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8/18/2019 Many Body Lecture 3 (1)
1/35
Many-body Green’s Functions
• Propagating electron or hole interacts with other e -/h +• Interactions modify renormali!e " electron or hole energies• Interactions produce finite lifetimes for electrons/holes #uasi-particles "• $pectral function consists of #uasi-particle pea%s plus &bac%ground’• 'uasi-particles well defined close to Fermi energy
• M(GF defined by
{ }
o
oHHo )t','(ψ̂t),(ψ̂)t','t,,G(
Ψ
ΨΨ= +
state)ground*eisenberge+acto,er a,eragedoperator fieldof functionncorrelatioi e
rrrr T i
8/18/2019 Many Body Lecture 3 (1)
2/35
Many-body Green’s Functions
• $pace-time interpretation of Green’s function• x,y" are space-time coordinates for the endpoints of the Green’s function• Green’s function drawn as a solid) directed line from y to x • .on-interacting Green’s function G o represented by a single line• Interacting Green’s Function G represented by a double or thic% single line
time
dd particle 0emo,e particle
t 1 t’t’
time
0emo,e particle dd particle
t’ 1 tt
x
y
y
)t'(t)t',(ψ̂t),(ψ̂ oHHo −ΨΨ + θ yx
t)(t't),(ψ̂)t',(ψ̂ oHHo −ΨΨ +
θ xy
x
G o x)y"x)ty)t’
G x)y"x)ty)t’
8/18/2019 Many Body Lecture 3 (1)
3/35
Many-body Green’s Functions
• 2ehmann 0epresentation F 34 M 534" physical significance of G
{ }
oo
onn
n
oSnnSo-o-n
oSnnSo
oHnnHooHHo
oHHooHHo
nn
n
o
oHHo
tiEtĤitiE-tĤi-
)t'tEi(E-
t'Ĥi-t'ĤitĤi-tĤi
ee ee
)'(ψ̂)(ψ̂)(e
)e'(ψ̂e)e(ψ̂e
)t','(ψ̂t),(ψ̂)t','(ψ̂t),(ψ̂
t)(t't),(ψ̂)t','(ψ̂-)t'(t)t','(ψ̂t),(ψ̂)t','t,,G(
)t','(ψ̂t),(ψ̂)t','t,,G(
Ψ=ΨΨ=Ψ
ΨΨΨΨ=
ΨΨΨΨ=
ΨΨΨΨ=ΨΨ
−ΨΨ−ΨΨ==ΨΨ
ΨΨ
ΨΨ=
++
++
+
+
++
++
+
rr
rr
rrrr
rrrrrr
1
rrrr
θ θ i
T i
formalismnumber occupationinoperator unit
number particleany)state)*eisenberge+act
state)ground*eisenberge+act
8/18/2019 Many Body Lecture 3 (1)
4/35
Many-body Green’s Functions
• 2ehmann 0epresentation physical significance of G"
{ }
onebyinnumber particlereduces ooS
oSoSSS
on
oSnnSo
on
oSnnSo
-o-noSnnSo
-o-noSnnSo
oHHo
ψ̂
ψ̂)1 N(ψ̂n̂ )(ψ̂)(ψ̂dn̂
δ)EE(εψ̂ψ̂
δ)EE(εψ̂ψ̂
e)t','t,,)G(t'-d(t),',G(
t)(t')(e)(ψ̂)'(ψ̂
-)t'(t)(e)'(ψ̂)(ψ̂)t','t,,G(
)t','(ψ̂t),(ψ̂)t','t,,G(
)t'(t
)t'tEi(E
)t'tEi(E-
ΨΨΨ−=Ψ=
−−+ ΨΨΨΨ++−− ΨΨΨΨ=
=
−ΨΨΨΨ−
−ΨΨΨΨ=
ΨΨ=
+
++
∞+
∞−
+
+
+
∫
∫ −++
rrr
rrrr
rr
rrrr
rrrr
ii
ii
i
T i
iε ε
θ
θ
8/18/2019 Many Body Lecture 3 (1)
5/35
Many-body Green’s Functions
• 2ehmann 0epresentation physical significance of G"
µ
µ
+−−−=−−
−−+−−−=−−
−ΨΨ=ΨΨΨΨ
++−+=−+
−+++−+=−++ΨΨ=ΨΨΨΨ
+
++
)1 N(E)1 N(E) N(E)1 N(E
) N(E)1 N(E)1 N(E)1 N(E) N(E)1 N(E
ψ̂ψ̂ψ̂
)1 N(E)1 N(E) N(E)1 N(E
) N(E)1 N(E)1 N(E)1 N(E) N(E)1 N(E
ψ̂ψ̂ψ̂
onon
ooonon
2
nSooSnnSo
onon
ooonon
2
oSnoSnnSo
statesparticle6.and.connects
statesparticle6.and.connects
8/18/2019 Many Body Lecture 3 (1)
6/35
Many-body Green’s Functions
• 2ehmann 0epresentation physical significance of G"• Poles occur at e act .+6 and .-6 particle energies• Ionisation potentials and electron affinities of the . particle system• Plus e citation energies of .+6 and .-6 particle systems
