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Nonequilibrium Green’s Function Method in Thermal Transport
Wang Jian-Sheng
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• Lecture 1: NEGF basics – a brief history, definition of Green’s functions, properties, interpretation.
• Lecture 2: Calculation machinery – equation of motion method, Feynman diagramatics, Dyson equations, Landauer/Caroli formula.
• Lecture 3: Applications – transmission, transient problem, thermal expansion, phonon life-time, full counting statistics, reduced density matrices and connection to quantum master equation.
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References
• J.-S. Wang, J. Wang, and J. T. Lü, “Quantum thermal transport in nanostructures,” Eur. Phys. J. B 62, 381 (2008).
• J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, “Nonequilibrium Green’s function method for quantum thermal transport,” Front. Phys. (2013).
• See also http://staff.science.nus.edu.sg/~phywjs/NEGF/negf.html
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Lecture One
History, definitions, properties of NEGF
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A Brief History of NEGF
• Schwinger 1961• Kadanoff and Baym 1962• Keldysh 1965• Caroli, Combescot, Nozieres, and Saint-James
1971• Meir and Wingreen 1992
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Equilibrium Green’s functions using a harmonic oscillator as an example
• Single mode harmonic oscillator is a very important example to illustrate the concept of Green’s functions as any phononic system (vibrational degrees of freedom in a collection of atoms) at ballistic (linear) level can be thought of as a collection of independent oscillators in eigenmodes. Equilibrium means that system is distributed according to the Gibbs canonical distribution.
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Harmonic Oscillator
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2 2 2 †
† †
1 ,2 2
1 1 1 ,2 2 2
, [ , ] , [ , ] 12
pH kx u x mm
kH u u a am
u a a x p i a a
k m
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Eigenstates, Quantum Mech/Stat Mech
†
† † † †
1, , 0,1,2,2
1 , 1 1
1,Tr
0, 1
1Tr ,1
n n
H
HB
H n E n E n n
a n n n a n n n
ek Te
aa a a a a aa f
fe
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Heisenberg Operator/Equation
†
† †
( )( ) 1 [ ( ), ]
( ) 1 1 1[ ( ), ] ( ), ( ) ( )2
( )
( ) , ( )
[ ]( )
Ht Hti i
i t i t
O t e OedO t O t H
dt i
da t a t H a t a t a tdt i i
i a t
a t ae a t a e
O: Schrödinger operatorO(t): Heisenberg operator
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Defining >, <, t, Green’s Functions
†
( ') ( ')
( , ') ( ) ( ') , 1
( ) ( ) ( ) , ( )2
( , ') (1 )2
i t
i t t i t t
ig t t u t u t i
u t a t a t a t ae
ig t t fe f e
t
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( , ') ( ') ( ) ( ', )
( , ') ( ) ( ') ( ') ( , ') ( ' ) ( , ')
( , ') ( ) ( ') ( ' ) ( , ') ( ') ( , ')
1, if 01( ) , if 020, if 0
t
t
ig t t u t u t g t t
ig t t Tu t u t t t g t t t t g t t
ig t t Tu t u t t t g t t t t g t t
t
t t
t
: time order: anti-time order
TT
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Retarded and Advanced Green’s functions
2
( , ') ( ') [ ( ), ( ')]
sin ( ')( ') ,
( , ') ( ' ) [ ( ), ( ')] ( ', )
( ) ( ) ( ), with ( ) 0 for 0
r
a r
r r r
ig t t t t u t u t
t tt t
ig t t t t u t u t g t t
g t g t t g t t
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Fourier Transform
2 2
*
[ ] ( ) , ( ) [ ]2
sin( )[ ] ( )
1 , 0
[ ] [ ] ,
[ ] ( ) (1 ) ( )
r r i t r r i t
r i t t
a r
dg g t e dt g t g e
tg t e dt
i
g gig f f
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Plemelj formula, Kubo-Martin-Schwinger condition
1 1 ( )
[ ] [ ] [ ] ( )
[ ] [ ],
( ) ( )
r a
P i xx i x
g g g f
g e g
g t g t i
Valid only in thermal equilibrium
P for Cauchy principle value
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Matsubara Green’s Function
( ') ( ')
0
1( , ') ( ) ( ')
1 (1 )2
where 0 , ' , ( ) ( )
( ) ( )
2[ ] ( ) , , , 1,0,1,2,
[ ] [ ]
n
M
H H
M M
iM Mn n
r Mn
g T u u
fe f e
u u i e ue
g g
ng i g e d n
g g i i
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Nonequilibrium Green’s Functions
• By “nonequilibrium”, we mean, either the Hamiltonian is explicitly time-dependent after t0, or the initial density matrix ρ is not a canonical distribution.
