Electron-electron and electron-phonon interactions in
time-dependent quantum transport:
Nonequilibrium many-body theory
Robert van Leeuwen
Niko Säkkinen
Department of Physics
Nanoscience Center
University of Jyväskylä
Finland
CECAM Lausanne, 9.11.2012
Overview
- Motivation
- Nonequilibrium Green’s functions and the Kadanoff-Baym equations
- Illustrative examples
- Electron-electron interactions
- Electron-vibron interactions (preliminary results)
The time-dependent quantum transport problem
Consider a molecule (or quantum dot) attached to leads
L
H LC
HCR
R
B.
Equation of
C
HCC
Problem:
HLL
HCL
HRC
HRR
FIG. 1: Sketch of the transport setup. The correlated central region (C) is coupled to semi-infinite left (L) and right
(R) tight-binding leads via tunneling Hamiltonians HαC and
HCα , α = L, R.
We assume the sy
librium at inverse te
described by Hamilt
tem is driven out of
we aim to study the
sity, current, etc.. In
namics in this syste
theory (for a review s
many-body effects in
Green’s function is
the contour-ordered
Calculate the time evolution of observables of this system when a
!
G
(z,
z
) =
rs
bias is applied.
−i
= −i
ation and annihilation operators respectively. The one-
where â and ↠are ei
Fundamental questions in quantum transport
- How long does it take before a steady state is reached (i.e what are the
switch times) and how does it depend on e-e and e-ph interactions?
- Are steady states always reached and are they unique ?
(bistability)
- What are the current and density distributions in the leads and the
molecule (i.e. where in the molecule does the current flow and can we
regulate this ) ?
- What is the influence of the contact region? (Image charge effect)
- Can we determine the level structure of the molecule from
transient spectroscopy?
Challenge for theoretical description, since we deal with :
- Open systems
- Nonequilibrium systems
- Electron-electron and electronic-vibrational interactions
electron-electron interactions
e-
e-
molecular vibrations
The Author
3,
2012
TimeNovember
evolution
of
a
many-body
November 3, 2012 system
The time-dependent electron-phonon Hamiltonian
Ĥ(t) =
Ĥph
Ĥel+(t)
+(t)
Ĥ+
Ĥ(t)
=+
Ĥph
Ĥel
Ĥe-ph
e-ph
=+
ĥ(t)
Ĥel (t)Ĥ=
ĥ(t)
Ŵ+ Ŵ
el (t)
Electron-phonon/vibron
! !
interactions
−
h(1))G(1,
2)
=
δ(1,
2)
+
d3
Σ[G,
D](1,
3)G(3,
2)
))G(1, 2) = δ(1, 2) + d3 Σ[G, D](1, 3)G(3, 2)
!
2
2
+ ω )D(1,
2) =energy
δ(1, 2)
t)D(1,
1
Kinetic
2) = δ(1, 2) +
+
!
d3 Π[G,
D](1, 3)D(3,interactions
2)
Electron-electron
d3 Π[G, D](1, 3)D(3, 2)
+
D(1,
2) = −i"TC [∆ûH (1)∆ûH (2)]#
time-dependent
D(1, 2) = −i"TC [∆ûH (1)∆ûH (2)]#
external potential
i = µi ti
(bias, gate voltage,
i = µlaser
i2ti etc.)
C (t)
−α(t−t0 )
2
−α(t−t0 )
# 2
e
$f (t)$C 2≤
sup e
$g(t )$
(t)
α t! ∈[t−α(t−t
t0 )
2
# 2
0)
0 ,t]
e
0
R
†
α
n̂i =Hĉi(2)]#
ĉi =
−i"TC [∆ûH (1)∆û
L
, 2) =
0
>
<
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1, 2)
ρ̂
=
2 1
t = −2 −β Ĥ 1 2
0
Tr
e
2
2
"
"
i = µ t$
t = −1 #
i i†
†
† is
C calculate the time-dependent
The
goal
expectation
of
Ĥ(t) = Ĥphvalues
+ Ĥel (t)
+ Ĥe-ph
û
n̂
â
â
−
g
= −(ĉ1 ĉ2 + ĉ2 ĉ1 ) + vδ(t)n̂
+
ω
i
i
i
1
0
" †
"i †
"
#=
Ô$
ρ̂i=1
V1,5LĤ=
Vω4,5R
=âi−0.5
observables
: 0= Tr
âi −
tÔ ĉi ĉj −i=1 ûi n̂i
Ĥel (t) = ĥ(t) + Ŵ
>
i
i
vii==
1.5
D (1, 2)
−i"∆û
(1)∆û!i,j"
!
H#
H (2)]#$
< Tr ρ̂ Ô† (t) (i∂t − h(1))G(1, 2) = δ(1, 2) + d3 Σ[G, D](1, 3)
<
#
Ô(t)$
=
1
TrW
(t1 ,Hût2i(2)∆û
)= âi H
D Im
(1, 2)
= −i"∆û
+(1)]#
âi
H
!
A(ω)/30
−β Ĥ0
−i[(t
†
0 −iβ)−t0 ]Ĥ0
2
2
e
=e
=
Û
(t
−
iβ,
t
)
−(∂
+
ω
)D(1, 2) = δ(1, 2) + d3 Π[G, D](1, 3)
n̂
=
ĉ
ĉ
0
0
t
i
i
1
i
ÔH (t) =
Û (t0 , t) Ô Û (t, t0 )
A(ω)
2 2) = −i"T [∆û (1)∆û (2)]#
2
ÔH (t) = "Û (t0 , t)Ô Û (t, t0 )
"
"
D(1,
C
H
H
"
†
†
†
µ
=
2.26
(t,
t
)
=
Ĥ(t)
Û
(t,
t
)
ûi n̂i
âi âi − g
Ĥ =i∂
−(ĉ
ĉ
+
ĉ
ĉ
)
+
vδ(t)n̂
+
ω
t Û
2
1
1
0
1
2
δni (t)
i = µi ti
χij"(t)
=
−β Ĥ0
i=1
i=1"
"
e
vÛ
j (t, t )Ĥ(t )
i∂t! Ûρ̂(t,
t
)
=
−
Ĥ0 = Ĥ(t0 )
initial
correlations
=D> (1,!2) = −i"∆û
H (1)∆ûH (2)]#
2 (t)
Tre−β Ĥ0
iωt
C
χ11 (ω)<Û
=(t,dt
χ11
(t)e
e−α(t−t0 ) $f (t)$2 ≤
sup e−α(t−t0 ) $g(t#
t)
=
1
D (1, 2) = −i"∆û (2)∆û (1)]#
−β Ĥ0
"e
Û (t, t ) =
!
H
t
−i[(t0 −iβ)−t0 ]Ĥ0
=e 1
T exp(−i
= Û (t0 −
) C 2 (t)
$fiβ,
(t)$t20≤
dτ Ĥ(τ ))
t!δni (t)
χij (t) =
α
H
vj Ĥ
−β
0
2
!
t
t! ∈[t0 ,t]
# α(t! −t0 ) −α(t! −t0 )
dt e
e
t0
#
2
−α(t! −t0 )
≤ C (t) sup $g(t )$ e
t ∈[t ,t]
Û (t0 − iβ, t0 ) =
! e
#
$
χ11 (ω) = dt χ11 (t)eiωt
Tr Û (t0 1− iβ, t0 )Û (t0 , t) Ô Û (t, t0 )
!
0
x ∈ [−1, 1]
!
"
t
t0
$g(t# )
# α(t! −t
dt e
#$
α(t−t0 )
α
≤e
t
"
"
i∂
Û
(t,
t
)
=
Ĥ(t)
Û
(t,
t
) to Ĥ0 on the contour ru
(t0 − iβ, t0 ) = e
, if we tdefine Ĥ(t) to be equal
time
contour
(L.V.Keldysh,
Sov.Phys.JETP20,
1018 (1965)
"
"our expression
"
mplexThe
time
plane.
