Correlated Electron Dynamics and Nonequilibrium Dynamical Mean-Field Theory

Transcription

1 Correlated Electron Dynamics and Nonequilibrium Dynamical Mean-Field Theory Marcus Kollar Theoretische Physik III Electronic Correlations and Magnetism University of Augsburg Autumn School on Correlated Electrons: DMFT at 25: Infinite Dimensions Forschungszentrum Jülich, September 15-19, 2014 FOR 1346

2 Outline 1. Quantum many-body systems in nonequilibrium 2. Nonequilibrium Green functions 3. Nonequilibrium Dynamical Mean-Field Theory 4. Interaction quench in the Hubbard model Review on Nonequilibrium DMFT: H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Rev. Mod. Phys. 86, 779 (2014)

3 1. Quantum many-body systems in nonequilibrium

4 How to put a quantum many-body system out of equilibrium and observe its relaxation

5 Time-resolved pump-probe spectroscopy Pump-probe setup: Pump laser pulse: puts system into nonequilibrium Probe laser pulse: looks at system after delay time probe laser nonlinear crystal delay ~ fs pump Various time-resolved probes: detector sample t.-r. ARPES: photoemitted electrons t.-r. optical spectroscopy: transmitted/reflected light t.-r. X-ray or electron diffraction: snapshots of atomic positions

6 Melting of a Charge Density Wave in TbTe 3 Schmitt, Kirchmann, Bovensiepen, Moore, Rettig, Krenz, Chu, Ru, Perfetti, Lu, Wolf, Fisher, Shen, Science '08 trarpes on TbTe 3 : 1.5-eV 50-fs pump pulse, 6-eV 90-fs probe pulse photodoping! closing of CDW gap! electron thermalization! vibrational excitation

7 Quenched Bose condensate Abrupt increase of interaction of 87 Rb atoms: Greiner, Mandel, Hänsch, Bloch 02 ˆb kˆb k ψ(0) = Bose condensate t=0µs t=100µs U t=150µs t=250µs H U i ˆn 2 i t=350µs t=550µs t=400µs ψ(t) = e iĥt ψ(0) oscillates ˆb kˆb k Relaxation 'collapse and and revival' t

8 Time scales in nonequilibrium dynamics Time scales in pump-probe experiments Excitation due to pump pulse Relaxation due to electron scattering ~ fs ~ fs Energy transfer to ion lattice ~ ps Time scales in cold-atom experiments Switching times ~ ms Relaxation times ~ ms! study relaxation of isolated quantum systems first!

9 How can an isolated system relax to an equilibrium state?

10 Time evolution of isolated systems Schrödinger equation: i d (t)i = H(t) (t)i dt Quantum quench: Prepare 0i and switch to H at t = 0 Time evolution for t 0: (t)i = e iht 0 i = X! n!!!!! hn 0i e ie n Energy after quench: Expectation values: E = h (t) H (t)i = h 0 H 0i hai t = h (t) A (t)i Thermalization: hai t t!1 -! hai therm = Tr Ae H Tr e H? with effective from hhi therm = E

11 The thermal state: Putting a system into equilibrium by coupling it to a heat bath

12 Gibbs ensemble for system + bath Maxwell 1866, Boltzmann 1872, Gibbs 1878 System + heat bath: E tot = E s + E b = const Boltzmann relation: S = ln, = 1 Obtain # of system states from bath: system!(e s )!! ln b (E tot E s ) = ln b (E tot ) + (E tot E s ) +... bath!(e b )! P(E s ) / b (E tot E s ) / exp( E s ) System in thermal state when in equilibrium with bath Microcanonical ensemble gives same results (in thermodyn. limit)

