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Many-particle systems far from equilibrium from Green functions to stochastic dynamics Michael Bonitz, Sebastian Hermanns, Christopher Hinz, Niklas Schlnzen and Denis Lacroix Institut fr Theoretische Physik und Astrophysik

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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

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Acknowledgements1

1thanks also to: D. Hochstuhl (MCTDHF, CI) and K. Balzer (NEGF)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 2 / 57

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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

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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

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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

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Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 6 / 57

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Quantum dynamics in second quantization (2)

Dynamics of the field operators

use Heisenberg representation of quantum mechanics: ˆ ciH(t) = U †(t, t0)ˆ ciU(t, t0)

with N-particle time evolution operator:

i∂tU(t, t′) = ˆ H(t)U(t, t′), and U(t, t) = ˆ 1 Heisenberg equation of motion: i∂t ˆ ciH(t) + [ ˆ HH(t), ˆ ciH(t)] = 0, ˆ ciH(t0) = ˆ ci

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 7 / 57

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Quantum dynamics in second quantization (2)

Dynamics of the field operators

use Heisenberg representation of quantum mechanics: ˆ ciH(t) = U †(t, t0)ˆ ciU(t, t0)

with N-particle time evolution operator:

i∂tU(t, t′) = ˆ H(t)U(t, t′), and U(t, t) = ˆ 1 Heisenberg equation of motion: i∂t ˆ ciH(t) + [ ˆ HH(t), ˆ ciH(t)] = 0, ˆ ciH(t0) = ˆ ci evaluate commutator: i∂t ˆ ciH(t) =

  • m
  • h0

im + vim,H(t)

  • ˆ

cmH +

  • mln

wilmn ˆ c†

lH ˆ

cnH ˆ cmH Effective single-particle (mean field) problem, nonlinear: i∂t ˆ ciH(t) =

  • m
  • h0

im + ˆ

veff

im,H(t)

  • ˆ

cmH(t)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 8 / 57

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How to proceed? Simple equation for a complicated object

i∂t ˆ ciH(t) =

  • m
  • h0

im + ˆ

veff

im,H(t)

  • ˆ

cmH(t)

Ensemble average

  • I. coordinate representation: replace ˆ

ψH(r, t) → ψ(r, t)

“quasi-classical” approximation (many particles in single state) Gross-Pitaevskii-type equation (bosons)

  • II. Fermions: ni = 0, 1, “quantum” treatment necessary.

Ensemble average: ˆ ciH = 0, ˆ c†

iH ˆ

cjH = ρij(t) = i|ˆ ρ1(t)|j Reduced density operators: ˆ c†

i1 . . . ˆ

c†

is ˆ

cjs . . . ˆ cj1 → ˆ ρ1...s(t) Equations of motion: BBGKY hierarchya

  • III. Ensemble average of two(many)-time operator products:

Nonequilibrium Green functions ˆ c†

H(t)ˆ

cH(t′) → G(1)(t, t′)

  • aM. Bonitz, Quantum Kinetic Theory, Teubner 1998
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 9 / 57

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Keldysh Green functions

[Keldysh, 1964] time-ordered one-particle Nonequilibrium Green function, two times z, z′ ∈ C (“Keldysh contour”), arbitrary one-particle basis |φi G(1)

ij (z, z′) = i

  • ˆ

T C ˆ ci(z)ˆ c†

j (z′)

  • Keldysh–Kadanoff–Baym equation (KBE) on C:
  • k
  • i ∂

∂z δik − hik(z)

  • G(1)

kj (z, z′) = δC(z, z′)δij − i

  • klm
  • C

d¯ z wiklm(z+, ¯ z)G(2)

lmjk(z¯

z; z′¯ z+) KBE: first equation of Martin–Schwinger hierarchy for G(1), G(2) . . . G(n)

  • C wG(2) →

C ΣG(1),

Selfenergy Nonequilibrium diagram technique Example: Hartree-Fock + Second Born selfenergy

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 10 / 57

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Real-time Dyson equation/ KBE

Contour Green function mapped to real-time matrix Green function Gij =

  • GR

ij

G<

ij

GA

ij

  • G<

ij (t1, t2) = ∓i

  • ˆ

c†

j (t2)ˆ

ci(t1)

  • G>

ij (t1, t2) = −i

  • ˆ

ci(t1)ˆ c†

j (t2)