We can
therefore
rewrite
for the expectat
i∂t! Û (t,Konstantinov
t ) = −Û (t,
t
)
Ĥ(t
)
and Perel’ , JETP 12,142(1961))
"
#
Û iβ,
(t, t)
Tr Û (t0 −
t0 =
)Û!1(t0 , t)ÔÛ (t, t0 )
t
"
#
"Ô# =
"
Û (t, t ) =
T
exp(−i
dτ
Ĥ(τ
))
Tr Û (t − iβ, t )
−β Ĥ0
0
t!
0
−β Ĥ0
Û
(t
−
iβ,
t
)
=
e
0
0
guments of the evolution operators
in
the
numerator of this$expression from
#
volves from t0 along the Tr
realÛtime
t after which the operator Ô act
(t0 −axis
iβ, tto
0 )Û (t0 , t) Ô Û (t, t0 )
e real axis from#Ô(t)$
time =
t to t0 and finally
parallel $
to the imaginary axis fro
#
r is displayed in Fig. 1. From this
Trobservation
Û (t0 − iβ, twe
0 ) see that we can write th
G(1, 2) =
†
<
−i#TC [ψ̂H (1)ψ̂H"(2)]$ = θ(t1 , t2 )G> (1, 2) + θ(t
,
t
)G
(1, 2)
2 1
#
$
†Ĥ(t̄)Ô(t)]
>
TC=
[exp(−i
d
t̄
GTr
(1, 2)
−i#ψ̂H (1)
ψ̂
C
H (2)$
"Ô(t)# =
"
#
†
<
G (1, 2)
Tr= i#
Ûψ̂(tH0(2)
− ψ̂
iβ,
t0 )
H (1)$
G(x1 t0 − iβ, 2) = −G(x1 t0 , 2)
volution operator on the contour as
G(1, x2 t0 ) = −G(1, x2 t0 − iβ)
!
†
†
2
G2 (1,Û2,(t3,
4) = (−i) #T C[ψ̂
H (1)ψ̂H (2)
exp(−i
dtψ̂Ĥ(t)).
H (3)ψ̂H (4)]$
0 − iβ, t0 ) = TC
!
t!
Û!(t,t tt" ) = TC exp(−i
dτ Ĥ(τ ))
"
t!
Û
(t,
t
dτ
Ĥ(τ
))
" ) = TC exp(−i
Û (t, t ) = TC exp(−i t!dτ Ĥ(τ ))
t! Û (t0 − iβ, t0 ) = e−β Ĥ0
Electron propagators
−β#Ĥ0
$
Û
(t
−
iβ,
t
)
=
e
−β
Ĥ0
0
0
Û (t0 − iβ, t0 ) = eTr Û (t − iβ, t )Û (t , t) Ô Û (t, t )
0
0$
0
0
##
$
# t0 )
$
#Ô(t)$
=
Tr
Û
(t
−
iβ,
t
)
Û
(t
,
t)
Ô
Û
(t,
0
0
0
We
function
as
:
Tr define
Û (t0 −the
iβ,Keldysh
t0 )Û (t0 , contour-ordered
t) Ô ÛTr(t, tÛ0 )(t0 − iβ,Green
t
)
0
#
$
Ô(t)$
=
#
$
(t)$ =
Tr ÛÛ(t(t0−−iβ,
iβ,†t t)0 )
Tr
G(1, 2) = −i#TC [ψ̂0H (1)ψ̂H0(2)]$ = θ(t1 , t2 )G> (1, 2) + θ(t2 , t1 )G< (1, 2)
†
>
<
†
>
<
#T
[
ψ̂
(1)
ψ̂
(2)]$
=
θ(t
,
t
)G
(1,
2)
+
θ(t
,
t
)G
(1,2)2)
†
> (1, 2) + θ(t ,2t )G
2
1
TCC[ψ̂HH(1)ψ̂HH(2)]$ = θ(t11, t2G
)G
(1,
(1, 2) = −i#ψ̂2H (1)
ψ̂
(2)$
1
H
†
>
†
> (1, 2) = −i#ψ̂H (1)ψ̂
†
G
(2)$
of a “particle” (added electron)
G (1, 2) = −i#ψ̂H (1)G
ψ̂<HH
(2)$
(1,
2) = Propagation
i#ψ̂H
(2)ψ̂H (1)$
†
<
† (2)
< (1, 2) = i#ψ̂
G
G(x
t(1)$
= −G(x1 t0of, 2)
1(1)$
0 − iβ, 2)
G (1, 2) = i#ψ̂HH(2)
ψ̂ψ̂HH
Propagation
a “hole” (removed electron)
G(x11tt00−−iβ,
iβ,2)2)==−G(x
−G(x
,)2)= −G(1, x2 t0 − iβ)
G(1,1xt102tt,002)
G(x
†
†
2
G(1,
x
t
)
=
−G(1,
x
t
−
iβ)
2
0
2
0
2, 3, 4)
2 (1,
G(1, x2 t0 )G=
−G(1,
x2=
t0 (−i)
− iβ)#TC [ψ̂H (1)ψ̂H (2)ψ̂H (3)ψ̂H (4)]$
Natural tool
in quantum transport;
†
†electrons are continuously added and
2
†
†
2
1,
4)==(−i)
(−i) #T
#TCC[ψ̂
[ψ̂HH(1)
(1)ψ̂ψ̂HH(2)
(2)ψ̂ψ̂H(3)
(3)ψ̂ψ̂H(4)]$
(4)]$
, 2,2,3,3,4)
H
H
removed from the central system
2
22
H(t) = Hph + Hel (t) + He-ph
Ĥel (t) = ĥ(t) + Ŵ
Ĥ
(t) = ĥ(t) + Ŵ
= Ĥ
Ĥph
+ Ĥ
Ĥel
(t)+
+Ĥ
Ĥe-ph
Ĥ(t) =
+
(t)
el
ph
el
e-ph
Phonon!propagators
!
1, 2) = δ(1, 2) +(i∂t1d3
D](1,
2) + d3 Σ[G, D](1, 3)G(3, 2)
−Σ[G,
h(1))G(1,
2)3)G(3,
= δ(1, 2)
Ĥel
(t) =
= ĥ(t)
ĥ(t)+
+Ŵ
Ŵ
Ĥ
el(t)
!!
!
!
2 2)
2 d3
the
Keldysh
contour-ordered
phonon
propagator
as
:
δ(1,
+
Σ[G,
D](1,
3)G(3,
2)
2)+=
=−(∂
δ(1,
2)
+
d3
Σ[G,
D](1,
3)G(3,
2)
+
ω
)D(1,
2)
=
δ(1,
2)
+
d3
Π[G,
D](1,
3)D(3,
2)
,−
2)h(1))G(1,
= We
δ(1, define
2)2)
d3
Π[G,
D](1,
3)D(3,
2)
t1
!!
>
<
>
<
D(1,
2)
=
−i"T
[∆û
(1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1, 2)
2
2
C+ (1,
H
HD](1,
1(1,
22)
2 1
1)∆û
(2)]#
=
θ(t
,
t
)D
2)
+
θ(t
,
t
)D
)D(1,
2)
=
δ(1,
2)
d3
Π[G,
3)D(3,
2)
+
ω
=
δ(1,
2)
+
d3
Π[G,
D](1,
3)D(3,
2)
H
1
2
2
1
t1
i = µi ti
>>
<<
i
=
µ
t
i
i
(1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1,
i"TC
[∆û
(1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1,2)2)
C
H
H
11 22
22 1 1
H
H
Ĥ0 = Ĥ(t0 )
(1, 2) = −i"∆ûH (1)∆û
ii =
iitt(2)]#
ii
= µµH
2 (t)
C
−α(t−t0 )
2
−α(t−t0 )
# 2
e
$f
(t)$
≤
sup
e
$g(t
)$
>
>(1, 2) =(2)∆û
−i"∆û
(1)∆û
(2)]#
(1, 2) =D
−i"∆û
(1)]#
H
H
HH (1)∆ûH (2)]# α t! ∈[t0 ,t]
D (1, 2)H= −i"∆û
Displacement correlation function
<
!
