Correla'ons within Non-equilibrium Greens Func'ons method Hossein - PowerPoint PPT Presentation

Correla'ons within Non-equilibrium Green’s Func'ons method Hossein Mahzoon MSU Pawel Danielewicz Arnau Rios (University of surrey)

• Introduction to Non-Equilibrium Green’s functions (NEGF) • Applications of NEGF • Infinite nuclear matter • Finite system

Why NEGF • Evolution of correlated/uncorrelated quantum many-body systems can be described in a consistent way in NEGF formalism A Φ ( x 1 ...x A ; t ) = 1 • TDHF : X Y ( − 1) sgn σ φ α ( x sgn σ , t ) A ! α =1 σ ∂ 2 ⇢ � − 1 i ∂ ∂ t φ α ( x, t ) = ∂ x 2 + U ( x ) φ α ( x, t ) 2 m • limitations on allowed excitations The validity of TDHF requires a negligible role played by correlations in the dynamics • NEGF is suitable for central reactions due to averaging over more than one-body effect

The Contour h O H ( t ) i = h U ( t 0 , t ) O I ( t ) U ( t, t 0 ) i R t R t D T a h ⇣ ⌘i O I ( t ) T c h ⇣ ⌘iE = exp t 0 d τ H ( τ ) exp t 0 d τ H ( τ ) − i − i R t U ( t 0 , t ) = T a h ⇣ ⌘i exp t 0 d τ H ( τ ) where t > t 0 i introducing a contour running along the time and a T operator ordering along the contour. t 0 t t 0 P. Danielewicz: Annals of physics 152. 239-304(1984)

Kadanoff-Baym Equations a † (1)ˆ G < ( x 1 , t 1 ; x 1 0 , t 1 0 ) ! G < (1 , 1 0 ) = i h ˆ a (1 0 ) i a † (1 0 ) i G > ( x 1 , t 1 ; x 1 0 , t 1 0 ) ! G > (1 , 1 0 ) = � i h ˆ a (1)ˆ + ~ 2 ∂ 2  � i ~ ∂ Z G ? = 1 Σ HF (1¯ 1) G ? (¯ 11 0 ) dx ¯ ∂ x 2 2 m ∂ t 1 1 Z t 1 Z t 1 0 d ¯ Σ > (1¯ 1) − Σ < (1¯ G ? (¯ d ¯ 1 Σ ? (1¯ G > (¯ 11 0 ) − G < (¯ ⇥ ⇤ ⇥ ⇤ 11 0 ) − 11 0 ) + 1 1) 1) t 0 t 0 + ~ 2 ∂ 2  � − i ~ ∂ Z G ? = 1 Σ HF (1¯ 1) G ? (¯ 11 0 ) dx ¯ ∂ x 0 2 ∂ t 0 2 m 1 1 Z t 1 Z t 1 0 d ¯ G > (1¯ 1) − G < (1¯ Σ ? (¯ d ¯ 1 G ? (1¯ Σ > (¯ 11 0 ) − Σ < (¯ ⇥ ⇤ ⇥ ⇤ 11 0 ) − 11 0 ) + 1 1) 1) t 0 t 0

Kadanoff-Baym Equations + ~ 2 ∂ 2  � i ~ ∂ Z G ? = 1 Σ HF (1¯ 1) G ? (¯ 11 0 ) dx ¯ ∂ x 2 2 m ∂ t 1 1 Σ HF Z t 1 Z t 1 0 d ¯ Σ > (1¯ 1) − Σ < (1¯ G ? (¯ d ¯ 1 Σ ? (1¯ G > (¯ 11 0 ) − G < (¯ ⇥ ⇤ ⇥ ⇤ 11 0 ) − 11 0 ) + 1 1) 1) t 0 t 0 Σ ?

HF approximation • In HF approximation : Σ HF (12) = δ ( t 1 − t 2 ) Σ HF ( x 1 , x 2 ) • KB equations reduces to: ∂ 2 ∂ 2  − 1 ∂ x 2 + U ( x, t ) + 1 � i ∂ ∂ tG < ( x, x 0 ; t ) = G < ( x, x 0 ; t ) ∂ x 0 2 − U ( x 0 , t ) 2 m 2 m ρ ( x, x 0 ; t ) = − iG < ( x, t ; x 0 , t )

Adiabatically switching • Adiabatic switching ( 1 , t → −∞ H ( t ) = F ( t ) H 0 + [1 − F ( t )] H 1 F ( t ) = 0 , t → t i 1 F ( t ) = f ( t ) − f ( t f ) f ( t ) = 1 + e t/ τ f ( t i ) − f ( t f ) • Preparing the initial state H 0 = 1 2 kx 2 H 1 = U mf U mf ( x ) = 3 4 t 0 n ( x ) + 2 + σ 16 t 3 [ n ( x )] σ +1