• We’ll show how to build nonequilibrium Green’s function from the equilibrium ones through product initial state or through the Dyson equation.
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Definitions of General Green’s functions
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( , ') ( ) ( ') , ( , ') ( ') ( )jk j k jk k ji iG t t u t u t G t t u t u t
( , ') ( ') ( , ') ( ' ) ( , '),
( , ') ( ' ) ( , ') ( ') ( , ')
t
t
G t t t t G t t t t G t t
G t t t t G t t t t G t t
( , ') ( ') ,
( , ') ( ' )
r
a
G t t t t G G
G t t t t G G
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Relations among Green’s functions
,
,
( , ') ( ', )
( , ') ( ', )
r a
t t r t
t t r a a t
jk kj
r ajk kj
G G G G
G G G G G G G
G G G G G G G
G t t G t t
G t t G t t
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Steady state, Fourier transform
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†
( , ') ( '),
[ ] ( ) ,
[ ] [ ]
i t
r a
G t t G t t
G G t e dt
G G
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Equilibrium Green’s Function, Lehmann Representation
( )
,
| | , ,
( ) , | | 1
( ) Tr ( ) (0)
1| ( ) (0) |
1| | | |
n
n
n mn
HE
nn
Ht Hti i
j jm
jk j k
Ej k
n
E E tE i
j kn m
eH n E n Z eZ
u t e u e m m
iG t u t u
i e n u t u nZ
i e n u m m u nZ
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Kramers-Kronig Relation
0
0
0
00
ˆ[ ] ( ) is analytic on the upper half plane of
ˆ[ ] [ ], 0
1 [ '] ( )[ ] 'P , lim'
1 [ '][ ] 'P'
izt r
r
xrr IR
x
rr RI
G z e G t dt z
G G z i
G f xG d P dxx x
GG d
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Eigen-mode Decomposition
( ) 2 ( ) (1) (2) ( )
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222
2
, , , , , , , , , ,
122 2
, ( , , , )
,
0 00 0
diag0 0
0
( ) diag ( )
1[ ] diag
j j Nj
T
Tj
N
t t r a t t r a T
r T
j
Ku u S u u u
u SQ S S I
S KS
G t S g t S
G S S i Ki
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Pictures in Quantum Mechanics
• Schrödinger picture: O, (t) =U(t,t0)(t0)
• Heisenberg picture: O(t) = U(t0,t)OU(t,t0) , ρ0, where the evolution operator U satisfies
'''''
( , ') ( , '),
( , ') , 't
tt
t
i H dt
U t ti H U t tt
U t t Te t t
See, e.g., Fetter & Walecka, “Quantum Theory of Many-Particle Systems.”
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Calculating correlation
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'
0 0 0 0 0
0 0 0
( )
0 '
( '') ''
( ) ( ') Tr ( ) ( ') '
Tr ( ) ( , ) ( , ) ( , ') ( ', )
Tr ( ) ( , ) ( , ') ( ', )
Tr ( ) ,
( , ') ,( , ') ( ', '') ( , '')
C
t
t
i H d
C t t
i H t dt
A t B t A t B t t t
t U t t AU t t U t t BU t t
t U t t AU t t BU t t
t T e A B
U t t TeU t t U t t U t t
t0 t’ t
B A
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Evolution Operator on Contour2
1
2 1 2 1
3 2 2 1 3 1 3 2 1
11 2 2 1 1 2
0 0
( , ) exp ,
( , ) ( , ) ( , ),
( , ) ( , ) ,
( ) ( , ) ( , )
ciU T H d
U U U
U U
O U t OU t
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Contour-ordered Green’s function
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( )
0 '
( , ') ( ) ( ')
Tr ( ) C
TC
i H dTC
iG T u u
t T u u e
t0
τ’
τ
Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho.
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Relation to other Green’s function
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'
( , ), or ,
( , ') ( , ') or
,
,
t
t
t t
G GG G t t G
G G
G G G G
G G G G
t0
τ’
τ
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An Interpretation
, 0 0 0
1 1 ,2 2
( ) ( )
Tr ( ) ( ) ( , )
exp ( ) ( , ') ( ') '2
T T T
T
M M
T
C C
H p p u Ku K K
V F u
U t t t U t t
i F G F d d
G is defined with respect to Hamiltonian H and density matrix ρ and assuming Wick’s theorem.