t
D
2)
Ĥ
=
D0<(1,
(1,Ĥ(t
2) =
=0 )−i"∆û
−i"∆ûHH2(2)∆û
(2)∆û2HH(1)]#
(1)]#
# α(t! −t0 ) −α(t! −t0 )
$f (t)$ ≤ C (t)
dt e
e
$g(t# )$2
2 (t)
Ĥ
Ĥ(t
t0
00 =
00))
Ĥ
=
Ĥ(t
C
! t
(t)$2 ≤
sup 2 e−α(t−t0 ) $g(t# )$2
! −t )
! −t )
2(t)
α
! ∈[tC
2
#
2
−α(t
#
α(t
−α(t−t
)
2
−α(t−t
)
#
2
0
0
C
(t)
0
0
t
,t]
−α(t−t0 )$f (t)$ 2 ≤ 0 ≤ C sup
−α(t−t
# 2
(t)
sup
$g(t
)$
e
dt
e
e
e
$g(t
)$
0)
e
$f (t)$ ≤ α ! supt! ∈[t
e ,t]
$g(t )$
t0
,t] 0
!Electrons
α tt!∈[t
∈[t00 ,t]
interact
via
phonon-propagators
t
"
#$
%
!
!
−t0 )
# 2
eα(t−t0 )
C 2 (t) 2 dt# e2 α(t !!−ttt0 ) e#−α(t
$g(t
)$
≤
α
α(t!!−t0 ) −α(t! −t
# 2
0)
!
$f
(t)$
dt
2t0≤ C 2(t)
#e α(t −t0 )e −α(t −t0 )$g(t )$
# 2
$f (t)$ ≤ C (t) t dt e
e
$g(t )$
0
x ∈ [−1, 1]
! t
t0
!!! t−t0 )
# 2 −α(t! −t0 )
#
α(t
2 )
! −te)
! −t
up
$g(t
)$
e
2
# 2 −α(tdt
α(t
0 v(x,tt) # =
0
sin
(πx) sin(ωt)
!
!
≤
C
dt
e
2 (t) sup $g(t #)$ 2e −α(t −t0 )
# α(t −t0 )
∈[t≤
t0
C (t) t! ∈[t
sup
$g(t
)$
e
dt
e
0 ,t]
,t]
t
1
0
0
"
#$
%
Brief
Article
The equation of motion for the propagators attain the form
Ĥ(t) = Ĥph + Ĥel (t) + Ĥe-ph
The ofAuthor
effect
e-e and e-ph interactions
on(t)
electronic
motion
Ĥ
=
ĥ(t)
+
Ŵ
el
November
3, 2012
!
(i∂t1 − h(1))G(1, 2) = δ(1, 2) + d3 Σ[G, D](1, 3)G(3, 2)
!
−(∂t21 + ω 2 )D(1,
2)Ĥ+ (t)d3
Π[G,
D](1, 3)D(3, 2)
Ĥ(t)2)==Ĥδ(1,
+
+
Ĥ
ph
el
e-ph
A space-time dependent nonlocal
>
<
D(1, 2) = −i"TC [∆ûH (1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1, 2)
2 effects of e-e 2 1
potential
ĤHel (t)describing
= ĥ(t) 1+ the
Ŵ
and e-ph interactions
i = µi t!
i
(i∂t1 − h(1))G(1,
2)
=
δ(1,
2)
+
d3
Σ[G,
D](1,
3)G(3,
2)
D > (1, 2) = −i"∆ûH (1)∆ûH (2)]#
!
2
2 D < (1, 2) = −i"∆ûH (2)∆ûH (1)]#
−(∂t1 + ω )D(1, 2) = δ(1, 2) + d3 Π[G, D](1, 3)D(3, 2)
Ĥ0 = Ĥ(t0 )
>
<
D(1, 2) = −i"TC [∆ûH (1)∆ûH (2)]# = θ(t1 , t2 )D (1, 2) + θ(t2 , t1 )D (1, 2)
2 (t) effect of e-e and e-ph interactions
C
−α(t−t0 )
# 2
e−α(t−t0 ) $f (t)$2 ≤ i = µi ton
sup
e
$g(t
i vibronic motion )$
α t! ∈[t0 ,t]
!
)w(1+ , 4)Γ(32; 4) − iδ(1, 2) d3 w(1, 3)G(3, 3+ )
By splitting the equation of motion in components, one obtains the set
! of Kadanoff-Baym equations. For example for the lesser component G< :
δM (1, 2)
d4d5d6d7
G(4, 6)G(7, 5)Γ(67; 3)
δG(4, 5)
Time-dependent
external
field
>
) + θ(t1 , t2 )Σ (1, 2) + θ(t2 , t1 )Σ< (1, 2)
!
!
<
(2)
!
! 3 ΣHF
! x3 t1 )G
! < (x3 t1 , 2)
!
(i∂Σt1 −(t,h(1))G
(1,
2)
−
dx
(1,
t)=
G
(2)
kl (t, t! )Gmn (t, t )Gpq (t , t)
! viqmk (2v!lnpj − v!nlpj )
ij
Σij (t, t ) =
Gkl (t, t )Gmn (t, t )Gpq (t , t) viqmk (2vln
! t2
! t0 −iβ
klmnpq
klmnpq <
<
>
<
>
"
"
Σ t(1,
2)[G
(3,
2)
−
G
(3,
2)]
+
G
(1,
3)Σ
(3, 2) "
t
2
1
"
<
<
> t2
<
t0=
t2)
0 −
d3[Σ> (1, 3) − Σt<1 (1, 3)]G
(3,
d2
Σ
(1,
3)[G
(3,
2)
−
G
>
<
<
< (3, 2)] >
d3[Σ (1, 3) − Σ t(1,
d2 Σ (1, 3)[G (3
=
t0
0 3)]G (3, 2) −
t0
t0
δΦ[G]
M (1, 2) =
δG(2, 1)
!
+
"
t0 −iβ
d3 Σ# (1, 3)G$ (3, 2)
t0
3)G(4, 2)Γ(34; 5)δv(5) = d5 Λ(12, 5)δv(5)
Collision or electron
correlation terms :
<
HF
<
Memory
(i∂tkernels
−
h)G
(1,
2)
−
Σ
(t
)
·
G
(t1 , t2Initial
)
correlations
1
1
! t2
! t0 −iβ
Σ< (1, 2)[G> (3, 2) − G< (3, 2)] +
G< (1, 3)Σ> (3, 2)
t
t
The corresponding self-energy diagrams to 2nd order are :
Σ=
+
+
+
The interaction lines can represent Coulomb interactions
or phonon propagators (also mixed diagrams)
If the equations of motion are solved self-consistently then
we can guarantee satisfaction of conservation laws
(electron particle number, energy, momentum)
!""#$%&'()&$*+,-$#,'(*./0$1.,
Conserving many-body approximations
+23-/2*2#4.
!"#$#%%&'()*
+,-./(#,
01
!
!
dr fxc (r, r ; !ω)∇ n(r ) = ∇vxc (r)
!1 )# =
"P(t
dx
"j(1)#
1
Kadanoff-Baym equations: practical solution
∇n(r)
!
!
dr$ fxc (r, r$ ; ω) = ∇vxc (r)
!
∂For
= − $ the
dxGreen
+
"j(1)#
×
B(1)]
t1 "P(t
1 )# solution
1 ["n(1)#E(1)
practical
function
is
expanded
into
one-particle states
$
dr fxc (r, r ; ω) = fxc (k = 0, ω)
G(1, 2) =
$
! ij (t1 , t2 )ϕ∗ (x2 )
ϕi (x1 )G
j
d2 Gs (1, 2)G(2, 1)vijxc (2) =
d2d3 Gs (1, 2)Σxc (2, 3)G(3, 1)
For the one-particle states we can, for instance,
† use the solutions
d)
G
(t
,
t
)
=
−i"T
â
(t
)â
(t
)#
xc
ij
1
2
1
2
C
i,H
ALDA
j,H
to the Hartree-Fock
or Kohn-Sham
(n(rt))
vxc
(rt) = equations
dn
2)
d
The e-eALDA
interaction$ attains
thexcform
$
$
$
fxc
(rt, r t ) =
(n0 (r))δ(r − r )δ(t − t )
2
dn
!