  • Propagators, spectral function

GR/A(t1, t2) = ±θ [±(t1 − t2)] {G>(t1, t2) − G<(t1, t2)} Correlation functions G≷ obey real-time KBE

[i∂t1 − h0(t1)] G<(t1, t2) =

  • dt3 ΣR(t1, t3)G<(t3, t2) +
  • dt3 Σ<(t1, t3)GA(t3, t2)

G<(t1, t2) [−i∂t2 − h0(t2)] =

  • dt3 GR(t1, t3)Σ<(t3, t2) +
  • dt3 ΣA(t1, t3)G<(t3, t2)
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 11 / 57

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Information in the Nonequilibrium Green functions

Time-dependent single-particle operator expectation value ˆ O(t) = ∓ i

  • dx
  • (x′t) G<(xt, x′t)
  • x=x′

Particle density ˆ n(x, t) = n(1) = ∓ i G<(1, 1) Density matrix ρ(x1, x′

1, t) = ∓ i G<(1, 1′)

  • t1=t′

1

Current density: ˆ j(1) = ∓ i

  • ∇1

2i − ∇1′ 2i + A(1)

  • G<(1, 1′)
  • 1′=1

Interaction energy (two-particle observable, [Baym/Kadanoff]) ˆ V12(t) = ± i V 4

  • d

p (2π)3

  • (i ∂t − i ∂t′) − p2

m

  • G<(

p, t, t′)|t=t′

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 12 / 57

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Numerical solution of the KBE

Full two-time solutions: Danielewicz, Schäfer, Köhler/Kwong, Bonitz/Semkat,

Haug, Jahnke, van Leeuwen, Stefanucci, Verdozzi, Berges, Garny, Balzer ...

1

Uncorrelated initial state

2

adiabatically slow switch-on of interaction for t, t′ ≤ t0 [1, 2]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45 50 Switching function Time t [arb.u.] t0

3

solve KBE in t − t′ plane for g≷(t, t′)

[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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Two-time simulations: Summary

1

perfect conservation of total energy

2

accurate short-time dynamics: phase 1: correlation dynamics 2: relaxation of orbital occupations Example: electrons in dense hydrogen, interaction quench [1]

3

accurate long-time behavior: spectral functions and high-order correlated spectra from real-time KBE dynamics (via Fourier transform) [2]

4

extended to optical absorption, double excitations [3] etc.

[1] MB and D. Semkat, Introduction to Computational Methods in Many-Body Physics, Rinton Press 2006, [2] N. Kwong and MB, PRL 84, 1768 (2000), [3] K. Balzer, S. Hermanns, MB, EPL 98, 67002 (2012)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 14 / 57

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NEGF for finite inhomogeneous systems: molecules2

few-electron atoms, molecules [PRA 81, 022510 (2010), PRA 82, 033427 (2010)]

1D He ground state

LiH, XUV-pulse excitation

2pioneered by N.E. Dahlen, R. van Leeuwen and K. Balzer

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 15 / 57

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SLIDE 16

Challenges of inhomogeneous NEGF calculations

Complicated structure of interaction wklmn and selfenergy Σ Collision intergrals involve integrations over whole past CPU time ~N 3

t , RAM ~N 2 t

Typical computational parameters

Spatial basis size: Nb = 70 Time steps: Nt = 10000 RAM consumption: 2 TB number of CPUs used: 2048 total computation time: 2-3 days Solutions3 Finite-Element Discrete Variable Representation [PRA 81, 022510 (2010)] Generalized Kadanoff–Baym ansatz [Phys. Scr. T151, 014036 (’12), JPCS 427, 012006 (’13)] Adiabatic switch-on of interaction [Phys. Scr. T151, 014036 (’12)] Parallelization [PRA 82, 033427 (2010)] and GPU computing

  • 3K. Balzer, M. Bonitz, Lecture Notes in Phys. vol. 867 (2013)
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 16 / 57

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SLIDE 17

Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 17 / 57

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SLIDE 18

To save CPU time and memory: Reduction to single-time propagation

recall TD equilibrium: G(p, ω) = A(p, ω)f (p) (“KB ansatz”) Generalize to non-equilibrium: ω → τ = t − t′, (Fourier trafo) new: dependence on T = t+t′

2

straightforward extension of KBA: G(p, τ; T) = A(p, τ; T)f (p, T)

