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7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Propagating electron or hole interacts with other e-
/h+
• Interactions modify (renormalize) electron or hole energies
• Interactions produce finite lifetimes for electrons/holes (quasi-particles)
• Spectral function consists of quasi-particle peaks plus ‘background’
• Quasi-particles well defined close to Fermi energy
• MBGF defined by
{ }
o
oHHo )t','(ψt),(ψ)t','t,,G(
Ψ
ΨΨ= +
state,groundHeisenbergexact
overaveragedoperatorfieldoffunctionncorrelatioi.e.
rrrr T i
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Space-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
• Non-interacting Green’s function Go represented by a single line
• Interacting Green’s Function G represented by a double or thick single line
time
Add particle Remove particle
t > t’t’
time
Remove particle Add particle
t’ > tt
x
y
y
)t'(t)t',(ψt),(ψ oHHo −ΨΨ + θ yx
t)(t't),(ψ)t',(ψ oHHo −ΨΨ+
θ xy
x
Go(x,y)
x,ty,t’
G(x,y)x,ty,t’
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Lehmann Representation (F 72 M 372) physical significance of G
{ }
oo
onn
n
oSnnSo
-o-n
oSnnSo
oHnnHooHHo
oHHooHHo
nn
n
o
oHHo
tiEtHitiE-tHi-
)t'tEi(E-
t'Hi-t'HitHi-tHi
ee ee
)'(ψ)(ψ)(
e
)e'(ψe)e(ψe
)t','(ψt),(ψ)t','(ψt),(ψ
t)(t't),(ψ)t','(ψ-)t'(t)t','(ψt),(ψ)t','t,,G(
)t','(ψt),(ψ)t','t,,G(
Ψ=ΨΨ=Ψ
ΨΨΨΨ=
ΨΨΨΨ=
ΨΨΨΨ=ΨΨ
−ΨΨ−ΨΨ=
=ΨΨ
Ψ
Ψ
ΨΨ=
++
++
+
+
++
++
+
rr
rr
rrrr
rrrrrr
1
rrrr
θ θ i
T i
formalismnumberoccupationinoperatorunit
numberparticleany,state,Heisenbergexact
state,groundHeisenbergexact
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Lehmann Representation (physical significance of G)
{ }
onebyinnumberparticlereduces ooS
oSoSSS
on
oSnnSo
on
oSnnSo
-o-noSnnSo
-o-noSnnSo
oHHo
ψ
ψ)1N(ψn )(ψ)(ψdn
δ)EE(εψψ
δ)EE(εψψ
e)t','t,,)G(t'-d(t),',G(
t)(t')(e)(ψ)'(ψ
-)t'(t)(
e)'(ψ)(ψ)t','t,,G(
)t','(ψt),(ψ)t','t,,G(
)t'(t
)t'tEi(E
)t'tEi(E-
ΨΨ
Ψ−=Ψ=
−−+ΨΨΨΨ+
+−−ΨΨΨΨ=
=
−ΨΨΨΨ−
−ΨΨΨΨ=
ΨΨ=
+
++
∞+
∞−
+
+
+
∫
∫ −+
+
rrr
rrrr
rr
rrrr
rrrr
ii
ii
i
T i
iε ε
θ
θ
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Lehmann Representation (physical significance of G)
µ
µ
+−−−=−−
−−+−−−=−−
−ΨΨ=ΨΨΨΨ
++−+=−+
−+++−+=−+
+ΨΨ=ΨΨΨΨ
+
++
)1N(E)1N(E)N(E)1N(E
)N(E)1N(E)1N(E)1N(E)N(E)1N(E
ψψψ
)1N(E)1N(E)N(E)1N(E
)N(E)1N(E)1N(E)1N(E)N(E)1N(E
ψψψ
onon
ooonon
2
nSooSnnSo
onon
ooonon
2
oSnoSnnSo
statesparticle1NandNconnects
statesparticle1NandNconnects
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Lehmann Representation (physical significance of G)• Poles occur at exact N+1 and N-1 particle energies
• Ionisation potentials and electron affinities of the N particle system
• Plus excitation energies of N+1 and N-1 particle systems
• Connection to single-particle Green’s function
FbelowstatesforasstatesunoccupiedtolimitedSum
unoccupied
stategroundg)interactin-(nonparticle-singletheis
ε
δ θ
θ
θ
ε
00c
n0cc0 )t'(t)e'(ψ)(ψ
)t'(t0)(t'c(t)c0)'(ψ)(ψ
)t'(t0)t','(ψt),(ψ0)t','t,,(G
0
n
mnnmn*
n
unocc
n
n
nm
*
n
nm,
m
HHo
)t'-(t-
=
∈=−=
−=
−=
+
+
+
++
∑
∑i
i
rr
rr
rrrr
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Gell-Mann and Low Theorem (F 61, 83)• Expectation value of Heisenberg operator over exact ground state
expressed in terms of evolution operators and the operator in question in
interaction picture and ground state of non-interacting system
oIo
oIIIo
oo
oHo
)-,(U
)(t,-U(t)Ot),(U(t)O
φ φ
φ φ
∞+∞
∞+∞=
ΨΨΨΨ
oφ
{ }
oo
oHHo
|
)t','(ψt),(ψ
)t','t,,G( ΨΨ
ΨΨ
=
+ rr
rr
T
i
FunctionsGreen'Body-Many
( ) [ ] ( )57F
)(t'
I
Hdt't
0
t
0
t
0
nI2I1I
t
0
n21
n
I e)(tH)...(tH)(tHdt...dtdtn!