• 7onnection to single-particle Green’s function
Fbelowstatesfor asstatesunoccupiedtolimited$um
unoccupied
stategroundg"interactin-nonparticle-singletheis
ε
δ θ
θ
θ
ε
00ĉ
n0ĉĉ0 )t'(t)e'(ψ)(ψ
)t'(t0)(t'ĉ(t)ĉ0)'(ψ)(ψ
)t'(t0)t','(ψ̂t),(ψ̂0)t','t,,(G
0
n
mnnmn*
n
unocc
nn
nm*n
nm,m
HHo
)t'-(t-
=
∈=−=
−=
−=
+
+
+
++
∑
∑i
i
rr
rr
rrrr
8/18/2019 Many Body Lecture 3 (1)
7/35
Many-body Green’s Functions
• Gell-Mann and 2ow 8heorem F 96) :5"• ; pectation ,alue of *eisenberg operator o,er e act ground state
e pressed in terms of e,olution operators and the operator in #uestion ininteraction picture and ground state of non-interacting system
oIo
oIIIo
oo
oHo
)-,(Û
)(t,-Û(t)Ôt),(Û(t)Ô
φ φ
φ φ
∞+∞∞+∞=ΨΨ
ΨΨ
oφ
{ }oo
oHHo )t','(ψ̂t),(ψ̂
)t','t,,G( ΨΨΨΨ
=
+ rr
rr
T
i
FunctionsGreen
8/18/2019 Many Body Lecture 3 (1)
8/35
Many-body Green’s Functions
• Perturbati,e ; pansion of Green’s Function F :5"
• ; pansion of the numerator and denominator carried out separately• ;ach is e,aluated using >ic%’s 8heorem• ?enominator is a factor of the numerator • @nly certain classes of connected " contractions of the numerator sur,i,e• @,erall sign of contraction determined by number of neighbour permutations• n A B term is Cust G o x)y"• x ) y are compound space and time coordinates i e x D ) y) !) t "
( )( ) [ ]
( ) ( ) [ ]o- -
nI2I1I-
on210n
n
oIo
o- -
nI2I1I-
on210n
n
oIo
)(tĤ)!!!(tĤ)(tĤdt!!!dtdtn"
,Û
)(ψ̂)(ψ̂)(tĤ)!!!(tĤ)(tĤdt!!!dtdtn",Û
1),G(
φ φ φ φ
φ φ φ φ
∫ ∫ ∫ ∑
∫ ∫ ∫ ∑∞+
∞
∞+
∞
∞+
∞
∞
=
+∞
∞
+∞
∞
++∞
∞
∞
=
−=−∞∞+
−−∞∞+
=
T i
T i
i yxyx
8/18/2019 Many Body Lecture 3 (1)
9/35
Many-body Green’s Functions
• Fetter and >alec%a notation for field operators F ::"
( )( )( )( ) +−
−+−
+++++
+++++
++
≤=>=≤=
+>=+≡+=
+≡+=
#̂ #̂- tt ),(G
tt 0)(ψ̂)(ψ̂
tt 0
$̂$̂ tt ),(G)(ψ̂)(ψ̂
#̂$̂)(ψ̂)(ψ̂)(ψ̂
#̂$̂)(ψ̂)(ψ̂)(ψ̂
%&o
%&)()(
%&
%&o)()(
(-))(
(-))(
yx
yx
yxyx
xxx
xxx
i
i
( )( )( ) ( )
0ψ̂ψ̂ 0ψ̂ψ̂ 0 #̂ #̂$̂ #̂ #̂$̂$̂$̂
#̂ #̂$̂ #̂ #̂$̂$̂$̂
#̂$̂ #̂$̂
ψ̂ψ̂ψ̂ψ̂ψ̂ψ̂ (-))()()(
======
+++=++≡++=
++++++
++++
++
+++−+++++
similarly
8/18/2019 Many Body Lecture 3 (1)
10/35
Many-body Green’s Functions
•.on!ero contractions in numerator of M(GF
-6"5 i"5, r )r ’"Go r ’)r " G o r )r ’" Go x)y"
-6"E i"5, r )r ’"Go r )r " G o r ’)r ’" Go x)y"
-6"= i"5, r )r ’"Go x)r " G o r ’)r ’" Go r )y"
-6"E i"5, r )r ’"Go r ’)r " G o x)r ’" Go r )y"
-6"9 i"5, r )r ’"Go x)r " G o r )r ’" Go r ’)y"
-6"3 i
"5
, r )r ’"Go r )r " G o x)r ’" Go r ’)y"( ) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
( ) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
( ) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
( ) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
(2) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
(1) )(ψ̂)(ψ̂)(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
+++
+++
+++
+++
+++
+++
8/18/2019 Many Body Lecture 3 (1)