Switching function time [fm/c] M. Watanaba et all, PRL 65,no. 26, page3301

Collision of two slabs A. Rios et al :Annals of Physics 326 (2011) 1274

Correlations • Equation incorporating the interactions: Z dp 1 dp 2 Σ ? ( p, t ; p 0 , t 0 ) = 2 π V ( p − p 1 ) V ( p 0 − p 2 ) G ? ( p 1 , t ; p 2 , t 0 ) Π ? ( p − p 1 , t ; p 0 − p 2 , t 0 ) 2 π Z dp 1 dp 2 Π ? ( p, t ; p 0 , t 0 ) = 2 π G ? ( p 1 , t ; p 2 , t 0 ) G ? ( p 2 − p 0 , t 0 ; p 1 − p, t ) 2 π 1 − 2 x 2 ✓ ◆ e − x 2 √ π ( η p ) 2 e − ( η p )2 V ( x ) = V 0 η 2 V ( p ) = V 0 4 η 2 The parameters are chosen to result reasonable physical quantities such as depletion number A. Rios et al :Annals of Physics 326 (2011) 1274

infinite nuclear matter Energy/particle 0 ∆ E corr V old − 5 20 Total Energy V new = V old − ∆ E corr Kinetic energy − 10 MF Correlaton Energy − 15 0 E/A [MeV] E/A [MeV] − 20 − 25 -20 − 30 − 35 -40 − 40 − 45 0 10 20 30 40 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 time [fm/c] n [fm − 3 ] Density in coordinate space +

Density in momentum space k [fm -1 ]

EOS in infinite nuclear matter 0 V old mf U old mf − 10 Etot − 20 E/A [MeV] − 30 − 40 − 50 − 60 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 n [fm − 3 ] Density in coordinate space

Finite nuclear matter • Starting from harmonic oscillator Hamiltonian • Adiabatically switching on mean-field and correlations • Technicalities: – Setting cut-off for energy ( dx ) and finding the appropriate dt – Starting from different initial ω HO – Friction term

Solving two-time equations t 2 G < ( t 1 , T + ∆ t ) t 2 T G > ( T + ∆ t, t 2 ) t 1 t 1 t 0 T Using symmetries: G 7 (1 , 2) = − [ G 7 (2 , 1)] ∗ G < ( t 1 , T + ∆ t ) = G < ( t 1 , T ) e i ε ∆ t − I < 1 − e i ε ∆ t � 2 ( t 1 , T ) ε − 1 � G > ( T + ∆ t, t 2 ) = e i ε ∆ t G > ( T, t 2 ) − 1 − e − i ε ∆ t � ε − 1 I > � 1 ( T, t 2 ) G < ( T + ∆ T, T + ∆ T )

Different starting points 40 ω 1.5 ω 30 2 ω 2.5 ω 20 E/A [MeV] 10 0 -10 -20 -30 0 20 40 60 80 100 time [fm/c] Starting from different frequencies, energy arrives to the same final value

Observables and central density 0.4 2 ω ω 1.5 ω 1.5 ω 1.8 0.35 2 ω 2 ω 2.5 ω 2.5 ω 1.6 0.3 <x> [fm] -3 ] 1.4 n [fm 0.25 1.2 0.2 1 0.15 0.8 0.1 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 160 180 200 time [fm/c] time [fm/c] • Comparing the time evolution of central density (in coordinate space) and the size of the system, for different initial cases, • They all converge to the same final value

Density Density n(x) [fm -3 ] x [fm] Time evolution of the density in the coordinate space,

Friction term • A time-dependent external potential A. Bulgac et. al U t ≡ U t ( x ) hNps://arxiv.org/abs/1305.6891 • As long as , U t ∝ ˙ ρ the local quantum friction potential cools the system c ˙ ρ • The friction term can be implemented in both momentum and coordinate space

Effect of friction term 0.35 friction No friction 0.3 -3 ] ρ (x=0) [fm 0.25 0.2 0.15 0 20 40 60 80 100 120 time [fm/c]

Effect of friction term 2 friction No friction 1.8 1.6 <x> [fm] 1.4 1.2 1 0 20 40 60 80 100 120 time [fm/c]

occupation number Occupation number Density n(x) [fm -3 ] x [fm]

What is next • Including isospin dependency in the formalism • Performing the collision of slabs

Thanks!

Occupation number 1.2 "nofric_w_N1_moreaccurate/ocnum.dat" 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20

Application: Metal Oxide Semiconductors(MOS) • The quantitative simulation tools for the new generation of devices will require atomic-level quantum mechanical models . • The NEGF provides a conceptual basis for this new simulators Fermi levels µ 2 µ 1 • The device is driven out of equilibrium by two contacts with different Fermi levels Device • NGF can be used to determine the density matrix contact2 contact1 Supriyo Datta : Superlattices and Microstructures, Vol. 28, No. 4, 2000

Scale ∆ ⌧ = ~ ✏ τ f τ f = ~ ∆ τ Γ Γ = ~ n σ v ∼ 50 MeV The energy, , is of the same order ✏