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Calculus on the contour
• Integration on (Keldysh) contour
• Differentiation on contour
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( ) ( ) ( ) ( )f d f t dt f t dt f t dt
( ) ( )df df td dt
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Theta function and delta function• Theta function
• Delta function on contour
where θ(t) and δ(t) are the ordinary theta and Dirac delta functions
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'
1 if is later than ' along the contour( , ')
0 otherwise
( , ') ( ')( , ') ( ' )
( , ') ( , ')( , ') 0( , ') 1
t t t tt t t t
t tt tt t
''
( , ')( , ') ( , ') ( ')d t t t td
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Transformation/Keldysh Rotation'
' '
' '
( , ') ( , ')
or
1 0 1 11, ,0 1 1 12
1 ,2
0
jj jj
t
t
Tz
r KT T
z K a
t t t t
t t t t
r K
a
A A A t t
A AA A A
A A
R RR I
A AA R AR R AR
A A
A A A A A A A AA A A A A A A A
G GG
G
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Convolution, Langreth Rule
2 3 1 2 2 3 1
11
( , ) ( , ) ( , )
0 0 0
or , , +
( )
n n n
r K r K r K
a a a
r r r a a a K r K K a
r r r r r r r r
K K r r K r K a K a a
AB D d d d A B D
C AB C AB C AB
C C A A B BC A B
C A B C A B C A B A B
G g g G G g g G G g
G g g G g G g G
,
(1 ) (1 )r r a a r aG G g G G G
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Lecture two
Equation of motion method, Feynman diagrams, etc
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Equation of Motion Method
• The advantage of equation of motion method is that we don’t need to know or pay attention to the distribution (density operator) ρ. The equations can be derived quickly.
• The disadvantage is that we have a hard time justified the initial/boundary condition in solving the equations.
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Heisenberg Equation on Contour
2
1
2 1 2 1
0 0
( , ) exp ,
( ) ( , ) ( , )
( ) [ ( ), ]
ciU T H d
O U t OU t
dOi O Hd
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Express contour order using theta function
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),'()()'()',()'()(
)'()()',(
TTT
TC
uuiuui
uuTiG
Operator A(τ) is the same as A(t) as far as commutation relation or effect on wavefunction is concerned
Iiuu T ])(),([
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Equation of motion for contour ordered Green’s function
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IKG
IuKuTi
uuiuuTi
uuiuui
uuiG
uuTi
uuiuui
uuiuuiG
TC
TTC
TTT
T
TC
TTT
TTT
)',()',(
)',())'()((
)',(])'(),([)'()(
),'()()'()',()'()(
)',()'()()',(
)'()(
),'()()'()',()'()(
),'()()'()',()'()()',(
2
2
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Equations for Green’s functions
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2
2
2' '
'2
2, , , ,
2
2
2
2, ,
2
( , ') ( , ') ( , ')
( , ') ( , ') ( ') , , '
( , ') ( , ') ( ')
( , ') ( , ') ( ')
( , ') ( , ') 0
r a t r a t
t t
G KG I
G t t KG t t t t It
G t t KG t t t t It
G t t KG t t t t It
G t t KG t tt
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Solution for Green’s functions
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2, , , ,
2
2 , , , ,
1, , 2
12
( , ') ( , ') ( ')
using Fourier transform:
[ ] [ ]
[ ] ( ) ( )
[ ] [ ] ( ) , 0
( ),
r a t r a t
r a t r a t
r a t
r a
r a
t r
G t t KG t t t t It
G KG I
G I K c K d K
G G i I K
G f G G G e G
G G G
c and d can be fixed by initial/boundary condition.