11/35
Many-body Green’s Functions
• .on!ero contractions- i"5, r )r ’"Go r ’)r " G o r )r ’" Go x)y" 6"
+ i"5, r )r ’"Go r )r " G o r ’)r ’" Go x)y" 4"
- i"5, r )r ’"Go x)r " G o r ’)r ’" Go r )y" 5"
+ i"5, r )r ’"Go r ’)r " G o x)r ’" Go r )y" E"
+ i"5, r )r ’"Go x)r " G o r )r ’" Go r ’)y" ="
- i"5, r )r ’"Go
r )r " Go
x)r ’" Go
r ’)y" 9"
y
x
r r ’
y
x
r r ’
x
y
r r ’
y
r r ’
x
y
r ’ r
x x
y
r ’ r
6" 4"
5" E"
=" 9"
8/18/2019 Many Body Lecture 3 (1)
12/35
• .on!ero contractions in denominator of M(GF• ?isconnected diagrams are common factor in numerator and denominator
Many-body Green’s Functions
(+) )(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
( ) )(ψ̂)'(ψ̂)'(ψ̂)(ψ̂
rrrr
rrrr
++
++-6"5 i"4, r )r ’"Go r ’)r " G o r )r ’"
-6"E i"4, r )r ’"Go r )r " G o r ’)r ’"
r r ’3"
r r ’:"
?enominator A 6 + + +
.umerator A 6 + + + H + + + H
8/18/2019 Many Body Lecture 3 (1)
13/35
• ; pansion in connected diagrams
• $ome diagrams differ in interchange of dummy ,ariables
• 8hese appear m ways so m term cancels• 8erms with simple closed loop contain time ordered product with e#ual times• 8hese arise from contraction of *amiltonian where adCoint operator is on left• 8erms interpreted as
Many-body Green’s Functions
∑ ∫ ∫ ∞
=
∞
∞−
∞
∞−
+−=0m connected
om111om1 )(ψ̂)(ψ̂)(tĤ !!!)(tĤ.dt!!!dtm")(
),G( φ φ yxyx T i
i
iG x) y" A + + +
{ }
densitychargeginteractin-non )(/)(ψ̂)(ψ̂
)t',(ψ̂t),(ψ̂),(G
ooo
ooim
'o
xxx
xxxx
−=−=
=
+
++→
φ φ
φ φ δ T i t t
8/18/2019 Many Body Lecture 3 (1)
14/35
• 0ules for generating Feynman diagrams in real space and time F J3"
• a" ?raw all topologically distinct connected diagrams with m interaction lines and4m+6 directed Green’s functions Fermion lines run continuously from y to or closeon themsel,es Fermion loops "
• b" 2abel each ,erte with a space-time point x A r )t"• c" ;ach line represents a Green’s function ) G o x)y") running from y to x• d" ;ach wa,y line represents an unretarded 7oulomb interaction• e" Integrate internal ,ariables o,er all space and time• f" @,erall sign determined as -6" F where F is the number of Fermion loops
• g" ssign a factor i"
m
to each mth
order term• h" Green’s functions with e#ual time arguments should be interpreted as G r )r ’)t)t+"where t + is infinitesimally ahead of t
• ; ercise K Find the 6B second order diagrams using these rules
Many-body Green’s Functions
8/18/2019 Many Body Lecture 3 (1)
15/35
• Feynman diagrams in reciprocal space
• For periodic systems it is con,enient to wor% in momentum space• 7hoose a translationally in,ariant system homogeneous electron gas "• Green’s function depends on x-y) not x)y• G x)y" and the 7oulomb potential) L) are written as Fourier transforms• E-momentum is conser,ed at ,ertices
Many-body Green’s Functions
( )
t-!! ddd
)e',()'-d()(
)eG(
2
d),G(
)'!(
)!(
ω ω
π
xk xk k k
rrrrq
k k
yx
rrq
yxk
≡≡=
=
∫ ∫
−
−
i-
i
Fourier 8ransforms
( ) ( *21*21 2eeed!!!