But: this is wrong

violates energy conservation violates causality in contradiction to (single-time) density matrix theorya

  • aM. Bonitz, Quantum Kinetic Theory
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 18 / 57

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SLIDE 19

The generalized Kadanoff-Baym ansatz

Idea of the GKBA: lowest order solution [1] G≷

GKBA(t1, t2) = −GR(t1, t2) ρ≷(t2) + ρ≷(t1)GA(t1, t2)

ρ<(t) = ρ(t) = ±iG<(t, t), ρ>(t) = 1 ± ρ<(t) correct causal structure, non-Markovian, no near-equilibrium assumption [2]

[1] P. Lipavsky, V. Spicka and B. Velicky Phys. Rev. B 34, 6933 (1986), [2] M. Bonitz, Quantum Kinetic Theory

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 19 / 57

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SLIDE 20

The generalized Kadanoff-Baym ansatz

Idea of the GKBA: lowest order solution [1] G≷

GKBA(t1, t2) = −GR(t1, t2) ρ≷(t2) + ρ≷(t1)GA(t1, t2)

ρ<(t) = ρ(t) = ±iG<(t, t), ρ>(t) = 1 ± ρ<(t) correct causal structure, non-Markovian, no near-equilibrium assumption [2], Reduction to single-time quantities by use of HF propagators GR/A

HF (t1, t2) = ∓iθ[±(t1 − t2)] exp

  • −i

t1

t2 dt3 hHF(t3)

  • HF-GKBA: same conservation properties as two-time approximation

damped propagators, local approximation violate E-conservation [3]

[1] P. Lipavsky, V. Spicka and B. Velicky Phys. Rev. B 34, 6933 (1986), [2] M. Bonitz, Quantum Kinetic Theory [3] M. Bonitz, D. Semkat, H. Haug, Eur. Phys. J. B 9, 309 (1999)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 20 / 57

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SLIDE 21

Peformance gain with the GKBA

time stepping along diagonal

  • nly. Full memory retained.

100 101 102 103 time steps T 10−2 10−1 100 101 102 103 104 105 106 elapsed time [s] ∝ T 3 ∝ T 2

full 2B 2B-GKBA

  • S. Hermanns, K. Balzer, and M. Bonitz, Phys. Scripta T151, 014036 (2012)

we use about 103 time steps for the adiabatic switching and 103 . . . 106 for the excitation and relaxation.

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 21 / 57

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SLIDE 22

Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 22 / 57

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SLIDE 23

The Hubbard model

Simple, but versatile model for solid state systems

  • ptical lattices, macromolecules...

single band, small bandwidth, parameters from ab initio simulations

⇒

ˆ H(t) = −J

ij, α hij ˆ

c†

iαˆ

cjα + U

i ˆ

c†

i↑ˆ

ci↑ˆ c†

i↓ˆ

ci↓ +

ij,αβ fij,αβ(t) ˆ

c†

iαˆ

cjβ hij = δi, j and δi, j = 1, if (i, j) is nearest neighbor, δi, j = 0 otherwise

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 23 / 57

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SLIDE 24
  • Noneq. initial state N = 8, half filling, U = 0.1

Sites 0 − 3 doubly occupied, 4 − 7 empty

20 40 60 80 100 time t [J−1] 0.0 0.5 1.0 1.5 2.0 density on site i = 0

2B-GKBA Exact HF

Rapid failure of HF (!), good performance of GKBA up to longer times (t ∼ 50) GKBA improves with particle number

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 24 / 57

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SLIDE 25

Long relaxation exact result vs. GKBA, N = 4, n = 1/2, U = 0.1

Sites 0 − 1 doubly occupied, 2 − 3 empty

100 200 300 400 500 time t [J−1] 0.0 0.5 1.0 1.5 2.0 density on site i = 0

2B-GKBA Exact

HF-GKBA: long-time stability, no divergencies. Qualitatively correct up to t ∼ 180

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 25 / 57

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SLIDE 26

Spectrum—from long real-time propagation

Response to weak short pulse4 ∼ δ(t) 10 . . . 1000 times longer propagation compared to two-time KBE Increased resolution of spectra. Capture double excitations

Real-time propagation following weak excitation and Fourier transform Example: N = 8, n = 1/2, U = 0.1

1 2 3 4 5 frequency ω [J] 10−9 10−7 10−5 10−3 10−1 spectrum [arb. u.]

