Many-particle systems far from equilibrium– from Green functions to stochastic dynamics Michael Bonitz, Sebastian Hermanns, Christopher Hinz, Niklas Schlünzen and Denis Lacroix ∗ Institut für Theoretische Physik und Astrophysik Christian-Albrechts-Universität zu Kiel, Germany ∗ Institut de Physique Nucléaire, IN2P3-CNRS, Université Paris-Sud, F-91406 Orsay Cedex, France Hamburg, 25/03 2014 M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 1 / 57
Acknowledgements 1 1 thanks also to: D. Hochstuhl (MCTDHF, CI) and K. Balzer (NEGF) M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 2 / 57
Research interests: Classical and quantum many-body systems in nonequilibrium I. First principle simulation of strongly correlated plasmas (MC, MD), analytical concepts: kinetic theory, fluid theory [A] II. QMC of correlated bosons and fermions (A. Filinov, [B]) III. Wave function based methods for atoms and molecules Solution of Schrödinger equation, Full CI Multiconfiguration time-dependent Hartree-Fock and time-dependent Restricted active space CI [1] (S. Bauch) [ A ] Introduction to Complex plasmas , M. Bonitz, N. Horing, and P. Ludwig (eds.), Springer 2010 and 2014 [ B ] A. Filinov, M. Bonitz, and Yu.E. Lozovik, Phys. Rev. Lett. 86 , 3851 (2001); A. Filinov, N. Prokof’ev, and M. Bonitz, Phys. Rev. Lett. 105 , 070401 (2010) [1] D. Hochstuhl, C. Hinz, and M. Bonitz, EPJ-ST 223 , 177-336 (2014), review M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 3 / 57
Research interests: Classical and quantum many-body systems in nonequilibrium I. First principle simulation of strongly correlated plasmas (MC, MD), analytical concepts: kinetic theory, fluid theory II. QMC of strongly correlated bosons and fermions (A. Filinov) III. Wave function based methods for atoms and molecules Solution of Schrödinger equation, Full CI Multiconfiguration time-dependent Hartree-Fock and time-dependent Restricted active space CI [1] (S. Bauch) IV. Statistical approaches (plasmas, atoms, condensed matter) Nonequilibrium Green functions (NEGF, 2-time fcts [2]) NEGF with generalized KB ansatz (GKBA, 1-time fcts [3]) Stochastic mean field approach [4] [1] D. Hochstuhl, C. Hinz, and M. Bonitz, EPJ-ST 223 , 177-336 (2014), review [2] K. Balzer, and M. Bonitz, Springer Lecture Notes in Physics 867 (2013) [3] M. Bonitz, S. Hermanns, and K. Balzer, Contrib. Plasma Phys. 53 , 778 (2013), arXiv:1309.4574 S. Hermanns, and M. Bonitz, Phys. Rev. B, sumbitted (2014), arXiv: 1402.7300 [4] D. Lacroix, S. Hermanns, C. Hinz, and M. Bonitz, Phys. Rev. Lett., submitted (2014), arXiv:1403.5098 M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 4 / 57
Correlated quantum systems in non-equilibrium High-intensity lasers, free electron lasers strong nonlinear excitation of matter high photon energy: core level excitation localized excitation: spatial inhomogeneity Ultra-short pulses (sub-)fs dynamics of atoms, molecules, solids sub-fs dynamics of electronic correlations Need: Nonequilibrium many-body theory conservation laws on all time scales linear and nonlinear response macroscopic to finite (inhomogeneous) systems M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 5 / 57
Outline Introduction 1 Quantum dynamics in second quantization 2 1. Dynamics of the field operators 2. Non-equilibrium Green functions (NEGF) 3. Generalized Kadanoff-Baym ansatz (GKBA) Excitation dynamics in Hubbard nanoclusters 3 1. Testing the GKBA 2. Relaxation Dynamics 3. Beyond weak coupling: T-matrix selfenergy with GKBA Stochastic Mean Field Approach 4 SMF–Numerical results Conclusions 5 M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 6 / 57
Quantum dynamics in second quantization (2) Dynamics of the field operators use Heisenberg representation of quantum mechanics: c iH ( t ) = U † ( t , t 0 )ˆ ˆ c i U ( t , t 0 ) with N -particle time evolution operator: i ∂ t U ( t , t ′ ) = ˆ H ( t ) U ( t , t ′ ) , and U ( t , t ) = ˆ 1 Heisenberg equation of motion: c iH ( t ) + [ ˆ i ∂ t ˆ H H ( t ) , ˆ c iH ( t )] = 0 , c iH ( t 0 ) = ˆ ˆ c i M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 7 / 57