13 An equivalent equilibrium formulation: Ensembles containing microstates with same a priori probabilities

14 Reformulation Equilibrium statistical with fundamental mechanics postulate Prediction for equilibrium state: Fundamental postulate: All accessible states equally probable S = Tr[ ˆρ ln ˆρ]= max  i conserved fix Tr[ ˆρ ensemble  i ] =  i t=0 ˆρ ensemble exp( i λ i  i ) Boltzmann-Gibbs ensemble Integrable systems: Ĥ eff = L α=1 ɛ α ˆn α Maxwell 1866, Boltzmann 1872, Gibbs 1878 von Neumann 1927, Jaynes 1957,..., Balian 1991 many constants of motion Generalized Gibbs ensembles: ˆρ GGE exp( α λ α ˆn α ) Jaynes 57 Girardeau 69 Rigol et al. 06 Cazalilla 06 Rigol et al. 07  t =  GGE for simple observables and initial states Kollar & Eckstein 08 Barthel & Schollwöck 08 von Neumann 1927, Jaynes 1957,... Balian 1991

15 Reformulation Equilibrium statistical with fundamental mechanics postulate Prediction for equilibrium state:! for!an!isolated!system! Fundamental postulate: All accessible states equally probable S = Tr[ ˆρ ln ˆρ]= max  i conserved fix Tr[ ˆρ ensemble  i ] =  i t=0 ˆρ ensemble exp( i λ i  i ) Boltzmann-Gibbs ensemble Integrable systems: Ĥ eff = L α=1 ɛ α ˆn α Maxwell 1866, Boltzmann 1872, Gibbs 1878 von Neumann 1927, Jaynes 1957,..., Balian 1991 many constants of motion Generalized Gibbs ensembles: ˆρ GGE exp( α λ α ˆn α ) Jaynes 57 Girardeau 69 Rigol et al. 06 Cazalilla 06 Rigol et al. 07  t =  GGE for simple observables and initial states Kollar & Eckstein 08 Barthel & Schollwöck 08 von Neumann 1927, Jaynes 1957,... Balian 1991

16 Why does a many-body system relax to the thermal state?

17 Eigenstate Thermalization Hypothesis (ETH) hn A ni A(E n ) + smaller, n-dep. terms Deutsch PRA '91, Srednicki PRE '94 Rigol, Dunjko, Olshanii, Nature '08 Nonintegrable Integrable

18 Eigenstate Thermalization Hypothesis (ETH) hn A ni A(E n ) + smaller, n-dep. terms Deutsch PRA '91, Srednicki PRE '94 Rigol, Dunjko, Olshanii, Nature '08 Energy dependence is that of microcanonical ensemble: A mic (E) = Tr[ mic (E n ) A] = 1 Z mic X E E<E n <E hn A ni = A(E) + smaller terms Long-time average tends to thermal value: A(t) = h (t) A (t)i = X n hn A ni {z } A mic (E n ) h (0) ni 2 {z } peaked at E A mic (E) ETH sufficient for thermalization!

19 Nonequilibrium: Thermalization is due to dependence of expectation values only on energy Equilibrium: Thermal Gibbs state is due to immersion in structureless heat bath

20 2. Nonequilibrium Green functions

21 Quantum time evolution Hamiltonian: Density matrix: Propagator: H (t) = H(t) (0) = 1 Z e (t) = U(t, 0) (0)U(0,t) Expectation value of observable A: µ N(t) H (0) = 1 Z d dt U(t, t0 ) = ih (t)u(t, t 0 ) 8 U(t, t 0 ) = >< T exp >: T exp i i Z t Z t X e n t 0 d t H ( t) t 0 d t H ( t) E n nihn!! for t>t 0 for t<t 0 h i hai t = Tr (t)a = 1 h i Z Tr U( i, 0)U(0, t) A U(t, 0)

22 Kadanoff-Baym formalism with time contour Expectation value of observable A: hai t = 1 Z Tr[U( i, 0)U(0, t) A U(t, 0)] Represent as integral over time contour C 1 + C 2 + C 3 : Im t C 3 C 1 t i C t Re t 2 t max 8 >< T exp i U(t, t 0 ) = >: T exp i Z t Z t t 0 d t H ( t) t 0 d t H ( t) Insert formal time dependence into Schrödinger operator A: hai t = Tr T C A(t) exp[ i R C d t H ( t)] Tr T C exp[ i R C d t H ( t)]!! for t>t 0 for t<t 0