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Feynman diagrammatic method
• The Wick theorem• Cluster decomposition theorem, factor
theorem• Dyson equation• Vertex function• Vacuum diagrams and Green’s function
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Handling interaction
41
Transform to interaction picture, H = H0 + Hn
0 00 0
0 00 0
0 00 0
0 0 00 0 0
( ) ( )
0 0 0
( ) ( ' )
( ) ( )
0 0
( ) ( ) ( )
0 0
( ) ( ) ( , ) ( , ) ( )
( , ') ( , ')
( ) ( , ) ( , )
( ) ( , ) ( ) (
H Hi t t i t t
I S H I
H Hi t t i t t
H Hi t t i t t
I H
H H Hi t t i t t i t t
I s H
t e t e U t t S t t t
S t t e U t t e
t e U t t U t t e
A t e A e e U t t A t U t
0
0( ), )
Hi t tt e
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Scattering operator S
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0 0 0 0
0 0 0 0( ) ( ) ( ' ) ( ' )
0 0 0 0
0 0
( ) ( ') Tr ( ) ( ') '
Tr[ ( , ) ( ) ( , ) ( , ') ( ') ( ', )]Tr[ ( , ) ( ) ( , ') ( ') ( ', )]
H H H
H H H Hi t t i t t i t t i t t
I
I
A t B t A t B t t t
U t t e A t e U t t U t t e B t e U t tS t t A t S t t B t S t t
0 00 0
'
( ) ( )
( '') ''
( , ') ( ) ( , '), ( )
( , ') , '
tIn
t
H Hi t t i t tI I Sn n n
i H t dt
i S t t H t S t t H t e H et
S t t Te t t
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Contour-ordered Green’s function
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CIn
C
dHiIk
IjCI
dHi
kjC
kjjk
euuT
euuTt
uuiG
)(
)(
',,0
)'()(Tr
)(Tr
)'()()',(
t0
τ’
τ13n ijk i j k
ijk
H T u u u
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Perturbative expansion of contour ordered Green’s function
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( '') ''
2
1 1 1 1 2 2
3
1 2 3 1 2 3
( , ') ( ) ( ')
1( ) ( ') 1 ( ) ( ) ( )2!
1 1( ) ( ') ( ) ( ') ( , , ) ( ) ( ) ( )3! 3
ni H d
jk C j k
C j k n n n
C j k C j k lmn l m n
iG T u u e
i i iT u u H d H d H d
i iT u u T u u T u u u
1 2 3
4 5 6 4 5 6 4 5 6
01 2 3 4 5 6
1 2 5 3 6 4
1 ( , , ) ( ) ( ) ( )3
( , ') ... ( ) ( ') ( ) ( ) ( ) ( ) ( ) ( )
(Wick's theorem)
( ) ( ) ( ) ( ) ( ) ( ) ( )
lmn
opq o p qopq
jk C j k l m n o p q
C j l C m p C n q C o
d d d
T u u u d d d
G T u u u u u u u u
T u u T u u T u u T u u
( ')k
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General expansion rule
1 2 1 2
0
1 2 1 2
( , ')( , , )
( , , , ) ( ) ( ) ( )n n
ijk i j k
j j j n C j j j n
GT
iG T u u u
Single line
3-line vertex
n-double line vertex
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Diagrammatic representation of the expansion
46
21221100 )',(),(),()',()',( ddGGGG n
= + 2iħ + 2iħ
+ 2iħ
= +
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Self -energy expansion
47
Σn =
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Explicit expression for self-energy
48
' ' ', 0, 0,
'' '' '', ' 0, 0,
, ''
4
'[ ] 2 [ '] [ ']2
'2 '' [0] [ ']2
( )
n jk jlm rsk lr mslmrs
jkl mrs lm rslmrs
ijk
di T T G G
di T T G G
O T
![Page 49: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/49.jpg)
Junction system, adiabatic switch-on
49
• gα for isolated systems when leads and centre are decoupled
• G0 for ballistic system• G for full nonlinear system
t = 0t = −
HL+HC+HR
HL+HC+HR +V
HL+HC+HR +V +Hn
gG0
G
Equilibrium at Tα
Nonequilibrium steady state established
![Page 50: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/50.jpg)
Sudden Switch-on
50
t = ∞t = −
HL+HC+HR
HL+HC+HR +V +Hn
g
Green’s function G
Equilibrium at Tα
Nonequilibrium steady state established
t =t0
![Page 51: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/51.jpg)
Three regions
51
RCLuuTiG
uuu
uuuu
u
TC
CL
L