2B-GKBA Exact HF

4Idea: Kwong, Bonitz, PRL 84, 1768 (2000)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 26 / 57

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SLIDE 27

Dynamics in larger inhomogeneous systems

N = 16, half filling, U = 0.1. Sites 0 − 7 doubly occupied, 8 − 15 empty

20 40 60 80 100 time t [J−1] 0.0 0.5 1.0 1.5 2.0 density on site i = 0

GKBA MCTDHF (M=10) HF

no FCI data, failure of HF (and MCTDHF), expect predictive capability of GKBA

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 27 / 57

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SLIDE 28

Fix problems of two-time calculations

Problems of NEGF in second Born5, N = 2, n = 1/2, U = 1 Strong excitation: fij,αβ(t) = w0δi,1δj,1δα,βΘ(t) , w0 = 5.0J −1

time-dependent density, KBE for various degrees of selfconsistency

  • artif. damping, mult. steady states
  • 5P. von Friesen, C. Verdozzi, and C.O. Almbladh, Phys. Rev. B (2010)
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 28 / 57

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SLIDE 29

Fix problems of two-time calculations

Problems of NEGF in second Born6, N = 2, n = 1/2, U = 1 Strong excitation: fij,αβ(t) = w0δi,1δj,1δα,βΘ(t) , w0 = 5.0J −1

time-dependent density, KBE for various degrees of selfconsistency

  • artif. damping, mult. steady states

GKBA: no damping selfconsistency problem cured

2 4 6 8 10 time t [J−1] 0.1 0.2 0.3 0.4 0.5 0.6 density on site i = 0

Exact GKBA

  • 6P. von Friesen, C. Verdozzi, and C.O. Almbladh, Phys. Rev. B (2010),
  • S. Hermanns, and M. Bonitz, Phys. Rev. B (2014), arXiv: 1402.7300
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 29 / 57

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SLIDE 30

Hartree-Fock-GKBA vs. 2-time NEGF

G = Gid + Gid

¯

ΣHF + ΣGKBA + ∆Σ

  • G,

GHF = Gid + Gid ¯ ΣHFGHF, GGKBA = GHF + GHFΣGKBAGGKBA, G = GGKBA + GGKBA∆Σ G.

ΣGKBA ≡ Σcor[f ≷, GR/A

HF ]

G: 2-time NEGF, contain in addition: ∆Σ: terms with 1...3 full propagators

  • S. Hermanns, and M. Bonitz, Phys. Rev. B (2014),

arXiv: 1402.7300

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 30 / 57

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SLIDE 31

Hartree-Fock-GKBA vs. 2-time NEGF

G = Gid + Gid

¯

ΣHF + ΣGKBA + ∆Σ

  • G,

GHF = Gid + Gid ¯ ΣHFGHF, GGKBA = GHF + GHFΣGKBAGGKBA, G = GGKBA + GGKBA∆Σ G.

ΣGKBA ≡ Σcor[f ≷, GR/A

HF ]

G: 2-time NEGF, contain in addition: ∆Σ: terms with 1...3 full propagators

  • S. Hermanns, and M. Bonitz, Phys. Rev. B (2014),

arXiv: 1402.7300

HF-GKBA reduces selfconsistency. Crucial for finite systems Not a weak coupling approximation. Applicable to arbitrary approximation for Σ

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 31 / 57

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SLIDE 32

Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 32 / 57

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SLIDE 33

Short-time dynamics – exact calculation, t = 0: sites 0 − 3 doubly occupied, 4 − 7 empty

0.0 0.5 1.0

n0

Ns = 8, N = 8 , U = 0.1

0.0 0.5 1.0

n7

0.0 0.5 1.0

n1

0.0 0.5 1.0

n6

0.0 0.5 1.0

n2

0.0 0.5 1.0

n5

20 40 60 80 100

time t [J−1]

0.0 0.5 1.0

n3

20 40 60 80 100 0.0 0.5 1.0

n4

  • Density wave to the right (diffusion)7
  • first: depopulation of n3 → n4(t) = 1 − n3(t)
  • delayed depopulation of n2, n1 (Pauli blocking)
  • decay of n0 when wave reflected at right boundary, n7(t) = 1 − n0(t)
  • interferences, relaxation, revivals. Systematics? time scales? pre-thermalization?