23 Contour calculus 8 >< g + (t) if t 2 [0,t max ] on C 1, g(t 2 C) = g (t) if t 2 [0,t max ] on C 2, >: g ( i ) if t = i on C 3, 2 [0, ], Z Z tmax Z tmax Z dt g(t) = dt g + (t) dt g (t) i d g ( C 0 Z 0 0 [a b](t, t 0 ) = d t a(t, t)b( t,t 0 ), C i t g(t) = g(t ± ) t 2 C 1,2, i@ g( i ) t = i 2 C 3 C (t, t 0 ) = ( 1 for t>c t 0, 0 otherwise, Im t C 1 t Re t Z d t C C(t, t 0 ) t C (t, t 0 ), C (t, t)g( t) = g(t) C 3 i t C 2 t max

24 Contour Green functions Contour time ordering: ( AB if t>c t 0 Im t C 1 t Re t T C A(t)B(t 0 ) = Action: S = ±BA if t< C t 0 Z i C dt H (t) C 3 i t C 2 t max Green function with 2 time arguments on branches C 1 or C 2 or C 3 : G(t, t 0 ) = ihc(t)c (t 0 )i = i h oi Z Tr T C nexp(s)c(t)c (t 0 ) Let G ab (t, t 0 ) have time arguments on branches a, b = 1, 2, 3 Symmetries: G 11 (t, t 0 ) = G 12 (t, t 0 ) for t apple t 0 etc. G 13 (t, 0 ) = G 23 (t, 0 ),

25 Keldysh Green functions Retarded, Advanced, Keldysh, Mixed, Matsubara, lesser, greater GFs: G R (t, t 0 ) = 1 2 (G 11 G 12 + G 21 G 22 ) = i (t t 0 )h{c(t), c (t 0 )}i G A (t, t 0 ) = 1 2 (G 11 + G 12 G 21 G 22 ) = i (t 0 t)h{c(t), c (t 0 )}i G K (t, t 0 ) = 1 2 (G 11 + G 12 + G 21 + G 22 ) = ih[c(t), c (t 0 )]i G (t, 0 ) = 1 2 (G 13 + G 23 ) = ihc ( 0 )c(t)i G (,t 0 ) = 1 2 (G 31 + G 32 ) = ihc( )c (t 0 )i G M (, 0 ) = ig 33 = ht c( )c ( 0 )i Im t G < (t, t 0 ) = G 12 = i hc (t 0 )c(t)i G > (t, t 0 ) = G 21 = i hc(t)c (t 0 )i C 3 i C 1 t C 2 t t max Re t

26 Noninteracting case and Self-energy Free electrons: H 0 (t) = X k [ k (t) µ]c kc k Contour GF: G 0,k (t, t 0 ) = i hc k (t)c k(t 0 )i EOM: + µ k (t) G 0,k (t, t 0 ) = C (t, t 0 ) Def. of inverse GF: Solution: G 0,k (t, t 0 ) = G 1 0,k (t, t0 ) = t + µ k (t) C(t, t 0 ) h i C (t, t 0 ) f( k (0) µ) ie i R t t 0 d t [ k ( t) µ] Self-energy : 1-particle irreducible amputated Feynman diagrams Dyson equation for full GF: G = G 0 + G 0 G Def. of inverse of full GF: G 1 = G 1 0

27 3. Nonequilibrium Dynamical Mean-Field Theory

28 The DMFT philosophy Start from limit of infinite lattice dimension d!1 Scale the kinetic energy, i.e., NN hopping amplitude t ij / 1 p d Map lattice problem onto dynamic single-site problem with self-consistency condition [ e.g. single-impurity Anderson model (SIAM) ] and solve numerically Extend to e.g. finite d using clusters, dual fermions, dyn. vertex approx.,... magnetic phases, phonons,... input from density functional theory,...