L
R
C
L
,,,,)'()()',(
,, 2
1
![Page 52: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/52.jpg)
Heisenberg equations of motion in three regions
52
,1 1 ,2 2
1 1 1[ , ], [ , ],
, ,
T LC T RCL C R L C R C n
T T
C CL CRC C C L R C n
CC
H H H H u V u u V u H
H u u u K u
u u H H K u V u V u u Hi i i
u K u V u L R
![Page 53: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/53.jpg)
53
Force Constant Matrix
0,
0
1 12 2
L LC
CL C CR
RC R
LT T T T
L C R C n
R
L
C
R
K VK V K V
V K
uH p p u u u K u H
uu
p u uu
![Page 54: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/54.jpg)
Relation between g and G0
54
Equation of motion for GLC
2
2
2
2
( , ') ( ) ( ') ,
( , ') ( ) ( ')
( , ') ( , '),
( , ') ( , '') ( '', ') '',
( , ') ( , ') ( , ')
TLC C L C
TLC C L C
L LCLC CC
LCLC L CC
LL L
iG T u u
iG T u u
K G V G
G g V G d
g K g I
![Page 55: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/55.jpg)
Dyson equation for GCC
55
RCR
CRLCL
CL
CCCCCC
CCRC
RCR
CCLC
LCL
CCC
RCCR
LCCL
CCC
TCCCCC
TCCCCC
VgVVgV
ddGggG
IdGVgV
dGVgVGK
IGVGVGK
IuuTiG
uuTiG
)',()',()',(
,)',(),(),()',()',(
),',('')',''()'',(
'')',''()'',()',(
)',()',()',()',(
)',()'()()',(
,)'()()',(
212211
2
2
![Page 56: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/56.jpg)
56
Equation of Motion Way (ballistic system)
2
2
2
2
( , ') ( , ') ( , ')
( , ') ( , ') ( , ')
0 0 0 0, 0 0 , 0
0 0 0 0
( , ') ( , ') '' ( , '') ( '', ')
L LC
C CL CR
R RC
C
G KG I
g Dg I
K VK D V D K V V V
K V
G g d g VG
![Page 57: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/57.jpg)
57
“Partition Function”
0 0 0
( ) ( )
0
Tr ( ) ( , ) ( , )
Tr ( )
1
nI I
C
i V H d
c
Z t U t t U t t
t T e
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58
Diagrammatic WayFeynman diagrams for the nonequilibrium transport problem with quartic nonlinearity. (a) Building blocks of the diagrams. The solid line is for gC, wavy line for gL, and dash line for gR; (b) first few diagrams for ln Z; (c) Green’s function G0
CC; (d) Full Green’s function GCC; and (e) re-sumed ln Z where the ballistic result is ln Z0 = (1/2) Tr ln (1 –gCΣ). The number in front of the diagrams represents extra combinatorial factor.
![Page 59: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/59.jpg)
The Langreth theorem
59
aaarrr
rrrr
ar
ar
rrrrrr
CBACBACBAD
CBAD
ddCBAD
BABAC
dtttBttAdtttBttAttC
BACdtttBttAttC
dtttBttAdBAC
,
)',(),(),()',(
][][][][][
'')',''()'',('')',''()'',()',(
][][]['')',''()'',()',(
'''')',''()'',('')',''()'',()',(
212211
''
'''''
![Page 60: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/60.jpg)
Dyson equations and solution
60
an
r
aan
rn
ran
r
rn
rr
ar
rCr
n
CC
GG
GIGGIGGG
GG
GGG
KIiG
GGGGGggG
)(
)()(
,)(
0,)(
,
0
110
000
120
00
00
![Page 61: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/61.jpg)
Energy current
61
0
( ', ) ( ', )( , ') ( , ') '
1 Tr [ ]2
1 Tr [ ] [ ] [ ] [ ]2
T LCLL L C
t ar L LCC CC
t
LCCL
r aCC L CC L
dHI u V udt
t t t ti G t t G t t dtt t
V G d
G G d
![Page 62: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/62.jpg)
Landauer/Caroli formula
62
0
1 Tr2
,2
,
( )
r aLL CC L CC R L R
r a
L RL
r aL L R R
a r r aL R
dHI G G f f ddt
i
I II
G G G i f f
G G iG G
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63
1D calculation
• In the following we give a complete calculation for a simple 1D chain (the baths and the center are identical) with on-site coupling and nearest neighbor couplings. This example shows the general steps needed for more general junction systems, such as the need to calculate the “surface” Green’s functions.
