamiltonian on a logarithmic energy mesh (Wilson 1975,Oliveira ) 0 z (z+1) (z+1) z Λ Λ Λ 0 Λ Λ Λ 0 2. mapping onto a semi-infinite chain impurity t 0 t 1 t m 1 t m+1 t N 1 H m R m,n 3. diagonalizing the Hamiltonian H N+1 using the recursion H N+1 = ΛH N + ) ξ Nα (f N+1α f Nα + f Nα f N+1α α 4. truncate the basis set, go back to step 3 Concepts in Electron Correlation, Hvar, 30/9/2005 p.9/18

11 Novel Many-Body Approach to NEQ of QIS impurity t 0 t 1 t m 1 t m+1 t N 1 H m R m,n Subsystem Environment use the NRG to generate a complete basis l, e; m H m l = E m l l, l eliminated state e R m,n 1 = m l, e; m l, e; m l,e Puls at t = 0: H = H i Θ( t) + H f Θ operator Ô: property of the subsystem S Concepts in Electron Correlation, Hvar, 30/9/2005 p.10/18

12 Novel Many-Body Approach to NEQ of QIS time-dependent NRG (TD-NRG) (FBA, A. Schiller, cond-mat/ , PRL 2005) calculate ρ red NEQ ] [ρô Ô = Tr = α Ô α ρ red αα,m m,αα Subsystem Environment ρ red αα,m = e i(e α E α )t e α,e; m ρ eq α, e; m Feynman 1972, White 1992, Hofstetter 2000, mimic bath contiuum: use Oliveira s z-trick evolves towards the new steady state: [H(t > 0), ρ( )] = 0 Trace over the environment: dissipation and decoherence! Concepts in Electron Correlation, Hvar, 30/9/2005 p.10/18

13 Spin-Boson Model H = ǫσ z 2 σ x + q ω q b ( ) qb q + σ z M q b q + b q q S x = 1 2 ( + ) Decoherence N s =150, N z =16, N b =8, N iter =14, T= , Λ=2 1/2, α damp =0.1, α=0.1, ε=0, 0 =0.,ω c =1 QuBit state S x s=1.5 s=1.5 (ana.) s=1.0 s=0.8 s=0.6 s=0.4 s= ( + ) exact solution P = e Γ Mon Sep 19 11:46: t*t Leggett et al., Unruh, Palma et al., Bulla et al. Concepts in Electron Correlation, Hvar, 30/9/2005 p.11/18

14 Spin-Boson Model H = ǫσ z 2 σ x + q ω q b ( ) qb q + σ z M q b q + b q q J(ω) = 2παω 1 s c ω s for 0 < ω < ω c ; Ohmic case: s = 1 Fixed point: delocalized localized 0<α<1/2 oszillatory Toulouse Point: 1/2<α<α( ) c overdamped α( ) c < α α=1/2 Concepts in Electron Correlation, Hvar, 30/9/2005 p.11/18

15 Spin-Boson Model H = ǫσ z 2 σ x + q ω q b ( ) qb q + σ z M q b q + b q q S z N s =100, N b =8, N iter =25, N z =16, Λ=2, 1 =0.2, ε 1 =0, α=0.1, ω c =1, s=1, T=3*10-8 α=0.1 α=0.3 α=0.5 α=0.7 α=1.0 α=1.1 α=1.2 α=1.3 α= t*ω c S z J(ω) = 2παωc 1 s ω s Ohmic Regime: s = 1 QPT at α c ( ) Toulouse point α = 1/2 oszillatory α < 1/2 overdamped α c > α > 1/2 localize: α > α c Concepts in Electron Correlation, Hvar, 30/9/2005 p.11/18

16 Charge Fluctuation in a Small Quantum Dot H>0 H=0 µ H = kσ ǫ kσ c kσ c kσ + σ [E d σh]d σd σ + Un d n d time Ε d + kσ ) V (c kσ d σ + d σc kσ impurity levels change of E d : change dynamics change of mag. field H: spin dynamics change of V : route to new equilibrium Concepts in Electron Correlation, Hvar, 30/9/2005 p.12/18

17 Charge Fluctuation in a Small Quantum Dot H>0 H=0 µ Ε d n d (a) Γ 0 = Γ 1 =2 =4 =6 =8 =10 =12 =18 time impurity levels n d (b) Γ 0 = t*γ 1 Charge relaxation time scale : t ch = 1/Γ 1 Concepts in Electron Correlation, Hvar, 30/9/2005 p.12/18