Quantum dynamics in second quantization (2) Dynamics of the field operators use Heisenberg representation of quantum mechanics: c iH ( t ) = U † ( t , t 0 )ˆ ˆ c i U ( t , t 0 ) with N -particle time evolution operator: i ∂ t U ( t , t ′ ) = ˆ H ( t ) U ( t , t ′ ) , and U ( t , t ) = ˆ 1 Heisenberg equation of motion: c iH ( t ) + [ ˆ i ∂ t ˆ H H ( t ) , ˆ c iH ( t )] = 0 , c iH ( t 0 ) = ˆ ˆ c i evaluate commutator: � � � � c † h 0 i ∂ t ˆ c iH ( t ) = im + v im , H ( t ) c mH + ˆ w ilmn ˆ lH ˆ c nH ˆ c mH m mln Effective single-particle (mean field) problem, nonlinear: � � � h 0 v eff i ∂ t ˆ c iH ( t ) = im + ˆ im , H ( t ) c mH ( t ) ˆ m M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 8 / 57
How to proceed? Simple equation for a complicated object � � � h 0 v eff i ∂ t ˆ c iH ( t ) = im + ˆ im , H ( t ) c mH ( t ) ˆ m Ensemble average I. coordinate representation: replace ˆ ψ H ( r , t ) → ψ ( r , t ) “quasi-classical” approximation (many particles in single state) Gross-Pitaevskii-type equation (bosons) II. Fermions: n i = 0 , 1 , “quantum” treatment necessary. c † Ensemble average: � ˆ c iH � = 0 , � ˆ iH ˆ c jH � = ρ ij ( t ) = � i | ˆ ρ 1 ( t ) | j � c † c † Reduced density operators: � ˆ i 1 . . . ˆ i s ˆ c js . . . ˆ c j 1 � → ˆ ρ 1 ... s ( t ) Equations of motion: BBGKY hierarchy a III. Ensemble average of two(many)-time operator products: c † c H ( t ′ ) � → G (1) ( t , t ′ ) Nonequilibrium Green functions � ˆ H ( t )ˆ a M. Bonitz, Quantum Kinetic Theory , Teubner 1998 M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 9 / 57
Keldysh Green functions [Keldysh, 1964] time-ordered one-particle Nonequilibrium Green function, two times z , z ′ ∈ C (“Keldysh contour”), arbitrary one-particle basis | φ i � � ˆ � G (1) c † ij ( z , z ′ ) = i j ( z ′ ) T C ˆ c i ( z )ˆ � Keldysh–Kadanoff–Baym equation (KBE) on C : � � � � � i � ∂ G (1) kj ( z , z ′ ) = δ C ( z , z ′ ) δ ij − i � z w iklm ( z + , ¯ z ) G (2) z ; z ′ ¯ z + ) ∂ z δ ik − h ik ( z ) d¯ lmjk ( z ¯ C k klm � C wG (2) → � C Σ G (1) , Selfenergy Nonequilibrium diagram technique Example: Hartree-Fock + Second Born selfenergy KBE: first equation of Martin–Schwinger hierarchy for G (1) , G (2) . . . G ( n ) M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 10 / 57
Real-time Dyson equation/ KBE Contour Green function mapped to real-time matrix Green function � � � � c † G < ij ( t 1 , t 2 ) = ∓ i ˆ j ( t 2 )ˆ c i ( t 1 ) G R G < ij ij � � G ij = G A c † G > 0 ij ( t 1 , t 2 ) = − i ˆ c i ( t 1 )ˆ j ( t 2 ) ij Propagators, spectral function G R/A ( t 1 , t 2 ) = ± θ [ ± ( t 1 − t 2 )] { G > ( t 1 , t 2 ) − G < ( t 1 , t 2 ) } Correlation functions G ≷ obey real-time KBE � � d t 3 Σ R ( t 1 , t 3 ) G < ( t 3 , t 2 ) + d t 3 Σ < ( t 1 , t 3 ) G A ( t 3 , t 2 ) [i ∂ t 1 − h 0 ( t 1 )] G < ( t 1 , t 2 ) = � � d t 3 G R ( t 1 , t 3 )Σ < ( t 3 , t 2 ) + d t 3 Σ A ( t 1 , t 3 ) G < ( t 3 , t 2 ) G < ( t 1 , t 2 ) [ − i ∂ t 2 − h 0 ( t 2 )] = M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 11 / 57
Information in the Nonequilibrium Green functions Time-dependent single-particle operator expectation value � � � � ˆ o ( x ′ t ) G < ( xt , x ′ t ) O � ( t ) = ∓ i dx x = x ′ Particle density Density matrix � n ( x , t ) � = n (1) = ∓ i G < (1 , 1) � ˆ ρ ( x 1 , x ′ 1 , t ) = ∓ i G < (1 , 1 ′ ) � t 1 = t ′ 1 �� � � 2 i − ∇ 1 ′ � ˆ ∇ 1 G < (1 , 1 ′ ) Current density: j (1) � = ∓ i 2 i + A (1) 1 ′ =1 Interaction energy (two-particle observable, [Baym/Kadanoff]) � � � ( i ∂ t − i ∂ t ′ ) − p 2 V 12 � ( t ) = ± i V d � p � ˆ p , t , t ′ ) | t = t ′ G < ( � 4 (2 π � ) 3 m M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 12 / 57
Numerical solution of the KBE Full two-time solutions: Danielewicz, Schäfer, Köhler/Kwong, Bonitz/Semkat, Haug, Jahnke, van Leeuwen, Stefanucci, Verdozzi, Berges, Garny, Balzer ... adiabatically slow switch-on of 2 interaction for t , t ′ ≤ t 0 [1, 2] 1 0 . 9 0 . 8 Switching function 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 t 0 0 . 2 0 . 1 0 0 5 10 15 20 25 30 35 40 45 50 Time t [arb.u.] solve KBE in t − t ′ plane for g ≷ ( t , t ′ ) Uncorrelated initial state 1 3 [1] A. Rios et al., Ann. Phys. 326 , 1274 (2011), [2] S. Hermanns et al., Phys. Scr. T151 , 014036 (2012) M. Bonitz (Kiel University) Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 13 / 57
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