!
vijkl = dx dx$ ϕ∗i (x)ϕ∗j (x$ )v(r − r$ )ϕk (x$ )ϕl (x)
The Kadanoff-Baym equations become equations for time-dependent
matrices
2
2
dn
dn
d
)
xc
ALDA
$
$
! (rt,
! r$ t$ ) =
fxc
(n
(r))δ(r
−
r
)δ(t
−
t
)
2
0
2
d
)
xc $
$ by
$
ALDA
$ $for
dn
$
∗
∗
$
$
The
electronic
self-energy
second
Born
is
e.g.
given
(n
(r))δ(r
−
r
)δ(t
−
t
)
f
(rt,
r
t
)
=
0
xc
=
dx
dx
ϕ
(x)ϕ
(x
)v(r
−
r
)ϕ
(x
)ϕ
(x)
!
!
2
kl
k
l
i
j
dn
!∗ $
$ !∗
$
$
vijkl = dx dx ϕi (x)ϕj (x $)v(r
−
r
)ϕ
(x
)ϕ
(x)
k
l
∗
∗ $
$
$
2B
$ijkl =
$
HF
(2)
$
v
dx
dx
ϕ
(x)ϕ
(x
)v(r
−
r
)ϕ
(x
)ϕl (x)
k
Σ (t, t ) = δ(t, t )Σ (t) i+ Σ j (t, t )
2B
$
$
HF
(2)
$
Brief
Σ (t, t ) = δ(t,
(t) + $Σ HF
(t, t )
2B t )Σ$
(2)Article
$
Σ (t, t ) = δ(t, t )Σ (t) + Σ (t, t )
"
"
HF
+
Σij (t) = −i
Gkl=(t,
t )(2vGilkj(t,−t+
viljk
)
ΣHF
(t)
−i
)(2v
− viljk )
kl
ilkjAuthor
ij
The
kl
Brief Article
(2)
$
Σij (t, t )
"
=
kl
$
G
12kl (t, t )Gmn
The klmnpq
Author12
$
January
(t, t )G (t , t) v 8, 2012
(2v
$
January 8, 2012
If we use the notation
f ·g =
f !g =
!
!
∞
f (t)g(t)
12
pq
f ·g =
f !g =
− vnlpj )
f (t)g(t)
t0
!
β
dτ f (τ )g(τ )
0
c(z, z " ) =
dτ f (τ )g(τ )
lnpj
(Nils Erik Dahlen, RvL,
Phys.Rev.Lett.!98,
∞153004 (2007))
t0
β
iqmk
!
γ
dz̄ a(z, z̄)b(z̄, z " )
January 8, 2012
then the full set of Kadanoff-Baym equations is compactly given as
!
"
i∂t1 G≶(t1 , t2 ) = hHF (t1 )G≶(t1 , t2 ) + Σ≶ · GA + ΣR · G≶ + Σ! · G" (t1 , t2 )
!
"
−i∂t2 G≶(t1 , t2 ) = G≶(t1 , t2 )hHF (t2 ) + G≶ · ΣA + GR · Σ≶ + G! · Σ" (t1 , t2 )
!
"
i∂t G! (t, τ ) = ΣR · G! + Σ! # GM (t, τ )
"
!
Brief
Article
−i∂t G" (τ,Brief
t) = Article
G" · ΣA + GM # Σ" (t, τ )
# ∞
The Author
The Author f · g =
f (t)g(t)
t0
January 8, 2012
where all products
are8,matrix
# β and the retarded and advanced functions
January
2012 products
are defined as
dτ f (τ )g(τ )
f #g =
0
#
>
$
$k < (t, t! )]
kRR(t, t! )! = θ(t − t! )[k
t!!) −
!
>(t,c(z,
<
! dz̄ a(z, z̄)b(z̄, z )
z
)
=
k (t, t ) = θ(t − t )[k (t, t ) − k (t, t )]
γ
!
k AA
(t, t! )! = −θ(t! !− t)[k>>(t,Mt!!) − k$ <
<(t, t! )]
k (t, t ) = −θ(t − t)[k (t,
) −τ k) =
(t,c(t
t )]0 − iτ, t0 − iτ $ )
c t(τ,
!!
"
"
"
HF
≶
≶
≶ − iτ,
" t)#
cA(τ, t)R= c(t
,
t
)
=
h
(t
)G
(t
,
t
)
+
Σ
HF
≶
≶ · G A + ΣR · G≶ 0+ Σ" · G# (t1 , t2 )
2
1
1
2
(t1 , t2 ) = h (t1 )G (t1 , t2 ) + Σ · G + Σ · G + Σ · G (t1 , t2 )
!!
"
!
"
c
(t,
τ
)
=
c(t,
t
−
iτ
)
≶≶
HF
≶ 0 ""
HF(t2 ) + G≶
≶ · ΣA
A + GR
R · Σ≶
## (t1 , t2 )
,
t
)
=
G
(t
,
t
)h
+
G
·
Σ
2
1
2
(t1 , t2 ) = G (t1 , t2 )h (t2 ) + G · Σ + G · Σ + G · Σ (t1 , t2 )
!!
cR (t, t$ ) =""cδ (t)δ(t − t$ ) + θ(t − t$ )[c> (t, t$ ) − c< (t, t$ )]
Time propagation of the Kadanoff-Baym equations
Solve equilibrium case
on the imaginary axis
Carry out time-stepping in the double-time
plane ( possibly with external field applied)
"
"
0
0
!
!
t2
τ2
τ1
−iβ
t1
−iβ
(Nils Erik Dahlen, RvL,
Phys.Rev.Lett. 98, 153004 (2007),
A.Stan, N.E.Dahlen, RvL,
J.Chem.Phys.130, 224101 (2009))
an equat
Eq.(7) ha
Quantum transport
central re
where ΣMB is the many-body self-energy, H(z) is the
fact that
matrix representation of the one-body part of the full
starts and
The
one-body part
theintegration
Hamiltonian is
is projected
onto
different
Hamiltonian
andofthe
performed
over
the
case is co
H
H
L
R
regions
B. Equation of m
Keldysh-contour. This equation
of
motion
needs
to
be
Eq.(2)). T
C
solved with the boundary conditions [cite KMS here]
of the G C
We assume
the
sys
that
in
th
librium at inverse tem
G(t0 , z ! ) = −G(t0 − iβ, z ! )
a
(9)
describedisbygiven
Hamilto
G(z, t0 ) = −G(z, t0 − iβ).
tem is driven out of e
dz̄ G(z, z̄)ΣMB (z̄, z)
+
LC
(8)
CR
we aim to study the t
sity, current, etc.. In
which follow directly from the definition of the Green
namics in this system
function Eq.(5). Explicitly, Hthe one-body Hamiltonian
theory (for a review se
CC
H for the case of Htwo
leads, Left (L) andHRRRight (R), ismany-body effects in
LL
HCL
HRC
Green’s function
is d
The
proje


the contour-ordered p
CC and C
(10)
!
T
!
G rs (z, z ) i∂
= −i
z1 −
,
= −i"
where the different block matrices describe the projecHC
annihilation
operators
The one- onto
tions of ation
the and
one-body
part
H of respectively.
the Hamiltonian
where â and â†αare eit
HLL
HLC
FIG. 1: Sketch of the
transport
setup. 0The correlated central region (C) iscoupled to semi-infinite left(L) and right
H
H
H
H
=
CL
CC
CR
(R) tight-binding leads via tunneling Hamiltonians
HαC and
0 HRC HRR
HCα , α = L, R.
ite
KMS
here]
thecentral
G CC the
only.