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Ballistic transport in a 1D chain
• Force constants
• Equation of motion64
kkkk
kkkkkkkk
k
K
0020
202
0
0
0
0
1 0 1(2 ) , , 1,0,1,2,j j j ju ku k k u ku j
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Solution of g
65
2
0
0
0
0
1 20
( ) , 0
2 02 0
0 20 0
, 0,1,2, ,
(( ) 2 ) / 0, | | 1
RR
R
jRj
i K g I
k k kk k k k
Kk k k k
k
g jk
i k k k
![Page 66: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/66.jpg)
Lead self energy and transmission
66
2 1
| |
1
20 0
0 00 0 0 00 0 0
0 0 0
( ) ,
1, 4[ ] Tr
0, otherwise
L
r CL R
j krjk
r aL R
k
G K
Gk
k k kT G G
T[ω]
ω
1
![Page 67: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/67.jpg)
Heat current and conductance
67
max
min
0
2 2
0
[ ]2
1lim ,2 1
, 0, 03
L R
L L R
L
T TL R
B
dI T f f
I f d fT T T e
k T T kh
![Page 68: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/68.jpg)
General recursive algorithm for g
68
10
1110
011110
0100
00
00
kkkkkk
kk
K R
1200
12
01
11
00
)(
)|(| while}
)(
{ do
sIig
gggee
gss
eIig
kkeks
T
14
5
10
10
![Page 69: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/69.jpg)
69
Lecture Three
Applications
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70
CARBON NANOTUBE (6,0), FORCE FIELD FROM GAUSSIAN
Disp
ersio
n re
latio
n
Transmission
Calculation was performed by J.-T. Lü.
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Carbon nanotube, nonlinear effect
71
The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300K. From J-S Wang, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006).
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72
u4 Nonlinear modelOne degree of freedom (a) and two degrees freedom (b) (1/4) Σ Tijkl ui uj uk ul
nonlinear model. Symbols from quantum master equation, lines from self-consistent NEGF. For parameters used, see Fig.4 caption in J.-S. Wang, et al, Front. Phys. Calculated by Juzar Thingna.
1
5
10
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Molecular dynamics with quantum bath2
2
, , , ,
( ) ( ) ( ') ( ') '
( ) ( ') ( '), , ,
2,
tC rC
C L R
T
r C r C r r rL R
d u t F t t t u t dtdt
t t i t t L R
V g V
See J.-S. Wang, et al, Phys. Rev. Lett. 99, 160601 (2007);Phys. Rev. B 80, 224302 (2009).
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Average displacement, thermal expansion
74
One-point Green’s function
0 0
( ) ( )
' '' ''' ( ', '', ''') ( ', '') ( ''', )
( 0) [ 0]
1
j C j
lmn lm njlmn
rj lmn lm nj
lmn
jj
j
iG T u
d d d T G G
G T G t G
dGiM x dT
![Page 75: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/75.jpg)
Thermal expansion
75
(a) Displacement <u> as a function of position x.
(b) as a function of temperature T. Brenner potential is used. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).
Left edge is fixed.
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Graphene Thermal expansion coefficient
The coefficient of thermal expansion v.s. temperature for graphene sheet with periodic boundary condition in y direction and fixed boundary condition at the x=0 edge. is onsite strength. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).
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77
Phonon Life-Time
( )2
2 2,
,
,
2
1 , ( )[ ]
Re [ ] 2 ,
Im [ ]
13
q qq
ti tr rq qr
q n q
rn q q q q
qrn q q
q
q q qq
G G t ei
c v
For calculations based on this, see, Y. Xu, J.-S. Wang, W. Duan, B.-L. Gu, and B. Li, Phys. Rev. B 78, 224303 (2008).
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Transient problems
781 2
0
0
0
,1 2
2
, 0( )
, 0
( , )( ) Im
L R
T LRL R
RL
Lt t t
H H H
H tH t
H u V u t
G t tI t kt
![Page 79: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/79.jpg)
Dyson equation on contour from 0 to t
79
1 1 1
1 2 1 1 2 2,
( , ') ( ) ( ')
( ) ( ) ( ')
RL R RL L
C
R RL L LR RL
C C
G d g V g
d d g V g V G
Contour C
![Page 80: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/80.jpg)
80
TRANSIENT THERMAL CURRENT
The time-dependent current when the missing spring is suddenly connected. (a) current flow out of left lead, (b) out of right lead. Dots are what predicted from Landauer formula. T=300K, k =0.625 eV/(Å2u) with a small onsite k0=0.1k. From E. C. Cuansing and J.-S. Wang, Phys. Rev. B 81, 052302 (2010). See also arXiv:1005.5014.
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81
FINITE LEAD PROBLEM
Top : no onsite (↥ k0=0), (a) NL = NR = 100, (b) NL=NR=50, temperature T=10, 100, 300 K (black, red, blue). TL=1.1T, TR = 0.9T.
Right : with onsite ↦ k0=0.1k and similarly for other parameters. From E. C. Cuansing, H. Li, and J.-S. Wang, Phys. Rev. E 86, 031132 (2012).