qqqxxqxqxq −−=+∫ δ π -i-ii
E-momentum 7onser,ation
q 6
q 4 q 5
8/18/2019 Many Body Lecture 3 (1)
16/35
• 0ules for generating Feynman diagrams in reciprocal space
• a" ?raw all topologically distinct connected diagrams with m interactionlines and 4m+6 directed Green’s functions Fermion lines run continuouslyfrom y to or close on themsel,es Fermion loops "
• b" ssign a direction to each interaction• c" ssign a directed E-momentum to each line• d"7onser,e E-momentum at each ,erte• e" ;ach interaction corresponds to a factor , q "
• f"Integrate o,er the m internal E-momenta• g" ffi a factor i"m/ 4π"Em -6"F• h" closed loop or a line that is lin%ed by a single interaction is assigned a
factor e iεδ G o k)ε"
Many-body Green’s Functions
8/18/2019 Many Body Lecture 3 (1)
17/35
[ ]
[ ])(ψ̂)(ψ̂
1)(ψ̂)(ψ̂ddĤ
)(ψ̂)(ψ̂1)(ψ̂dĤ,ψ̂ψ̂t
)(ψ̂)(ˆ)(ψ̂dĤ
)(ψ̂)(ˆ Ĥ,ψ̂ψ̂t
1H2H21
2H1H21H
H2H2
2H2HHH
1H11H1H
HHHH
2
1rr
rrrrrr
rrrr
rr
rrrr
rr
−=
−==∂∂
=
==∂∂
++
+
+
∫ ∫ ∫
∫
for
for
i
i
;#uation of Motion for the Green’s Function
• ;#uation of Motion for Field @perators from 2ecture 4"
{ }oo
oHHo )t','(ψ̂t),(ψ̂)t','t,,G(
ΨΨΨΨ
=+ rr
rrT
i
8/18/2019 Many Body Lecture 3 (1)
18/35
;#uation of Motion for the Green’s Function
• ;#uation of Motion for Field @perators
[ ] [ ]
t),(ψˆ
t),(ψˆ1
t),(ψˆ
dt),(ψˆ
t),(ˆ
t
t),(ψ̂t),(ψ̂1
t),(ψ̂dt),(ψ̂t),(ˆ
tĤe)(ψ̂)(ψ̂
1)(ψ̂d
tĤe
tĤe)(ψ̂)(
ˆ
tĤe
tĤeĤ,ψ̂tĤet),(Ĥt),,(ψ̂t),(ψ̂t
H2H2
2H2H
H2H2
2H2H
22
22
SSHHH
rrrrrrrr
rrrr
rrrr
rrrrrrrr
rrr
−=−∂∂
−+=
−−
++
−+=
−+==∂∂
+
+
+
∫
∫ ∫
i
iiii
iii
8/18/2019 Many Body Lecture 3 (1)
19/35
;#uation of Motion for the Green’s Function
• ?ifferentiate G wrt first time argument{ }
[ ][ ] )t'-(t)-()t'-(t)t',(ψ̂t),,(ψ̂
)t'-(t)t',(ψ̂t),,(ψ̂
(t'-t)t
t),(ψ̂)t',(ψ̂-)t'-(t)t',(ψ̂t
t),(ψ̂
)t'-(tt),(ψ̂)t',(ψ̂--)t'-(t)t',(ψ̂t),(ψ̂
(t'-t)t
t),(ψ̂)t',(ψ̂-)t'-(t)t',(ψ̂
tt),(ψ̂
(t'-t)t),(ψ̂t
)t',(ψ̂-)t'-(t)t',(ψ̂t),(ψ̂t
)t',t,,G(t
)t',(ψ̂t),,(ψ̂)t',t,,G(
oooHHo
oHHo
oH
HHH
o
oHHHHo
oH
HHH
o
oHHHHo
oHHo
δ δ δ
δ
θ θ
δ δ
θ θ
θ θ
yxyx
yx
xyyx
xyyx
xyy
x
xyyxyx
yxyx
ΨΨ=ΨΨ
ΨΨ
+Ψ∂∂∂∂Ψ=
ΨΨ+
+Ψ∂∂
∂∂Ψ=
Ψ∂∂
∂∂Ψ=∂
∂
ΨΨ=
++
++
++
++
++
++
+
i
T i
8/18/2019 Many Body Lecture 3 (1)
20/35
;#uation of Motion for the Green’s Function
• ?ifferentiate G wrt first time argument
[ ]
[ ]
)t'-(t)-(
)t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂1
d)t',t,,G(ˆt
)t'-(t)-(
)t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂1