¿From
it
tion
G CCfrom
projected
the
region.
We these considerations
lated
G. Fortoofinstance,
time-dependent
density
that
in
the
one-particle
basis
the
matrix
structure
ant
to
extract
from
the
block
matrix
structure
! matrix
functionasand the self-energy attain the form
is given
z ) The Green
is given as
en function (9)
β).


nrs (t) =−iG rs (t− , t+
) 0
(6)
0
0
G LL G LC G LR
MB
MB
As


ion G
of =
the
Green
G

Σ
=
.
0
Σ
[G
]
0
G
G
.
(14)
CC
CL
CC
RC
CC
where
the
times
t
lie
on
the
lower/upper
branch
of
the
±
ther
body Hamiltonian
0
0
0
G
G
G
RL equations
CR
RR of motion for the Green function
contour.
The
nd Right (R), is
occu
of
the
full
system
can
be
easily
derived
from
the
definition
The
projection
of
the
equation
of motion (7) onto
Gre
nfor G CC . The many-body self-energy in
with equations
of motion
for
the
complete
system
Eq.(5)
and
read
andfor
Cαindices
yields in the
nonvanishing entriesCC
only
ther

(10)
! consequence
"
ion. This is an
immediate
of the
for
!
!
!
!z )
!
i∂
G(z,
z
)
=
δ(z,
z
)1
+
H(z)G(z,
z
i∂
1
−
H
(z)
G
(z,
z
)
=
δ(z,
z
)1 +
CC
he diagrammatic expansionz #of theCC
self-energy
#
MB in our
ends with and interaction
line
which
!
,
+
dz̄ Σ (z,! z̄)G(z̄, z )MB (7)
!
scribe
the
projecH
G
(z,
z
)
+
dz̄
Σ
(z,
z̄)G
(z̄,
z
Cα last
αC term of
CC
fined in the central region (see
CC
Hamiltonian onto
MBα
!
!a functional!
!
his also implies
that
Σ
[G
]
is
−i∂z! G(z, z ) = CC
δ(z, z )1 + G(z, z )H(z )
#
only. ¿From these considerations
it follows
MB
an
MB
+ structure
dz̄ G(z, of
z̄)Σ
(8)
one-particle basis the matrix
Σ (z̄, z)
MB

where Σ 0 is the
self-energy, H(z) is the
0 many-body
0
Eq.
cent
fact
in th
ding self-energies
Σem,α are independent of the
electronic
!
!
G αC (z,and
z ) =hence
dz̄ gof
z̄),Hand
(z̄,therefore
z ),
(18)
αα (z,
αC G CC
the s
interactions
G
are
comCC
The equation of motion
Even
pletely known once the lead Hamiltonians Ĥα of Eq.(3)
the integral
is along
theback
Keldysh
contour.
wethe
diffic
arewhere
specified.
Inserting
(20)
to (16)
thenHere
gives
FIG. 3: D
defined gofααmotion
as theprojected
solution of
Theequation
equation
on
the
central
region
has
the
form
propa
of motion
proximati
!
"
bias c
!
"
!
!
i∂z 1 − Hαα (z) g αα!(z, z ) = δ(z, z )1,
(19)
To s
i∂z 1 − HCC (z) G CC (z, z )
MB
find
#
in ΣCC
,
& function g αα is
% Eq.(9). The
with boundary
conditions
MB
MB
!
!
Σ
Σ
. In
(z,
z̄)
G
(z̄,
z
).
=
δ(z,
z
)1
+
dz̄
Σ
+
Σ
em
em
CC
the Green function of the isolated
and biased α-lead. The
CC
This
solve
the
first term on the righthand side of Eq.(16) then becomes
(22) the matic
imagi
#
$
tionfrom
in tim
!
!
HCα
Gthe
z can
) = similarly
dz̄self-energy
Σem (z,
z have
), Eq.(8).
(20)
αC (z,
CC (z̄,
An adjoint
equation
bez̄)G
derived
where
on top
of
a many-body
we
alsofrom
to study
ward
α
anEquation
effective embedding
self-energy
(22) is an
exact equation for the Green func- bedding
funct
tion
G CCwe
, for
class of the
Hamiltonians
of Eq.(1), pro- propagat
where
havethe
introduced
embedding self-energy
powe
MB
of
vided that an exact
$ expression for
$ ΣCC [G CC ] as a func- track
syste
!
!
!
ing with
Σemof
(z, G
z CC
) = is inserted.
Σem,α (z, zIn
) =practical
HCα gimplementations
tional
αα (z, z )HαC ,
biase
α
Eq.(22) is converted
to a set of αcoupled real-time equa- initially u
This clas
(21)
tions, known as the Kadanoff-Baym equations (see Ap- quantum
which accounts for the tunneling of electrons from the
pendix).
These
equations
are
solved
by
means
of
timetions
and
central region to the leads
and
vice
versa.
The
embedpropagation techniques31 . For the case of unperturbed
#
−hC (t1 ))GCC (t1 , t2 ) = δ(t1 , t2 )+ dt3 (Σ[GCC ](t1 , t3 )+Σem (t1 , t3 ))GCC (t3 , t2
!
C
dN
(t)
α
!
<
CalculatingIα
the
current
"
=
−2
ReTr
[G
(t,
t)H
]
(t)
=
C
αC
(i∂
−h
(t
))G
(t
,
t
)
=
δ(t
,
t
)+
dt
(Σ[G
](t
,
t
)+Σ
(t
(t3
Cα
3
CC
1
3
em
1 , t3 ))G
CC
t
C
1
CC
1
2
1
2
1 ))G
−h
(t
(t
,
t
)
=
δ(t
,
t
)+
dt
(Σ[G
](t
,
t
)+Σ
(t
,
t
))G
(t
,
t
)
CC 3 2
C 1
CCΣem
1 (t
2 1 , t2 ) =
1 dt
2
HCα3 gCαα (tCC
1
1 , t2 )1 H3αC em 1 3
C"
The total current flowing outα=L,R
of reservoir α is given by :
"
HCα
(t
1 , t2 ) HαC
Σem (t1Σ, em
t2 ) (t
=1 , t2 ) = HCα gαα
(t1 ,gtαα
)
H
2
αC
dN (t)
Iα (t) =
α
α=L,R
α=L,R
= −2 ReTrC [G<
dt dNα (t)
dN (t)
Cα (t, t)HαC ]
<
α
=
−2
ReTr
[G
(t, ]t)HαC ]
I
(t)
=
<
C
α
Cα
=
−2
ReTr
[G
(t,
t)H
I
(t)
=
C
αC
This gives αafter somedtmanipulations:
dt
Cα
α
α
!
#
$
t
! t
#
!
A
!
R
!
< $
!
!
<
t
)Σ
(t
,
t)
+
G
(t,
t
)Σ
(t,
t
)
Iα (t) = −2ReTrC! <dt G!CC (t,
A
!em,α
R
!
<
! ,α
CC
em
(t, t )Σem,α (t , t) + GCC (t, t )Σem,α (t, t )
dt G
α (t) = −2ReTrC
0 CC
0
! −iβ
#
$
! −iβ
#
$ !
#
$
!
!
$ t )Σem
−2ReTrC ! #dt GCC
! (t,
! ,α (t , t)
dt0 GCC (t, t )Σem,α (t , t)
−2ReTrC
0
Σem,α (t1 , t2 ) = HCα gαα (t1 , t2 ) HαC
Σem,α (t1 , t2 ) = HCα gαα (t1 , t2 ) HαC
Memory of initial correlations
Long time limit leads under some assumptions to Meir-Wingreen formula
tC = −1
UL (t) = −UR (t) = U θ(t − t0 )
A(ω)/30
µtα==2.26
−2
V1,5L = V4,5R = −0.5
A(ω)
−β
Ĥ
The spectral
function
0
e −1
tC =
vii = 1.5
µ = 2.26<
ρ̂ =
0
V1,5L = VTre
=Ĥ−0.5
Im TrW
(t1 , t2 )
4,5R−β
−β
Ĥ
0
eis defined as
function
for
a
nonequilibrium
system
†
!The spectral
!