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82
Full counting statistics (FCS)• What is the amount of energy (heat) Q transferred in a
given time t ?• This is not a fixed number but given by a probability
distribution P(Q)• Generating function
• All moments of Q can be computed from the derivatives of Z.
• The objective of full counting statistics is to compute Z(ξ).
dQQPeZ Qi )()(
![Page 83: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/83.jpg)
A brief history on full counting statistics
• L. S. Levitov and G. B. Lesovik proposed the concept for electrons in 1993; rederived for noninteracting electron problems by I. Klich, K. Schönhammer, and others
• K. Saito and A. Dhar obtained the first result for phonon transport in 2007
• J.-S. Wang, B. K. Agarwalla, and H. Li, PRB 2011; B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 2012.
83
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Definition of generating function based on two-time measurement
84
( )
2
/
Tr '
'
| || |( ) (0, ) ( ,0)
e.g., ( ,0)
L Li H i H t
a aa
a a
L
L L
iHt
Z e e
P P
P P a aH a a aH t U t H U t
U t e
Consistent history quantum mechanics (R. Griffiths).
![Page 85: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/85.jpg)
Product initial state
85
, ,
( )
/ 2 / 2
/ 2 / 2
, [ , ] 0, '
Tr
Tr (0, ) ( ,0)
Tr (0, ) ( ,0)
Tr (0, ) ( ,0)
( , ') ( , ')
L L
L L
L L L
L L
HL L
L C R
i H i H t
i H i H
i H i H i H
ixH ixHx
e P H
Z e e
e U t e U t
e U t e U t e
U t U t
U t t e U t t e
![Page 86: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/86.jpg)
86
'
'
( )
( )
0 0
( , ')
, '
( ) ( )
( )
( )
( )
ti
tL L
ti
xt
L L
L L
L L
L L
H t dtixH ixHx
H t dt
ixH ixHx
ixH ixHLC RCL C R
ixH ixHRC LCL C R
RC L T LC C xx I
ixH ixHL L Lx
U t t e Te e
Te t t
H t e H t e
e H H H H H e
H H H H e H e
H H u V u H H
u e u e u x
![Page 87: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/87.jpg)
Schrödinger/Heisenberg/interaction pictures
87tHitHi
I
dttHi
I
I
tHi
I
I
HHH
HH
i
AeetA
ttTettSttStHtttSi
tttStet
HHH
tHtAt
tAi
tAUtUtA
ttUtHt
ttUi
iiwtttUt
t
t I
00
'
0
)(
',)',(),',()()',(
)'(|)',()(|)(|
)](),([)()0(||),0,(),0()(
)',()()',(
||)'(|)',()(|
'')''(
0
Schrödinger picture
Heisenberg picture
Interaction picture
![Page 88: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/88.jpg)
Compute Z in interaction picture
88
LxL
ALRLCC
AL
ALCR
xLC
LCxL
CLC
C C
xI
xIC
xIC
dHi
C
GggG
G
GggZ
VgVg
ddHHidHiT
eT
tUtUZ
C
xI
,,
)]1ln[(Tr21
)]1)(1det[(ln21)](1[lnTr
21ln
][Tr211
')'()(21)(1Tr
]Tr[
)]0,(),0([Tr
00
0
0
2
)(
2/2/
-ξ/2
+ξ/2
![Page 89: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/89.jpg)
Important result
89
otherwise00if2/)(
0if2/)()(
)',()'('),()',(
,,
)1ln(Tr21ln
00
0
M
M
LLAL
LxL
ALRLCC
AL
tttx
tttxx
xx
GggG
GZ
![Page 90: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/90.jpg)
Long-time result, Levitov-Lesovik formula
90
)/(1),1/(1
,),(
)1]([),1]([
where
)2()1()1(detln4
01lnTr
21
)1ln(Tr21ln
00
0
0
0
22
22
Tkef
RLi
ebea
ffeefefeGGIdt
GGG
GZ
Ba
ar
iL
iL
RLii
Ri
Li
Ra
Lr
M
a
Kr
AL
baba
baba
![Page 91: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/91.jpg)
Arbitrary time, transient result
91 time)long in(
)(ln
)(Tr
21
)(ln
)(ln
)1ln(Tr21ln
2
2222
00
0
0
ItQ
iZQQQ
iG
iZQQ
iZQ
GZ
M
AL
n
nn
AL
![Page 92: Nonequilibrium Green’s Function Method in Thermal Transport](https://reader035.vdocuments.us/reader035/viewer/2022062315/568163e6550346895dd548ab/html5/thumbnails/92.jpg)
Numerical results, 1D chain
92
1D chain with a single site as the center. k= 1eV/(uÅ2), k0=0.1k,TL=310K, TC=300K, TR=290K. Red line right lead; black, left lead. B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 85, 051142, 2012.