(Berges, Eckstein, Kehrein, ...)

  • 7S. Hermanns, and M. Bonitz, PRB (2014), arXiv: 1402.7300
  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 33 / 57

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SLIDE 34

Short-time dynamics: four stages exact calculation, N = 8, n = 1/2, U = 0.1

Sites 0 − 3 doubly occupied, 4 − 7 empty

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Energies [J]

Etot Ekin EHF Ecorr

0.0 0.5 1.0 n0 0.0 0.5 1.0 n7 0.0 0.5 1.0 n1 0.0 0.5 1.0 n6 0.0 0.5 1.0 n2 0.0 0.5 1.0 n5 time t [J−1] 0.0 0.5 1.0 n3 20 40 60 80 100 0.0 0.5 1.0 n4

I: t ≤ 3, ballistic expansion (feature of inhomogeneity) II: t ≤ τcor ∼ 40, correlation build-up/saturation of HF energy III: t ≤ 50, one-particle equilibration (occupations) IV: t ≥ 50, weak revivals of occupations,

PRB (2014), arXiv: 1402.7300

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 34 / 57

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SLIDE 35

Short-time dynamics (U = 1.0): four stages exact calculation: N = 8, n = 1/2

Sites 0 − 3 doubly occupied, 4 − 7 empty

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Energies [J]

Ns = 8, N = 8 , U = 1.0

Ekin EHF Ecorr 0.0 0.5 1.0

n0

0.0 0.5 1.0

n1

0.0 0.5 1.0

n2

20 40 60 80 100

time t [J−1]

0.0 0.5 1.0

n3

I: t ≤ 3, ballistic expansion II: t ≤ τcor ∼ 10, with τcor ∼ 1/U

[MB, and D. Kremp, Phys. Lett. A 212, 83 (1996)]

III: t ≤ 50, one-particle equilibration (occupations) IV: t ≥ 50, weak revivals of occupations,

PRB (2014), arXiv: 1402.7300

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 35 / 57

slide-36
SLIDE 36

GKBA calculation: Nonequilibrium initial state N = 16, n = 1/2, U = 0.25

Sites 0 − 7 doubly occupied, 8 − 15 empty

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Energies [J]

Etot Ekin EHF Ecorr

0.0 0.5 1.0 n0 0.0 0.5 1.0 n7 0.0 0.5 1.0 n1 0.0 0.5 1.0 n6 0.0 0.5 1.0 n2 0.0 0.5 1.0 n5 time t [J−1] 0.0 0.5 1.0 n3 20 40 60 80 100 0.0 0.5 1.0 n4

GKBA: correctly describes time-scales of stages I-III shows incorrect return to non-equilibrated state

PRB (2014), arXiv: 1402.7300

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 36 / 57

slide-37
SLIDE 37

Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 37 / 57

slide-38
SLIDE 38

T-Matrix selfenergy

ΣT

ik(t¯

t) = i

  • C

d¯ t1d¯ t2 T ±

ij′kl(t¯

t1,¯ t¯ t2)Glj′(¯ t2¯ t1) modified T-matrix T ± is connected to original T-matrix T via:

  • Tijkl(t1t2, t′

1t′ 2)

:= Tijkl(t1t2, t′

1t′ 2) ∓ wijkl(t1t2)δC(t1 − t′ 1)δC(t2 − t′ 2)

  • T ±

ijkl(t1t2, t′ 1t′ 2)

:=

  • Tijkl(t1t2, t′

1t′ 2) ±

Tijlk(t1t2, t′

2t′ 1)

Lippmann-Schwinger equation for T ± on Keldysh-Contour C:

  • T ±

ijkl(t1t2, t′ 1t′ 2)

= ±i wij¯

k¯ l(t1t2)G¯ km(t1t′ 1)G¯ ln(t2t′ 2)wmnkl(t′ 1t′ 2)

+ i wij¯

k¯ l(t1t2)G¯ km(t1t′ 2)G¯ ln(t2t′ 1)wmnlk(t′ 2t′ 1)

+ i

  • C

d¯ t1d¯ t2 wij¯

k¯ l(t1t2)G¯ km(t1¯

t1)G¯

ln(t2¯

t2) T ±

mnkl(¯

t1¯ t2, t′

1t′ 2)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 38 / 57

slide-39
SLIDE 39

Keldysh Matrix elements of GKBA-collision integral

I (1)≷

ij

(tt) =

t

d¯ t ΣTR

ik (t¯

t)G≷

kj(¯

tt) +

t

d¯ t ΣT≷

ik (t¯

t)GA

kj(¯

tt) → Keldysh components of T ± are required in case of fermionic Hubbard clusters these components become:

  • T −A

ij

(tt′) = i U(t)

  • G>

ij (tt′)G> ij (tt′) − G< ij (tt′)G< ij (tt′)

  • U(t′)

+ i U(t)

t′

t

d¯ t

  • G<

ik (t¯

t)G<

ik (t¯

t) − G>

ik (t¯

t)G>

ik (t¯

t)

  • T −A

kj (¯

tt′)

  • T −≷

ij

(tt′) = −i U(t)G≷

ij (tt′)G≷ ij (tt′)U(t′)

+ i U(t)

t

d¯ t

  • G>

ik (t¯

t)G>

ik (t¯

t) − G<

ik (t¯

t)G<

ik (t¯

t)

  • T −≷

kj

(¯ tt′) +

t′

d¯ t G≷

ik(t¯

t)G≷

ik(t¯

t) T −A

kj (¯

tt′)

  • G≷ reconstructed via HF-GKBA, recover density operator result

Kremp, Bonitz, Kraeft, Schlanges, Ann. Phys. 258, 320 (1997)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 39 / 57

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SLIDE 40

Nonequilibrium initial state N = 2, U = 1

T-matrix with HF-GKBA compared to two-time T-matrix result (Friesen et al.)

0.3 0.4 0.5 2 4 6 8 10 density on site i=0 time t/J-1 exact T (Friesen) GKBA+T U=1 w0=8

HF-GKBA removes artificial damping. Good agreement of main frequency Agreement improves for larger N ⇒ can access strong coupling (low n)!

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 40 / 57

slide-41
SLIDE 41

Outline

1

Introduction

2

Quantum dynamics in second quantization

  • 1. Dynamics of the field operators
  • 2. Non-equilibrium Green functions (NEGF)
  • 3. Generalized Kadanoff-Baym ansatz (GKBA)

3

Excitation dynamics in Hubbard nanoclusters

  • 1. Testing the GKBA
  • 2. Relaxation Dynamics
  • 3. Beyond weak coupling: T-matrix selfenergy with GKBA

4

Stochastic Mean Field Approach SMF–Numerical results

5

Conclusions

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 41 / 57

slide-42
SLIDE 42

How to proceed? Simple equation for a complicated object

i∂t ˆ ciH(t) =

  • m
  • h0

im + ˆ

veff

im,H(t)

  • ˆ

cmH(t)

Ensemble average ⇒ simple(r) objects with complicated equations

  • I. Fermions: ni = 0, 1, “quantum” treatment necessary.

Ensemble average: ˆ ciH = 0, ˆ c†

iH ˆ

cjH = ρij(t) = i|ˆ ρ1(t)|j Reduced density operators: ˆ c†

i1 . . . ˆ

c†

is ˆ

cjs . . . ˆ cj1 → ˆ ρ1...s(t) Equations of motion: BBGKY hierarchy

  • II. Ensemble average of two(many)-time operator products:

Nonequilibrium Green functions ˆ c†

H(t)ˆ

cH(t′) → G1(t, t′) Equations of motion: Martin-Schwinger hierarchy

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 42 / 57

slide-43
SLIDE 43

How to proceed? Simple equation for a complicated object

i∂t ˆ ciH(t) =

  • m
  • h0

im + ˆ

veff

im,H(t)

  • ˆ

cmH(t)

Ensemble average ⇒ simple(r) objects with complicated equations

  • I. Fermions: ni = 0, 1, “quantum” treatment necessary.

Ensemble average: ˆ ciH = 0, ˆ c†

iH ˆ

cjH = ρij(t) = i|ˆ ρ1(t)|j Reduced density operators: ˆ c†

i1 . . . ˆ

c†

is ˆ

cjs . . . ˆ cj1 → ˆ ρ1...s(t) Equations of motion: BBGKY hierarchy

  • II. Ensemble average of two(many)-time operator products:

Nonequilibrium Green functions ˆ c†

H(t)ˆ

cH(t′) → G1(t, t′) Equations of motion: Martin-Schwinger hierarchy

Simple equations for simple objects? Avoid the ensemble average!