d),G(ˆ
)t'-(t)-(
(t'-t)t),(ψ̂t),(ψ̂t),(ψ̂)t',(ψ̂-1
d
)t'-(t)t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂1
d
(t'-t)t),(ψ̂)t',(ψ̂-)t'-(t)t',(ψ̂t),(ψ̂ˆ)t',t,,G(t
oHH1H1Ho1
1
oHH1H1Ho1
1
oH1H1HHo1
1
oHH1H1Ho1
1
oHHHHo
δ δ
δ δ
δ δ
θ
θ
θ θ
yx
yxrrrx
ryx
yx
yxrrrrryx
yx
xrryrx
r
yxrrrx
r
xyyxyx
=
ΨΨ−+
−∂∂
+ΨΨ−−−=
+
ΨΨ−−
+ΨΨ−−
+ΨΨ−=∂∂
++
++
++
++
++
∫
∫
∫
∫
T ii
T iii
i
i
ii
8/18/2019 Many Body Lecture 3 (1)
21/35
;#uation of Motion for the Green’s Function
• ;,aluate theT
product using >ic%’s 8heorem
• 2owest order terms
• ?iagram J" is the *artree-Foc% e change potential G o r 6)y"• ?iagram 6B" is the *artree potential G o x)y"• ?iagram J" is con,entionally the first term in the self-energy• ?iagram 6B" is included in * o in condensed matter physics
[ ]connectedoHH1H1Ho
11 )t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂
1d ΨΨ−
++∫ yxrrrxr T
)t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂ HH1H1H yxrr ++
)t',(ψ̂t),(ψ̂t),(ψ̂t),(ψ̂ HH1H1H yxrr ++
i"4, x)r 6"Go x)r 6" G o r 6)y"
i"4, x)r 6"Go r 6)r 6" G o x)y"
x
y
r 6
6B"
J"y
r 6
x
8/18/2019 Many Body Lecture 3 (1)
22/35
;#uation of Motion for the Green’s Function
• @ne of the ne t order terms in theT
product
• 8he full e pansion of the T product can be written e actly as
i"5, 1)2" , x)r 1"Go 1)x" G o r 1)2" G o 2)r 1" G o 1)y"
)(ψ̂)(ψ̂)(ψ̂)(ψ̂)(ψ̂)(ψ̂)(ψ̂)(ψ̂-1
-1
ddd HH1H1HHHHH1
1 yxrr1221rx21r21 ++++∫
66"
G o 1)y"y
1
x
Σ x)1"2
r 6
diagramsproper iteratingbygeneratedarelatter 8he diagrams
andintodiagramsorder higher di,idesndistinctio8his
lineGsingleacuttingbytwointocutbecannotdiagramsni#ueuni#ueareothersandrepeatedarediagramssomeordershigher t
diagram"thisin ,ariabledummyais
energy-self theis
o
improper
proper
1x
yxxxx
'
),'()G',('d o ΣΣ∫
8/18/2019 Many Body Lecture 3 (1)
23/35
;#uation of Motion for the Green’s Function
• 8he proper self-energy ΣN F 6B=) M 6:6"• 8he self-energy has two arguments and hence two &e ternal ends’• ll other arguments are integrated out• Proper self-energy terms cannot be cut in two by cutting a single G o• First order proper self-energy terms ΣN 6"
• *artree-Foc% e change term *artree 7oulomb" term
; ercise K Find all proper self-energy terms at second order ΣN 4"
r 6
x
x’ 6B"J"x’
x
8/18/2019 Many Body Lecture 3 (1)
24/35
;#uation of Motion for the Green’s Function
• ;#uation of Motion for G and the $elf ;nergy[ ]