18 Spin Fluctuation in a Small Quantum Dot H>0 H=0 µ s z =2 =4 =6 =8 =10 =12 =18 (c) Γ 0 = Γ 1 Ε d 0 time 0.2 impurity levels s z 0.1 (d) Γ 0 = t*γ 1 Concepts in Electron Correlation, Hvar, 30/9/2005 p.13/18

19 Spin Fluctuation in a Small Quantum Dot H>0 H=0 µ Ε d s z (a) Γ 0 = Γ 1 =2 =4 =6 =8 =10 =12 =18 time 0.2 impurity levels s z 0.1 (b) Γ 0 = t/t K Spin relaxation time scale : t sp 1/T K Concepts in Electron Correlation, Hvar, 30/9/2005 p.13/18

20 T SIAM Time-Evolution start: high temperature temperature T temperature T Free Impurity V 0 = 0 Kondo Regime Mixed Valence V 0 0 no universality time t time t time evolution depends on boundary conditions Concepts in Electron Correlation, Hvar, 30/9/2005 p.14/18

21 Kondo Model H = H cb + H z 2 σ z + H x σ x + JS 2 loc s cb Spin Precession J=0.2 Spin Precession: H x = 0.1 H z Sz <S xyz > 0 S x S y S z Fit: -0.5*cos(H z *t)*e *t 0 0 Sy Sx t*d Fri Jun 10 15:03: external magnetic field: x z-axis Concepts in Electron Correlation, Hvar, 30/9/2005 p.15/18

22 TD-NRG algorithm: Outlook Summary and Outlook complete basis set needed for the time evolution dissipation and decoherence due to the bath Spin-Boson Model benchmark: decoherence and non-ohmic baths oscillatory vs overdamped regime SIAM: two relaxation time scale for spin and charge dynamics Kondo Model: spin relaxation and spin precession Bosonic QI models: charge transfer in bio-molecules, photosynthesis description of steady state currents Concepts in Electron Correlation, Hvar, 30/9/2005 p.16/18

23 Benchmark: Resonant Level Model t*γ n d (a) E d 1 /Γ = -1 E d 1 /Γ = -2 n d (t=0) E d 1 /Γ = N - m n d 0.6 (b) T/Γ = 0.1 T/Γ = 0.5 T/Γ = 1 T/Γ = t*t Concepts in Electron Correlation, Hvar, 30/9/2005 p.17/18

24 Anisotropic Kondo Model H = H cb + H z 2 + kk J perp 2 σ z + H x σ x + 2 kk αβ ( c k c k S imp + c k c k S+ imp J z 2 c kα c k β σ zs z imp ) Concepts in Electron Correlation, Hvar, 30/9/2005 p.18/18

25 Anisotropic Kondo Model spin relaxation: J 0 = 0, J 1 S z S z J z =-0.1 J =0.15 (analytic) J =0.15(ana) O(t 2 ) = 0.15D = const ,5 0,4 0,3 0,2 0,1 0 (a) (b) J z =0.15 J z =0.1 J z =0.05 J z = t*d two regimes: short time scale analytical result: G(x) = S z S z (0) (2ρJ 1 )2 [G(2Dt) 2G(Dt)] X l=1 ( 1) l+1 (2l)!2l(2l 1) x2l external magnetic field: z-axis Concepts in Electron Correlation, Hvar, 30/9/2005 p.18/18

26 Anisotropic Kondo Model spin relaxation: J 0 = 0, J 1 S z 0,5 0,4 0,3 0,2 0,1 0 = 0.15D = const t*t K J z =0.15 J z =0.1 J z =0.05 J z =0.0 J z =-0.1 two regimes: short time scale analytical result: S z S z (0) (2ρJ 1 )2 [G(2Dt) 2G(Dt)] G(x) = X l=1 ( 1) l+1 (2l)!2l(2l 1) x2l long time: t long 1/T K Concepts in Electron Correlation, Hvar, 30/9/2005 p.18/18

27 Anisotropic Kondo Model ferromagnetic regime, spin relaxation: H z = t*d S z /S z (0) (a) 0.4 S z /S z (0) (b) J=-0.1, T=0.13 J=-0.1, T=0.018 J=-0.1, T= J=-0.1, T=4.0*10-5 J=-0.2, T=0.13 J=-0.2, T=0.018 J=-0.2, T= J=-0.2, T=4.0* t*t*j(t) external magnetic field: z-axis Concepts in Electron Correlation, Hvar, 30/9/2005 p.18/18