29 The cavity method I Time-dep. Hubbard model: H(t) = X ij t ij (t) c i c j + U(t) X i n i" n i# Pick out single site i=0 from lattice action: S = S 0 + S + S (0), Z apple S 0 = i dt U(t)n 0" (t)n 0# (t) µ X n 0 (t), S = S (0) = Integrate out rest of lattice: Z = Tr Z i Z i C C C dt 2 4 X iî0, dt H (0) (t) n oi ht C exp(s 0 + S + S (0) ) h n oi = Tr 0 T C exp(s 0 )Tr rest exp( S + S (0) ) h oi = Tr 0 T C nexp(s 0 + S) Z S (0) 3 t i0 (t)c i (t)c 0 (t) + h.c. 5, S eff = S 0 + S Equil.: Georges et al. RMP 1996 Noneq.: Gramsch et al PRB 2014

30 The cavity method II Result of integration over lattice sites i 0: 1X X Z Z S = i dt 1... dtn C C n= Hybridization functions: 1... Power counting for d!1: 0 n 0 (t n 1,...,tn)c (t 1 )...c 0 0 n (tn) 0 0 (t n 1,...,tn) 0 = ( i)n 1 X n! 2 t 0i1 (t 1 ) t jn 0(tn)G 0 (0),c 0 i 1 1,...,j n n (t 1,...,tn) 0 i 1,...,j n cavity!green!func7on! (site!i=0!removed)! (t n 1,...,tn 0 ) / X i 1,...,j n {z } / d 2n t 0i1 (t 1 )...t jn 0(t 0 n ) {z } /( p d) 2n G (0),c 0 (i 1 1 ),...,(j n n ) (t 1,...,tn) 0 {z } /( p d) 2(2n 1) / 1 d n 1 Only one-particle Green functions (n=1) remain in hybridization!

31 The DMFT action Action for the cavity site i=0: S eff = Z i apple dt U(t)n " (t)n # (t) µ X n (t) C Z Z X i dt 1 dt 2 (t1,t 2 )c (t 1 )c (t 2 ) C C Hybridization function: (t, t 0 ) = X i,j t 0i (t)g (0),c ij (t, t 0 )t j0 (t 0 ) Self-consistency for NN hopping on the Bethe lattice t ij = v p Z (t, t 0 ) = v(t)g (t, t 0 )v(t 0 )

32 Local self-energy and self-consistency Lattice and impurity Green function: G ij (t, t 0 ) = i hc i (t)c j (t0 )i S G(t, t 0 ) = G 00 (t, t 0 ) Lattice and impurity self-energies: (G 1 lat ) ij(t, t 0 ) = [ ij (i@ t + µ) t ij (t)] C (t, t 0 ) ( lat ) ij (t, t 0 ) G 1 (t, t 0 ) = (i@ t + µ) C (t, t 0 ) (t, t 0 ) (t, t 0 ) For DMFT action: ( lat ) ij (t, t 0 ) = ij (t, t 0 ) local self-energy! Self-consistency conditions: lattice and impurity Dyson equation Z X h i dt 1 [ il (i@ t + µ) t il (t)] C (t, t 1 ) (t, t 1 ) G lj (t 1,t 0 ) = ij C (t, t 0 ) C Z l dt 1 [(i@ t + µ) C (t, t 1 ) (t, t 1 ) (t, t 1 )]G(t 1,t 0 ) = C (t, t 0 ) C

33 Solution of DMFT equations by iteration DMFT iteration: start from a hybridization function obtain impurity Green function G(t, t 0 ) = ihc(t)c (t 0 )i obtain self-energy from impurity Dyson equation obtain new local Green function G(t, t 0 ) from lattice Dyson eq. obtain new hybridization function Must solve Volterra-type integro-differential eqs. Can be implemented as time-propagation scheme see RMP 2014 & references

34 Real-time impurity solvers Many-body perturbation theory Weak-coupling perturbation theory [ sample code available as Supp.Mat. for RMP 86, 779 (2014) ] Strong-coupling perturbation theory Continuous-time Quantum Monte Carlo Hamiltonian-based methods / exact diagonalization Falicov-Kimball model see Lecture Notes for references

35 4. Interaction quench in the Hubbard model H = X ij t ij c i c j + U(t) X i n i " n i# {z } = X k k c k c k

36 Strong-coupling regime: collaps-and-revival oscillations

37 Interaction Hubbard interaction quench the quench: Hubbard Collapse model & revival Hubbard model in DMFT: (bandwidth = 4, density n = 1) Large interaction quench from! to U = 5 c kσ c kσ t n(ε,t) 1 Eckstein, Eckstein Kollar, et Werner al., PRL PRL '09, 09, PRB PRB '10 10 [CT-QMC] t time ε -2 momentum k Collapse-and-revival oscillations due to vicinity of atomic limit (U = )