ρ̂ = A(ω)/30
1.5
ii =
(t), âj,H (t )}|Ψ0 $
j (t, t ) = #Ψ0v|{â
i,H
−β Ĥ0
Tre
<
Im
TrW
(t1 , t2 )t! )
!
A(ω) †
!
A(t, t ) = TrA(t,
Aij (t, t ) = #Ψ0 |{âi,H (t), âj,H (t! )}|Ψ0 $
A(ω)/30
µ = 2.26 !
!
A(t, t ) = TrA(t,
t
)
−β Ĥ0
A(ω)
e
In equilibrium the spectral function only depends
ρ̂ =on the
µ=
−β Ĥ0
difference
of2.26
the time coordinates and can be Fourier
Tretransformed
to give
†
Aij (t, t ) = #Ψ0 |{âi,H (t), âj,H (t! )}|Ψ0 $
ρ̂ = 1
Tre−β Ĥ0
A(t, t! )1 = TrA(t, t! )
†
%
%
!
†
Aij (t, t! ) = #Ψ0A
|{â(ω)
(t),
â
(t
)}|Ψ
$
N −1
i,H = j,H|#ΨN +10|â |Ψ $|2 δ(ω + E N − E N +1 ) +
|#Ψ
|âi |Ψ0 $
ii
0
0
i
k
k
k
A(t, t! ) = TrA(t, kt! )
k
%
N ω)
−1 = A(ω)
N A(T,
lim
|2 δ(ω + E0N − EkN +1 ) +
+
E
)
|#ΨkN −1 |âi |Ψ0 $|2 δ(ω −TE
0
k
→∞
1
k
e−β Ĥ0
!
lim A(T, ω) = A(ω)
T →∞
1
It shows peaks
at electron addition and removal energies
t = −2
by an electron-hole
propagator
with a single polarization
tC =
−1
(with fullycase
dressed
Green’s functions)
contains
In bubble
the nonequilibrium
it
is
convenient
to
Fourier
transform
V1,5L = V4,5R = −0.5
therespect
essential
physics
of the
problem. We also wish to emwith
to the
relative
times:
=
1.5
ii THE
AUTHOR
phasize that the v2B
approximation
includes the so called
! < (tdiagram
second-orderIm
exchange
which
is also quadratic
TrW
dω 1 , t2 )t
t iωt
A(T
+ , T −is )less
e relevant due to
A(T, ω) = This
in the interaction.
diagram
A(ω)/30
2π
2
2
the restricted phase-space that two electrons in the chain
A(ω)
have to scatter and
exchange.
which
be calculated
from
the Green
function
as function
= 2.26
Wecan
then
turnµour
attention
to the
spectral
which is defined as e−β Ĥ0
ρ̂ = !
−β
dt Ĥ0iωt >
t
t
Tre
<
A(T, ω) =
−ImTrC
e †[G CC! − G CC ](T + , T − ).
!
ction
2
2
Aij (t, t ) = #Ψ0 |{âi,H2π
(t), âj,H (t )}|Ψ0 $
basis
(40)
!
!
= TrA(t,
t)
Bias
For values A(t,
of Tt )after
the transients
have died out the
%
N +1 † Inspectral
N +1spectral
N −1 becomes
N −1
2For
N
long+
time
the
function
independent
function
independent
of
T
.
such
|âi |Ψ0the
$|2 δ(ω
E0Nlimit
− Ebecomes
)
+
+
E
)
|#Ψ
|â
|Ψ
$|
δ(ω
−
E
i
0
0
k
k
k
of T when a steady state is being
reached
k
lim A(T, ω) = A(ω)
T →∞
1
 3)G(3,
vii
2
h(1))G(1,
2)
=
δ(1,
2)
+
d3
Σ[G](1,
2)
i
=
"
j
!
!
2π
(ω
−
ε
−
nU
−
Λ(ω))
2|i−j|
1.
formulas
0
ω
ΓL (ω)fL (ω)
+
Γ
(ω)f
(ω)
R
R
!
! d2 Σ(1, 2)G(2, 1 ) = −i d2 w(1, 2) G2 (1,µ2, 2+ , 1! )
−ω
t
2! R
2
Simple
interactions
g
(ω)f
(ω)
+Γ
(ω)f
(ω)
LΛ(ω))
R
!
π (ωdω
− ε0 example:
−ΓLnU
−e-e
+
(Γ(ω)/2)
U!L (t) =V
−U
U θ(t
−e
t0 )
R (t) ==
(t)
V
g
g
!
α G (1,22, 2+ , 1! )
2
−ω
t
d2
Σ(1,
2)G(2,
1
)
=
−i
d2
w(1,
2)
2π
(ω
−
ε
−
nU
−
Λ(ω))
+
(Γ(ω)/2)
g2) +
t =23)G(3,
−2 2)
0
(i∂t1 −
h(1))G(1,
2)
=
δ(1,
d3
Σ[G](1,
V
(t)
=
V
e
Interaction
g
g
! vijkl
−ωg t
=
v
δ
δ
t
=
−1
!
C
ij
il
jk
V
(t)
=
V
e
g
g
dω
Γ
(ω)f
(ω)
+
Γ
(ω)f
(ω)
v
=
v
δ
δ
L
L
R
R
ij
ijkl
il
jk
(i∂
−
h(1))G(1,
2)
=
δ(1,
2)
+
d3
Σ[G](1,

V1,5L = V4,5R =3)G(3,
−0.5 2)
n =t1  v
2 + (Γ(ω)/2)2
=
v
δ
δ
ij
ijkl
il
jk
2π
(ω
−
ε
−
nU
−
Λ(ω))
0
! 
U
i
=
j
v
=
1.5

U
i
=
j
ii
 dω V (t)ΓL=(ω)f
(ω)
L−ω
g t + ΓR (ω)fR (ω)
iVg=e j
n=
g vii

2 + (Γ(ω)/2)2
2π (ω − ε0 − nU v
− Λ(ω))
vij =
=
ij
v
=
v
δ
δ
ij
ijkl
il
jk
vij= U 
−ωg t

V
(t)
=
V
e
U

vii gi "= j g
2|i−j|
v
i
=
j
i
=
"
j
i
=
"
j

ii
2|i−j|
vijkl = vij δil δjk
2|i−j|
vij = µ 
 vii
L
H LC
HCR
R
+ (Γ
B.
C
HCC
HLL
HCL
HRC
HRR
FIG. 1: Sketch of the transport setup. The correlated central region (C) is coupled to semi-infinite left (L) and right
(R) tight-binding leads via tunneling Hamiltonians HαC and
HCα , α = L, R.
viii "=i j= j

2|i−j|
Time-dependent bias
vij = µ
 vii
i
=
"
j
2|i−j|
UL (t) = −UR (t) = U θ(t − t0 )
µ
UL (t) = −UR (t) = U θ(t − t0 )
Hoppings:
α
t = −2
tC = −1
V1,5L = V4,5R = −0.5
1
ation and annihilation operators respectively. The onebody part of the Hamiltonian hij (t) may have an arbitrary time-dependence, describing, e.g., a gate voltage or
pumping fields. The two-body part accounts for interactions between the electrons where vijkl are, for example
in the case of a molecule, the standard two-electron integrals of the Coulomb interaction. The lead Hamiltonians
have the form
µ
Ĥα (t) = Uα (t)N̂α +
!
†
hα
ij ĉiσα ĉjσα ,
(3)
ij,σ
where the creation and annihilation operators for the
"
leads are denoted by ĉ† and ĉ. Here N̂α = i,σ ĉ†iσα ĉiσα
is the operator describing the number of particles in lead
α. The one-body part of the Hamiltonian hα
ij describes
metallic leads and can be calculated using a tight-binding
representation, or a real-space grid or any other convenient basis set. We are interested in exposing the leads
to an external electric field which varies on a time-scale
much longer than the typical plasmon time-scale. Then,
the coarse-grained time evolution can be performed assuming a perfect instantaneous screening in the leads and
the homogeneous time-dependent field Uα (t) can be interpreted as the sum of the external and the screening
field, i.e., the applied bias. This effectively means that
We
libriu
descr
tem
we a
sity,
nami
theor
many
Gree
the c
G
wher
and t
and s
speci
tour.