potentialncorrelatio-exchangetheis
heresuppresseddependencetime
indirectputtoisphysicsmatter condensedin7on,ention
direct
e+changedirect
)',(
,,
)-(),'(G)',('d),G(3ˆt
)',(3)',()',(
Ĥ )(
)',(3),(G)'('1d)',)((
)()(
),'(G)',('d)(ψ̂)(ψ̂)(ψ̂)(ψ̂1
d
1
oH
H
o)1(
H11o1
1)1(
)1()1()1(
ooHH1H1Ho1
1
xx
ryx
yxyxxxxyx
xxxxxx
xxrrxxrx
rxx
yxxxxyxrrrx
r
∑
=∑+
−−∂∂
−∑→∑∑
=−−=∑
∑+∑=∑
∑=ΨΨ−
∫
∫
∫ ∫ ++
δ
δ
ii
iT i
8/18/2019 Many Body Lecture 3 (1)
25/35
;#uation of Motion for the Green’s Function
• ?yson’s ;#uation and the $elf ;nergy
),''(G)'','()',(G''d'd),(G),G(
3Ĥ Ĥ
)-(),(G3ˆt
)-(),'(G)',('d),G(3ˆt
ooo
Ho
oH
oH
;#uations?yson<
"incl systemginteractin-nonfor Gfor Motionof ;#uation
systemginteractinfor Gfor Motionof ;#uation
o
yxxxxxxxyxyx
yxyx
yxyxxxxyx
∫ ∫
∫
∑+=
=
= −−∂∂
=∑+
−−∂∂
δ
δ
i
ii
8/18/2019 Many Body Lecture 3 (1)
26/35
;#uation of Motion for the Green’s Function
• Integral ;#uation for the $elf ;nergy
e#uations?yson
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•?yson’s ;#uation F 6B9"
• In general) Σ∗ is energy-dependent and non-*ermitian• (oth first order terms in Σ are energy-independent• 'uantum 7hemistry K first order self energy terms included in Ho• 7ondensed matter physics K only &direct’ first order term is in Ho• $ingle-particle band gap in solids strongly dependent on &e change’ term
;#uation of Motion for the Green’s Function
∫ ∫ ∫ ∫
Σ+=
Σ+=
),''()G'','()',(G''d'd),(G),G(
),'')G('','()',(G''d'd),(G),G(
ooo
*oo
yxxxxxxxyxyx
yxxxxxxxyxyx
G x)y" A A + + +
Σ x’)x’’"A + +
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• @ne of the 6B second order diagrams for the self energy• 8he first energy dependent term in the self-energy• ;,aluate for homogeneous electron gas M 63B"
;,aluation of the $ingle 2oop (ubble
( )
( )
( ) oooo2
o
ooo
o
oo
2o
GGGG
GG
),(
),(G),(G2d
2
d(-1)!2!&
&))3((),(G2d
2d
iiii
i
i
ii
ii
=−=−
−=−=
++
−−−=
∫ ∫ ∫ ∫
π
π α π
β α β π β
π
α ω π α
π
8heorems>ic%<
q
q
qqk q
α+β) ℓ+qβ) ℓ
α+β) ℓ+qβ) ℓω−α) k-q
α) q
α) q
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•Polarisation bubble K fre#uency integral o,er β
• Integrand has poles at β A ε ℓ - iδ and β A -α + ε ℓ+q + iδ • 8he polarisation bubble depends on q and α • 8here are four possibilities for ℓ and q
;,aluation of the $ingle 2oop (ubble
δ ε α β β α
δ ε β β
β α β π β
i
ii
i
ii
ii
±−+=++±−=
++
+
∫
q
q
q
),(G ),(G
),(G),(G2d
oo
oo
44
44
44
44
k qk
k qk