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 43 / 57

slide-44
SLIDE 44

Statistical approaches versus Stochastic Mean Field

Dynamics of a correlated N-particle system with N-particle state |Ψ(t) ւ

1

Ensemble average

2

Many-body approximation

beyond mean-field (MF): Vxc[ρ], ρ12[ρ], Σ[G] etc. 3

ρ(t0)

4

i ˙

ρ = I MF[ρ] + I cor[ρ]

ց ց

1

Specify Gaussian ensemble

  • ¯

ρ, δρijδρkl

  • 3

Sample initial state (M realizations):

ˆ n(1)(t0) . . . ˆ n(M)(t0) 4

i ˙

ˆ n(n) = I MF[ˆ n(n)] ; n = 1 . . . M 5

Ensemble average ∀ t:

1 M

M

n=1 ˆ

n(n)(t)

ւ

density matrix ρ(t), observables A(t)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 44 / 57

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SLIDE 45

Stochastic concepts for quantum systems

  • Schwinger 1951: operator correlation function in QED
  • Klimontovich 1957: microscopic phase space density
  • Kadomtsev, Dubois: correlation functions of EM field

fluctuations

  • Gurevich, Kogan...: occupation number fluctuations
  • Balian, Veneroni: variational principles
  • Stochastic Schrödinger equation
  • Diagrammatic/real-time quantum Monte Carlo
  • ...

Goal here: stochastic approach to many correlated fermions in nonequilibrium that is, both, accurate and practically feasible

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 45 / 57

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SLIDE 46

Stochastic approach to correlated N-body dynamics8

Density matrix operator (not averaged): ˆ nij ≡ ˆ a†

i ˆ

aj → ˆ n Heisenberg dynamics: i∂tˆ nij(t) = U †(t, t0)[ˆ nij, ˆ H(t)]U(t, t0), ˆ nij(t0) = ˆ nij

U: time evolution operator, ˆ H(t) system hamiltonian in second quantization: ˆ H = ˆ T + ˆ V + ˆ W , ˆ T + ˆ U =

∞

  • i,j=1

ˆ a†

i (tij + vij(t)) ˆ

aj ˆ W = 1 2

∞

  • i,j,k,l=1

ˆ a†

i ˆ

a†

j wijkl ˆ

alˆ ak. Exact equation for density matrix operator: i∂t^ n(t) =

  • ^

n(t),

  • t∗ + v∗

H(t) + ^

U±

H(t)

  • ,

ˆ U ±

kj =

  • ln

wjnkl ± wjnlk 2 {ˆ nnl ∓ δln}

8Single-time version. Two-time version analogous

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 46 / 57

slide-47
SLIDE 47

Ensemble average. Correlations vs. fluctuations

Average, fluctuations and correlation functions: ˆ A ≡ A, δA = ˆ A − A ˆ Aˆ B = AB + δ ˆ Aδ ˆ B Ensemble average of equation of motion for ˆ n(t) : i∂tn(t) − n(t), t∗ + v∗

H(t) + U± H(t)

= δ^ n(t), δ ˆ U±

H(t)

≡ I(t) r.h.s.: collision integral (interactions beyond mean field, correlations) ⇒ determined by fluctuations of DM operator. Formal solution: n(t) = UHF†(t, t0) n0 UHF(t, t0) + nI(t), nI(t) = 1 i

t

t0

d¯ t U HF†(t,¯ t) I(¯ t) U HF(t,¯ t), i∂tU HF(t, t0) = h±

H(t)UHF(t, t0),

U HF(t, t) = 1. Stochastic solution: ˆ n(t) = lim

M→∞ M

  • n=1

ˆ n(n)(t), ˆ n(n)(t0) = ˆ n(n)

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 47 / 57

slide-48
SLIDE 48

Stochastic Mean Field9

Replace I by initial state fluctuations: I(t) → ˜ I(t0)δ(t − t0) n(t) = ˆ n(t) = UHF†(t, t0)

  • ˆ

n0 + ˆ ˜ I(t0)

  • U HF(t, t0)

pure Hartree-Fock dynamics from modified initial state. M random TDHF-trajectories: ˆ n(n)(t) = UHF†