Keld
real t
t0 to
the t
of th
All t
lated
matr
wher
conto
tion
defin
s initial embedding effects
n only the latter). At t = 0
Matsubara Green’s function
es all elements of G (t, τ )
of the value of τ . This beeffects die out in the longy rate is directly related to
y state. A very similar beB and GW approximation
of the oscillations.
*+,-
!"#"%
all times both in the weak and strong bias regime while
the HF current deviates from the correlated results already The
after Green
few time function
units. This result indicates that
a chain of 4 atoms is already long enough for screening
./"#%
!"#"$
effects to play a crucial role. The 2B and GW approximations
have
in common
first three diagrams
of theorbital the Green function
For
the
highesttheoccupied
molecular
perturbative expansion of the many-body self-energy il!the first order
matrix
thethat
following
structure (imaginary part displayed)
lustrated
in Fig. element
3. We thus has
conclude
8
!"
!"
!&"
!'"
!("
!$"
!)
exchange diagram (Fock) with an interaction
CC screened
,
uency of which corresponds to the removal energy of
lectron from the HOMO level, leading to a distinct
FIG. 6: Transient currents flowing into the right lead for
k in the spectral function (see Section III B below).
HF, 2B and GW approximations with the applied bias U
!
0.8 (three lowest curves) and U = 1.2.
he imaginary part of G CC,HH (t, τ ) within the HF apximation is displayed in Fig. 5 for real times between
0 and t = 30 and imaginary times from τ = 0 to
by an electron-hole propagator with a single polarizat
5. This mixed-time Green’s function accounts for
bubble (with fully dressed Green’s functions) conta
al correlations as well as initial embedding effects
the essential physics of the problem. We also wish to
hin the HF approximation only the latter). At t = 0
phasize that the 2B approximation includes the so ca
have the ground-state Matsubara Green’s function
!
second-order exchange diagram which is also quadr
as the real time t increases all elements of G CC (t, τ )
in the interaction. This diagram is less relevant due
roach zero independently of the value of τ . This bethe restricted phase-space that two electrons in the ch
or indicates that initial effects die out in the longhave to scatter and exchange.
e limit and that the decay rate is directly related to
We then turn our attention to the spectral funct
time for reaching a steady state. A very similar bewhich is defined as
or is found within the 2B and GW approximation
with a stronger damping of the oscillations.
!
dt iωt >
t
<
A(T, ω) = −ImTrC
e [G CC − G CC ](T + , T −
FIG. 4: The imaginary part of the lesser Green’s function
2π
2
<
!
FIG. 5: The imaginary part of the mixed Green’s function
G CC,HH B.
(t1 , t2Time-dependent
) of
in molecular orbital basis
(
<the central region
current
!
G CC,HHFor
(t, τ values
) of the
central
region
molecular orbital
basis. out
corresponding to the HOMO level
CC,HH
1 of the
2 central chain. Bias
of
T after
thein transients
have died
CC,HH
Bias voltage U = 1.2, HF approximation.
voltage U = 1.2, HF approximation.
spectral function becomes independent of T . For s
he time-dependent current at the right interface be-
FIG.
5:
The
imaginary
4: The imaginary part of the lesser Green’s
ndentFIG.
current
τ ) of theorbi
cen
G
(t , t ) of the central Gregion in(t,molecular
Bias
voltage
U = 1.2,cha
H
corresponding
to
the
HOMO
level
of
the
central
en the chain and the two-dimensional lead is shown
CC,HH
Bias voltage U = 1.2, HF approximation.
The transient currents
01
'2
34
!"#&'
./&#'
*+,-
!"#"%
./"#%
!"#"$
!"
!"
!&"
!'"
!("
,
!$"
!)"
V1,5L = V4,5R = −0.5
the 2B peaks remains small during the initial transient
vii = 1.5
regime
(up functions
to T = 70) to then<increase
dramatically.
This
The spectral
Steady state
regime
Im TrW (t1 , t2 )
A(ω)/30
U=0.8 (solid line)
U=1.2 (dashed line)
A(ω)
µ = 2.26
HF
10
5
0
2B
0
1
2
0
1
2
0
1
2
3
4
5
3
4
5
3
4
5
1
0
GW
1
1
0
FIG.
8:
Real-time
evolution
of
the
spectral
function
A(T,
ω)
Time-dependent buildup of the I-V curves
for the HF (left panel) and the 2B approximation (right panel)
for an applied bias of U = 1.2. On the horizontal axis the time
Hartree-Fock
second Born
T and the
vertical axis the frequency ω.
FIG. 9: Transient right current IR (U, t) as a function of apcorrelations
wash
out panel)
I-V features
pliedelectron
bias voltage
andbeyond
time mean-field
in the HF
(left
and 2B
(right panel) approximations.
the peak intensities. The peak of the ∆!L4 (d) transition
is very strong due to the sharpness of that particular resThe time-dependent dipole moment
U=1.2
0.04
L3
1R
2R
0
-0.02
c) HOMO-LUMO
L4
d)
e)
d(!)
d(t)
0.02
b)
L2,3R
a) 4R
HF
2B
GW
-0.04
0
0
10
20
0
1
30
t
2
40
3
50
4
60
Hel (t) = h(t) + W
(i∂t1 − h(1))G(1,
! 2) = δ(1, 2) +
d3 Σ[G, D](1, 3)G(3, 2)
Ĥel (t) = ĥ(t) + Ŵ
!
! systems)
interactions
)G(1, Electron-phonon
2) = δ(1, 2) +
d3
Σ[G, D](1,(test
3)G(3,
2)
2
G(1, 2)−(∂
= tδ(1,
d32)
Σ[G,
D](1,2)3)G(3,
+2)
ω 2+)D(1,
= δ(1,
+ 2)
d3 Π[G, D](1, 3)D(3, 2)
1
!
!
D(1, 2) = δ(1, 2) +
d3 Π[G, D](1, 3)D(3, 2) >
<
1,
2)
=
−i"T
[∆û
(1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(very
preliminary
from Friday last
D(1, 2)Holstein
= δ(1,Clattice
2) + Hsystem
d3 Π[G,
D](1,
3)D(3,1 2)results
H
2
2 1week)(1, 2)
>
<
(1)∆û
(2)]#
=
θ(t
,
t
)D
(1,
2)
+
θ(t
,
t
)D
(1, 2)
H
H
1 2>
2 1<
i
=
µ
t
i
i
H (1)∆ûH (2)]# = θ(t1 , t2 )D (1, 2) + θ(t2 , t1 )D (1, 2)
" †
" †
"
i=
µ=
i tiω
âi âi − t
Ĥ
ĉi ĉj −
ûi n̂i
0
i
=
µ
t
i
i
" †
" †
"
" †â â −"
i"
i
!i,j"
t
=
ω
ĉ
ĉ
−
û
n̂
†
i
0
j
i
i
â âi − t
=ω
ĉ ĉ i −
û n̂
0
i
i
i i
i j
!i,j"
!i,j"
†
†
i =
i +
âi âi
ûiû=
âiâ+
†
†
i =
n̂in̂=
ĉi ĉĉD
i ĉ>i (1, 2)
i
i
ûi
i i
=
†
âi
+ âi
displacement
†
n̂i = ĉi ĉi
densityH (1)∆ûH (2)]#
=site
−i"∆û
>> (1, 2) = −i"∆û (1)∆û (2)]#
D
D (1, 2) = −i"∆ûHD(1)∆û
(2)]#
H<
H
(1, 2) H= −i"∆ûH (2)∆ûH (1)]#
<<
(1,2)2)==−i"∆û
−i"∆û
DD (1,
H (2)∆û
H (1)]#
H (2)∆û
H (1)]#
Ĥ0 = Ĥ(t0 )
ĤĤ
==
Ĥ(t
Ĥ(t
0 0
0)0)
2 (t)
C
−α(t−t
2
−α(t−t0 )
% 2
0)
2 (t)
eC
$f
(t)$
≤
sup
e
$g(t
)$
2
0) 0
α) % )$t!2∈[t
) (t)$2 2≤ C (t) sup e−α(t−t
−α(t−t
% 02,t]
$f
$g(t
D > (1, 2) = −i"∆ûH (1)∆ûH (2)]#
Example 1 : Formation
polaron after
the switch-on
of the
D < (1, 2)of=a−i"∆û
(2)∆û
(1)]#
H
H
electronic-vibrational interaction
Ĥ0 = Ĥ(t0 )
Ĥ(t) = ω0 ↠â + %0 ĉ† ĉ − gθ(t)(↠+ â)ĉ† ĉ
2 (t)
C
−α(t−t0 )
2
−α(t−t0 )
% 2
e
$f (t)$ ≤
sup e
$g(t )$
α t! ∈[t0 ,t]
single mode
sudden switch-on of electron-phonon
! t
2
2
% α(t! −t0 interaction
) −α(t! −t0 )
% 2
$f (t)$ ≤ C (t)
dt e
e
$g(t )$
t0
We use a self-energy in the self-consistent Born approximation
1
vj "
2
2
"
†
!