k qk
k qk
>+<+> y
δ ε α β i++−= +qδ ε β i−=
44 k qk
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• Integral may be e,aluated in either half of comple plane
;,aluation of the $ingle 2oop (ubble
y
δ ε α β i++−= +qδ ε β i−=
44 k qk
( )0
1
ee2ed
2d
2
im
=∝∝
=+=
∫ ∫
∫ ∑∫ ∫
∞→
−
∞
∞−
r r
i
r
ir
i
ii
i
r φ φ
φ
π π β
π planehalf upper incirclesemi
-planehalf pper cloc%wise nti residues
( )( )
[ ]( ) #$
1
#5$51
6(5)
−=→
−−=
a!atf !"residue
( )( )
( )δ ε ε α δ ε δ ε α
δ ε α β δ ε α β δ ε β
i
i
ii
i
ii
i
i
i
+−+−=
−−++−=
++−=−−++−
++
++
22
atpolefor residue
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• From 0esidue 8heorem
• ; ercise K @btain this result by closing the contour in the lower half plane
;,aluation of the $ingle 2oop (ubble
δ ε ε α
δ ε ε α π π
β α β π β
i
i
i
iii
−+−=
+−+−−=++
+
+∫
q
q
q
122
),(G),(G2d
oo
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• Polarisation bubble K continued
• For
• (oth poles in same half plane• 7lose contour in other half plane to obtain !ero in each case
• ; ercise K For
• $how that
• nd that
;,aluation of the $ingle 2oop (ubble
44 k qk >+
δ ε ε α β α β π β
i
i
ii ++−−
=++ +∫ qq ),(G),(G2d
oo
( ) ( ) δ ε ε α π δ ε ε α π α π
i
i
i
ii
−+−−
++−=−
++∫ ∫
q2
2
d2
2
d),(o
44 k qk + Boiπ
),(G),(G2d
oo β α β π β ++∫ q ii
44 k qk >+<
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( ) ( )
( ) ( )
( ) ( )
( ) ( )
planehalf lower inpolesbothotherwisebemust
andatpoles
4
2
7o
27
8o
7o
2
oo2
o
εεε
εε2
ε)3(
2d
2
d
2
d
),(ε
)3(2d
2
d
),(),())3((ε
2d
2d
),(G),(G2d
2
d))3((),(G
2d
2
d-2
k qk
qqq
qqqq
qqqk q
qqk qk
qqk qk
qk qk
qk qk
>−−−=±−=
++−±−−=
−±−−=Σ
+−−±−−=
++−−−=Σ
+−−
+−−
−−
−−
∫ ∫ ∫
∫ ∫
∫ ∫
∫ ∫ ∫ ∫
δ α δ ω α
δ α δ α ω π α
π π
α π δ α ω π
α π
α π α π δ α ω π
α π
β α β π β
π α ω
π α
π
ii
i
i
i
i
ii
i
iiii
i
iiii
• $elf ;nergy
;,aluation of the $ingle 2oop (ubble
44 k qk >+<
β) ℓω−α) k-q
α) q
α) q
α+β) ℓ+q
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( ) ( )
( ) ( )
dependent,ector wa,eandenergyisenergy$elf
atresidue
δ ω π π
δ ω π π
δ ω
δ ω α δ α δ α ω
iii
iii
i
ii
i
i
i
−−−+−=Σ−
+−−+−=Σ−
>>+<+−−+
−=
+−→++−+−−
−−
−+
−+
−+−
∫ ∫
∫ ∫
qk q
qk q
qk q
qk qqk
k q-k k qk
εεε1)3(
2d
2d2
εεε1
)3(2
d
2
d2
,,εεε
2
εεε
2ε
28
27
444
• $elf ;nergy K continued
;,aluation of the $ingle 2oop (ubble
444 , , k qk k qk >−>+<
444 , , k qk k qk
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