(n) (t, t0) ˆ

n(n) UHF

(n) (t, t0),

n = 1 . . . M “Classical” average: ˆ n(t) = lim

M→∞ M

  • n=1

ˆ n(n)(t), ˆ n(n)(t0) = ˆ n(n) Select ˆ n(n) from Gaussian ensemble: n(n)

ij

= niδij δn(n)

ij δn(n) kl

= 1 2ni(1 − nj)δilδjk Simple TDHF-dynamics, correlations via efficient Monte Carlo sampling of trajectories

9Ayik 2008; Lacroix, Hermanns, Hinz and Bonitz, arXiv: 1403.5098

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 48 / 57

slide-49
SLIDE 49

Short-time dynamics: N = 8, n = 1/2, U = 0.1

0.0 0.5 1.0 1.5 2.0

density site 1

a) 20 40 60 80 100

time t [J−1]

0.0 0.5 1.0 1.5 2.0

density site 1

b)

Exact SMF NEGF TDHF

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 49 / 57

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SLIDE 50

Short-time dynamics: N = 16, n = 1/2, U = 0.1

20 40 60 80 100 time t [J−1] 0.0 0.5 1.0 1.5 2.0 density site 1

1D, n = 16, N = 16, U = 0.1 SMF 2B-GKBA TDHF

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 50 / 57

slide-51
SLIDE 51

Short-time dynamics: N = 8, n = 1/2, U ≤ 0.5

0.0 0.5 1.0 1.5 2.0

density site 1

U/J = 0.25 20 40 60 80 100

time t [J−1]

0.0 0.5 1.0 1.5 2.0

density site 1

U/J = 0.5

Exact SMF NEGF

Present SMF becomes worse for increasing U accurate for initial relaxation phase, t τcor ∼ 1/U, captures correlation buildup

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 51 / 57

slide-52
SLIDE 52

Short-time dynamics of long Hubbard chains

n = 1/2, N = 64, 256, 512, occupation of site N/4, U = 0.1

20 40 60 80 100 time T [t−1] 0.0 0.5 1.0 1.5 2.0 density site N/4

1D, U = 0.1 N=64 N=256 N=512

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 52 / 57

slide-53
SLIDE 53

Space resolved dynamics of long Hubbard chains

n = 1/2, N = 256, U = 0.1

20 40 60 80 100 time T [t−1] 0.0 0.5 1.0 1.5 2.0 density

1D, n = 256, N = 256, U = 0.1 site 1 site 64 site 128

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 53 / 57

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SLIDE 54

Dynamics of 3D Hubbard cluster, 4 × 4 × 4

Influence of dimensionality: 3D vs. 2D and 1D, U = 0.1

20 40 60 80 100

time t [J−1]

0.0 0.5 1.0 1.5 2.0 2.5

density

3D SMF TDHF

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 54 / 57

slide-55
SLIDE 55

Dynamics of 3D Hubbard cluster, 4 × 4 × 4

Influence of dimensionality: 3D vs. 2D and 1D, U = 0.1

20 40 60 80 100

time t [J−1]

0.0 0.5 1.0 1.5 2.0 2.5

density

1D 2D 3D

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 55 / 57

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SLIDE 56

Conclusions and Outlook

Correlated quantum systems in non-equilibrium – Goals: self-consistent description of correlation, exchange and nonlinear response to fields; short-time to long-time dynamics NEGF: can treat mixed and pure states, conserving

1 advantageous scaling with N (limitation: basis size) 2 GKBA ⇒ efficiency gain, no artificial damping 3 T-matrix selfenergy with GKBA: access to strong coupling

Stochastic Mean Field: Monte Carlo sampling of TDHF trajectories

1 highly efficient, large systems, arbitrary geometry

Response of finite Hubbard clusters to strong excitation

1 non-trivial dynamics of occupations, correlations, coherences 2 extension to materials via DMFT-type schemes

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 56 / 57

slide-57
SLIDE 57

Thank you for your attention!

References

  • M. Bonitz and D. Semkat, Introduction to Computational Methods in

Many-Body Physics, Rinton Press 2006

  • K. Balzer, and M. Bonitz, Lecture Notes in Physics 867 (2013)
  • D. Hochstuhl, C. Hinz, and M. Bonitz, EPJ-ST 223, 177-336 (2014)

www.itap.uni-kiel.de/theo-physik/bonitz/index.html

  • M. Bonitz (Kiel University)

Nonequilibrium Quantum dynamics Hamburg, 25/03 2014 57 / 57