û
n̂
âi âi −
g
+ vδ(t)n̂1 + ω0
i
i
Spectral
functions
iωt
ω) = dt χ11 (t)e
i=1
i=1
D < (1, 2) = −i"∆ûH (2)∆ûH
ω0 = 0.5
) = −i"∆ûH (1)∆ûH (2)]#
g = 0, 0.1, 0.2, . . . 1.0
A(ω)
) = −i"∆ûH (2)∆ûH (1)]# A(T, ω)
A(T, ω)
Ĥ0 = Ĥ(t0 )
δni (t)
χij (t) =
Ĥ(t) = ω0 ↠â + "0 ĉ† ĉ − gθ(t)(↠+ â)ĉ† ĉ
vj
2 (t)
C
1
!
sup e−α(t−t ) "g(t# )"2
e−α(t−t ) "f (t)"2 ≤
α t ∈[t ,t]
iωt
(ω) = dt χ11 (t)e
! t
0
0
!
2
0
# α(t! −t0 ) −α(t! −t0 )
2
"f (t)"
C (t)
Polaron shift
and ≤side
bandsdt e
A(ω)
ω = 0.5
0
e
t0
2
#
2
−α(t! −t0 )
≤ C (t) sup "g(t )" e
0 . . . 1.0
g = 0,&0.1,
0 =0.2,
A(T, ω)
Ĥ0 =1Ĥ(t0 )
t! ∈[t0 ,t]
Polaron
x ∈ [−1, 1]
!
t
" t0
"g(t# )"2
# α(t! −t0 )
dt e
#$
α(t−t0 )
α
≤e
2
v(x,
t)
=
sin
(πx) sin(ωt)
†
†
†
† g=0.5
0 â â + "0 ĉ ĉ − gθ(t)(â + â)ĉ ĉ
− 12
(1−x ) + 1)
Ψ
(x)
=
A(e
0
2 (t)
C
0)
1&
sup e−α(t−t&
"g(t# )"2†
"2 ≤
hij (t) âi âj +
wij n̂i n̂j
α t! ∈[t ,t] Ĥ(t) =
0
%
δni (t)
χij (t) =
vj
!
Uncoupled electron
χ11 (ω) = dt χ11 (t)eiω
A(T, ω)
Ĥ0 = Ĥ(t0 )
1
ûi = âi + âi
Optical
!i
n̂i = ĉ†i(i∂
ĉi t1 − h(1))G(1, 2)†= δ(1, 2) + d3 Σ[G, D](1, 3)G(3, 2)
ûi = âi + âi
!
response of 2a Holstein
dimer
2 2
2
"
"
d3 Π[G, D](1, 3)D(3, 2)
−(∂t1 +
ω )D(1,
2)
=
† δ(1, 2) +
†
(ĉ†1 ĉ2 + ĉ†2 ĉ1 ) + vδ(t)n̂1 + ω0
D
>
i
!i,j"
âi ân̂i i−=gĉi ĉi ûi n̂i
>
<
D(1, 2) = −i"Ti=1
(2)]# ="
2)
+
θ(t
,
t
)D
(1, 2)
2
2θ(t1 , t2 )D (1,
C [∆ûH (1)∆ûHi=1
2
1
"
†
+ iω=
0 µi ti âi âi
" †
" †
i=1
†
†
Ĥ ==−i"∆û
−(ĉ1 ĉ2H+
ĉ2 ĉ1 )H+(2)]#
vδ(t)n̂1
(1, 2)
(1)∆û
−g
ûi n̂i
"i=1
ĉi ĉj −
ûi n̂i
Ĥ(1)]#
= ω0
âi âi − t
D < (1, 2) = −i"∆ûH (2)∆û
H
>
D (1, 2) = −i"∆û
i H (1)∆û
H (2)]# i
!i,j"
δni<(t)
†
χij (t) = D (1, 2) = −i"∆ûHû(2)∆û
â(1)]#
i
i = âi +
H
vj
†
!
Ĥ0 = Ĥ(t0n̂)i = ĉi ĉi
χ11 (ω) = dt χ11 (t)eiωt
2 †
2
†
†
†
"
"
Ĥ(t)Ĥ==ω−(ĉ
â †âĉ ++%ĉ0†ĉĉ ĉ) +
−vδ(t)n̂
gθ(t)(â+ ω+ â)ĉâ†ĉâ − g
0
ûi n̂i
This amounts to a
1
0
1 2
2 1
i i
2 (t)
i=1
i=1
C
Ĥ−α(t−t
= Ĥ(t
)
solution of
0the
0
2
−α(t−t0 )
% 2
0)
e
$f (t)$ ≤D> (1, 2) =sup
e (1)∆û$g(t
)$
−i"∆û
(2)]#
H
α
†
† eqn
†
† t! ∈[t0 ,t] H
Bethe-Salpeter
Ĥ(t) = ω0 â â + &0 ĉ ĉ − gθ(t)(â +<â)ĉ ĉ
! Dt (1, 2) = −i"∆ûH (2)∆ûH (1)]#
with phonon-dressed
2
2
% α(t! −t0 ) −α(t! −t0 )
% 2
$f
(t)$
≤
C
(t)
dt
e
e
$g(t
)$
δn
(t)
Green functions
i
1
χij (t) =
t0
vj
χ111(ω) =
!
dt χ11 (t)eiωt
Ĥ0 = Ĥ(t0 )
Ĥ(t) = ω0 ↠â + &0 ĉ† ĉ − gθ(t)(↠+ â)ĉ† ĉ
Outlook
- We will study transient phenomena in time-dependent transport
involving both electron-electron and electron-phonon interactions
(for example SSH + e-e interactions)
Open issues, to be studied:
- What is the combined effect for e-e and e-ph interactions?
a) Structure of transients, I-V curves
b) Bistability
c) Gap closing/image charge effects with phonons
d) What level of many-body perturbation theory do we need?
(vertices beyong SCBA, mixed e-e / e-ph diagams)
- Towards realistic systems, can we cut memory depth to save
computational cost without loosing accuracy?
Questions for the discussion
- The SCBA approximation is valid in the weak coupling limit.
How to get beyond it?
Or, in general, what is the importance of terms that are
higher order in the electron-phonon interaction?
(we also need to expand the Hamiltonian to higher order
in the displacements which leads to new vertices)
- What is the interplay between e-e and e-ph interactions?
(Bistability, gap closing, image charge effects, etc.)
- What is the importance of solving the Kadanoff-Baym
equations for the phonon propagators as well?
(heating effects)