38 Weak-coupling regime: metastable prethermalized state

39 Interaction Hubbard interaction quench the quench: Hubbard Prethermalization model Hubbard model in DMFT: (bandwidth = 4, density n = 1) Eckstein, Kollar, Werner PRL 09, PRB 10 c kσ c kσ t Small interaction quench from! to U = 2 n(ε,t) t 4 5 time ε -2 momentum k Slow relaxation: Prethermalization plateaus due to vicinity of free system (U = 0) Moeckel & Kehrein 08 Berges et al. PRL '04 Moeckel & Kehrein PRL '08

40 Hubbard interaction quench from 0 to U Eckstein, Kollar, Werner, PRL 09, PRB 10 Eckstein et al., PRL ' (a) U=0.5 U=1 (b) U=8 Double occupation d(t) 0.17 U=1.5 U=2 U= U=2.5 U=3 U=3.3 U=5 U=4 Fermi surface discontinuity n(t) (c) U=1.5 U=2 U=2.5 U= t U=0.5 U=1 4 (d) U=8 U=6 U=5 U= t U=3.3

41 Prethermalization regime Weak interactions: H 0 = X c c H = H 0 + gh 1, g 1 H 1 = X c c c c Unitary pert. theory: trafo, evolve, backtrafo Fermi surface discontinuity n(t) a) U=0.5 U=1 U=1.5 U=2 U=2.5 V Moeckel & Kehrein PRL '08 Eckstein et al. PRB '10 Kollar et al. PRB '11 hai t = hai 0 + g 2 F(W t) + O(g 3 ) t!1 -! A pretherm + g2 cos( t+ ) t "! "! "! "! #! const (W t) Prethermalization plateau described by generalized Gibbs ensemble with approximate constants of motion! GGE «/ exp X en wit

42 n n n Prethermalization in one and two dimensions Prethermalization less pronounced in low dimensions d = 1, U from 0 to 1 d = 2, U from 0 to 2 DMRG 2 DMFT a DCA Nc 2 DCA Nc 4 DCA Nc 8 DCA Nc DCA Nc 64 b tj n n n a Tsuji, Barmettler, Aoki, Werner, PRB '14 DMFT 2 DCA Nc 2 2 DCA Nc 4 4 DCA Nc 8 8 DCA Nc b Π 2, Π 2 2 DMFT Π, 0 DCA Nc DCA Nc tj

43 Prethermalization in one and two dimensions Prethermalization less pronounced in low dimensions short-time expansion, Hamerla and Uhrig, PRB '14 d = 1 and d = 2 d = 2 Fermi surface discontinuity n(t) U=0.1 W U=0.5 W U=0.7 W U=1.0 W 1d results t [1/W] f kf (t) [1/W 2 ] k F =(π,0) k F =(π/2,π/2) ln( t 2 ) ln( t 2 ) t [1/W] n(t) perturbative correction does not reach a plateau

44 Intermediate-coupling regime: fast thermalization

45 Interaction Hubbard interaction quench the quench: Hubbard Thermalization model Hubbard model in DMFT: (bandwidth = 4, density n = 1) Eckstein, Kollar, Werner PRL 09, PRB 10! Intermediate interaction quench from to U = 3.3 c kσ c kσ t n(ε,t) t time ε -2 thermal momentum distribution (T = 0.84) momentum k Fast thermalization at intermediate U: both prethermalization and oscillations disappear at U dyn 3.2V c U dyn c ~3.4 well-captured by time-dependent Gutzwiller approximation: Schiro & Fabrizio PRL 2010

46 Summary and Outlook

47 Summary and Outlook Thermalization of correlated systems in nonequilibrium Eigenstate thermalization hypothesis: Thermalization is due to energy eigenstates that contribute only according to their energy Thermalization does not occur in integrable systems Nonequilibrium dynamical mean-field theory Controlled approximation for nonequilibrium problems Many applications: quenches, pulses, periodic driving,... cluster extensions,... magnetic phases; phonons; bosons;...