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Cumulants in Finite SystemPlot of the culumants of heat <<Qn>> for n=1,2,3, and 4 for one-atom center connected with two finite lead (one-dimensional chain) as a function of measurement time tM. The black and red line correspond to NL =NR =20 and 30, respectively. The initial temperatures of the left, center, right parts are 310K, 360K, and 290K, respective. We choose k=1 eV/(uÅ2) and k0=0.1 eV/(uÅ2) for all particles. From J.-S. Wang, et al, Front. Phys. 2013.
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FCS for coupled leads
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,
LC LRL
CL CRC
RL RCR
K V VK V K V G g gVG
V V K
1 1
ln ln det ( 1) (1 ) ( 1) (1 ) [ ]4
[ ] , , ,
If center is absent 0, then
[ ] [ ]
i iM L R R L g
r a a rg LR R RL L
LC RC
r ag I RR R RR L
dZ t I e f f e f f T
T G G i g g L R
V V
T T G G
From H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. E 86, 011141 (2012); H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. B 86, 165425 (2012). For multiple leads, see B. K. Agarwalla, et al, arXiv:1308.1604
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FCS in nonlinear systems
• Can we do nonlinear?
• Yes we can. Formal expression generalizes Meir-Wingreen
• Interaction picture defined on contour
• Some example calculations
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Vacuum diagrams, Dyson equation, interaction picture transformation on
contour
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,
[
Lj,
0 0
( )I1 2 1 2
0 0
( )
ln 1 Tr( ) 2 ( )
1( , ) Tr ( ) ( )
, (0 , ) ( ,0 ) /
( ) (0 , ) ( ) ( ,0 )
( ,0 )
InC
x T LC T CRL C C R
C
AL n
i H dI I TC
n
In M M
I
i u V u u V u d
C
Z Gi i
G G G G
iG T u u eZ
Z Z Z V t V t Z
O V O V
V T e
,0 ]
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Nonlinear system self-consistent results
97
Single-site (1/4)λu4 model cumulants. k=1 eV/(uÅ2), k0=0.1k, Kc=1.1k, V LC
-1,0=V CR
0,1=-0.25k, TL=660 K, TR=410K.
From H. Li, B. K. Agarwalla, B. Li, and J.-S. Wang, arxiv::1210.2798
( , ') 3 ( , ') ( , ')n i G
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Quantum Master Equations
• When we tried to calculation the reduced density matrix from Dyson expansion, we found that the 2nd order (of the system-bath coupling) diagonal terms do not agree with master equation results, and there are also divergences in higher orders. Similarly, the currents also diverge at higher order. We were quite puzzled. The following is our effort to solve this puzzle and found an expression of the high order current which is finite.
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Quantum Master Equations0
( )
0
( )
0
2 42 4 6
0
( ) Tr ( , ) ( ) ( , )
Tr ( ) ,
( ) Tr ( ) [ ( ), ( )] ,
( )2! 4!
| |, Tr
C
C
IH B
V dI
B c
V dI
H B c
T T T
Inm
O t S t O t S t
iT O t e
d O t T O t O t V t edt
X X V X V O
X n m
.B cT
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Unique one-to-one map ρ ↔ ρ0
2 4 42 4 2 2
0 2
42 3
42
4 42 3 2 6
2! 4! 2!
[ , ] [ , ]3!
[ , ]2!
( )3! 2!
I T T T T
T T
T T
TH
T LCL C
X V X V X V X V
d i X V V X V Vdt
X V X V V
I VV VV X V VV
V p V u
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Group & AcknowledgementsGroup members: front (left to right) Mr. Zhou Hangbo, Prof. Wang Jian-Sheng, Dr./Ms Lan Jinghua, Mr. Wong Songhan; Back: Mr. Li Huanan, Dr. Zhang Lifa, Dr. Bijay Kumar Agarwalla, Mr. Kwong Chang Jian, Dr. Thingna Juzar Yahya. Picture taking 17 April 2012. Others not in the picture: Qiu Hongfei. Thanks also go to Jian Wang, Jingtao Lü, Jin-Wu Jiang, Edardo Cuansing, Meng Lee Leek, Ni Xiaoxi, Baowen Li, Peter Hänggi, Pawel Keblinski.http://staff.science.nus.edu.sg/~phywjs/