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

• TDHF :
• 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

Φ(x1...xA; t) = 1 A! X

σ A

Y

α=1

(−1)sgnσφα(xsgnσ, t) i ∂ ∂tφα(x, t) = ⇢ − 1 2m ∂2 ∂x2 + U(x)

• φα(x, t)

The Contour

t0

t0

t

U(t0, t) = T a h exp ⇣ i R t

t0 dτH(τ)

⌘i

= D T a h exp ⇣ −i R t

t0 dτH(τ)

⌘i OI(t) T c h exp ⇣ −i R t

t0 dτH(τ)

⌘iE

hOH(t)i = hU(t0, t)OI(t)U(t, t0)i where

introducing a contour running along the time and a T operator

• rdering along the contour.
• P. Danielewicz: Annals of physics 152. 239-304(1984)

t > t0

Kadanoff-Baym Equations

 i~ ∂ ∂t1 + ~2 2m ∂2 ∂x2

1

• G? =

Z dx¯

1ΣHF (1¯

1)G?(¯ 110)

 −i~ ∂ ∂t0

1

+ ~2 2m ∂2 ∂x02

1

• G? =

Z dx¯

1ΣHF (1¯

1)G?(¯ 110) + Z t1

t0

d¯ 1 ⇥ G>(1¯ 1) − G<(1¯ 1) ⇤ Σ?(¯ 110) − Z t10

t0

d¯ 1G?(1¯ 1) ⇥ Σ>(¯ 110) − Σ<(¯ 110) ⇤ + Z t1

t0

d¯ 1 ⇥ Σ>(1¯ 1) − Σ<(1¯ 1) ⇤ G?(¯ 110) − Z t10

t0

d¯ 1Σ?(1¯ 1) ⇥ G>(¯ 110) − G<(¯ 110) ⇤

G<(x1, t1; x10, t10) ! G<(1, 10) = ihˆ a†(1)ˆ a(10)i G>(x1, t1; x10, t10) ! G>(1, 10) = ihˆ a(1)ˆ a†(10)i

Kadanoff-Baym Equations

ΣHF

Σ?

 i~ ∂ ∂t1 + ~2 2m ∂2 ∂x2

1

• G? =

Z dx¯

1ΣHF (1¯

1)G?(¯ 110)

+ Z t1

t0

d¯ 1 ⇥ Σ>(1¯ 1) − Σ<(1¯ 1) ⇤ G?(¯ 110) − Z t10

t0

d¯ 1Σ?(1¯ 1) ⇥ G>(¯ 110) − G<(¯ 110) ⇤

HF approximation

• In HF approximation:
• KB equations reduces to:

ΣHF (12) = δ(t1 − t2)ΣHF (x1, x2)

i ∂ ∂tG<(x, x0; t) =  − 1 2m ∂2 ∂x2 + U(x, t) + 1 2m ∂2 ∂x02 − U(x0, t)

• G<(x, x0; t)

ρ(x, x0; t) = −iG<(x, t; x0, t)

Adiabatically switching

• Adiabatic switching
• Preparing the initial state

H(t) = F(t)H0 + [1 − F(t)]H1

F(t) = ( 1, t → −∞ 0, t → ti

F(t) = f(t) − f(tf) f(ti) − f(tf)

f(t) = 1 1 + et/τ

H0 = 1 2kx2

H1 = Umf

Umf(x) = 3 4t0n(x) + 2 + σ 16 t3 [n(x)]σ+1

time [fm/c]

• M. Watanaba et all, PRL 65,no. 26, page3301

Switching function

Collision of two slabs

• A. Rios et al :Annals of Physics 326 (2011) 1274

Correlations

• Equation incorporating the interactions:

Π?(p, t; p0, t0) = Z dp1 2π dp2 2π G?(p1, t; p2, t0)G?(p2 − p0, t0; p1 − p, t)

V (p) = V0 √π(ηp)2e− (ηp)2

4

V (x) = V0 ✓ 1 − 2x2 η2 ◆ e− x2

η2

The parameters are chosen to result reasonable physical quantities such as depletion number

• A. Rios et al :Annals of Physics 326 (2011) 1274

Σ?(p, t; p0, t0) = Z dp1 2π dp2 2π V (p − p1)V (p0 − p2)G?(p1, t; p2, t0)Π?(p − p1, t; p0 − p2, t0)

infinite nuclear matter

Energy/particle

Density in coordinate space

+

0.00 0.05 0.10 0.15 0.20 0.25 0.30 n [fm−3] −45 −40 −35 −30 −25 −20 −15 −10 −5 E/A [MeV]

∆Ecorr Vold Vnew = Vold − ∆Ecorr

10 20 30 40 time [fm/c]

• 40
• 20

20 E/A [MeV] Total Energy Kinetic energy MF Correlaton Energy

Density in momentum space

k [fm-1 ]

EOS in infinite nuclear matter

0.00 0.05 0.10 0.15 0.20 0.25 0.30 n [fm−3] −60 −50 −40 −30 −20 −10 E/A [MeV]

Vold

mf

Uold

mf

Etot

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 – Friction term

ωHO

Solving two-time equations

G<(t1, T + ∆t) G>(T + ∆t, t2)

t2 t1

G7(1, 2) = −[G7(2, 1)]∗

Using symmetries:

G<(t1, T + ∆t) = G<(t1, T)eiε∆t − I<

2 (t1, T)ε−1

1 − eiε∆t G>(T + ∆t, t2) = eiε∆tG>(T, t2) −

• 1 − e−iε∆t

ε−1I>

1 (T, t2)

t2 t1

T

T

G<(T + ∆T, T + ∆T)

t0

Different starting points

20 40 60 80 100 time [fm/c]

• 30
• 20
• 10

10 20 30 40 E/A [MeV] ω 1.5ω 2ω 2.5ω

Starting from different frequencies, energy arrives to the same final value

Observables and central density

20 40 60 80 100 120 140 time [fm/c] 0.1 0.15 0.2 0.25 0.3 0.35 0.4 n [fm

• 3]

ω 1.5ω 2ω 2.5ω 20 40 60 80 100 120 140 160 180 200 time [fm/c] 0.8 1 1.2 1.4 1.6 1.8 2 <x> [fm] ω 1.5ω 2ω 2.5ω

• 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

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

Friction term

• A time-dependent external potential
• As long as ,

the local quantum friction potential cools the system

• The friction term can be implemented in both momentum and

coordinate space

Ut ≡ Ut(x)

• A. Bulgac et. al

hNps://arxiv.org/abs/1305.6891

Ut ∝ ˙ ρ

c ˙ ρ

Effect of friction term

20 40 60 80 100 120 time [fm/c] 0.15 0.2 0.25 0.3 0.35 ρ(x=0) [fm

• 3]

friction No friction

Effect of friction term

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

• ccupation 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

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

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

Device contact2 contact1

µ1

µ2

Fermi levels

• The device is driven out of

equilibrium by two contacts with different Fermi levels

• NGF can be used to determine

the density matrix

Supriyo Datta : Superlattices and Microstructures, Vol. 28, No. 4, 2000

Scale

∆τ τf

∆⌧ = ~ ✏

τf = ~ Γ

Γ = ~nσv ∼ 50MeV

The energy, , is of the same order

✏