(t,0)U∫ −
=−= ∫ ∫ ∫ i
T T i
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Perturbative Expansion of Green’s Function (F 83)
• Expansion of the numerator and denominator carried out separately
• Each is evaluated using Wick’s Theorem
• Denominator is a factor of the numerator
• Only certain classes of (connected) contractions of the numerator survive
• Overall sign of contraction determined by number of neighbour permutations
• n = 0 term is just Go(x,y)
• x, y are compound space and time coordinates i.e. x ≡ (x, y, z, tx)
( )
( ) [ ]
( )( ) [ ] o
- -
nI2I1I
-
on21
0n
n
oIo
o
- -
nI2I1I
-
on21
0n
n
oIo
)(tH)...(tH)(tHdt...dtdt
n!
,U
)(ψ)(ψ)(tH)...(tH)(tHdt...dtdtn!,U
1),G(
φ φ φ φ
φ φ φ φ
∫ ∫ ∫ ∑
∫ ∫ ∫ ∑
∞+
∞
∞+
∞
∞+
∞
∞
=
+∞
∞
+∞
∞
++∞
∞
∞
=
−=−∞∞+
−
−∞∞+=
T i
T i
i yxyx
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Fetter and Walecka notation for field operators (F 88)
( )
( )
( )
( ) +−
−+−
+++++
+++++
++
≤=
>=
≤=
+>=
+≡+=
+≡+=
bb- tt ),(G
tt 0)(ψ)(ψ
tt 0
aa tt ),(G)(ψ)(ψ
ba)(ψ)(ψ)(ψ
ba)(ψ)(ψ)(ψ
yxo
yx
)()(
yx
yxo
)()(
(-))(
(-))(
yx
yx
yxyx
xxx
xxx
i
i
( )( )
( ) ( )
0ψψ 0ψψ 0bbabbaaa
bbabbaaa
ba ba
ψψψψψψ (-))()()(
======
+++=
++≡
++=
++++++
++++
++
+++−+++++
similarly
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
•Nonzero contractions in numerator of MBGF
(-1)3 (i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y)
(-1)4(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y)
(-1)5(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y)
(-1)4(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y)
(-1)6(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y)
(-1)7
(i)
3
v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)(6) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(5) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(4) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(3) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(2) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(1) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
+++
+++
+++
+++
+++
+++
7/28/2019 Many_Body_Lecture_3.pdf
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Many-body Green’s Functions
• Nonzero contractions
-(i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y) (1)
+(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y) (2)
-(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y) (3)
+(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y) (4)
+(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y) (5)
-(i)3v(r,r’)Go
(r,r) Go
(x,r’) Go
(r’,y) (6)
y
x
r r’
y
x
r r’
x
y
r r’
y
r r’
x
y
r’ r
xx
y
r’ r
(1) (2)
(3) (4)
(5) (6)
7/28/2019 Many_Body_Lecture_3.pdf
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• Nonzero contractions in denominator of MBGF• Disconnected diagrams are common factor in numerator and denominator
Many-body Green’s Functions
(8) )(ψ)'(ψ)'(ψ)(ψ
(7) )(ψ)'(ψ)'(ψ)(ψ
rrrr
rrrr
++
++(-1)3(i)2v(r,r’)Go(r’,r) Go(r,r’)
(-1)4(i)2v(r,r’)Go(r,r) Go(r’,r’)
r r’(7)
r r’
(8)
Denominator = 1 + + + …
Numerator = [ 1 + + + … ] x [ + + + … ]
7/28/2019 Many_Body_Lecture_3.pdf
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•Expansion in connected diagrams
• Some diagrams differ in interchange of dummy variables
• These appear m! ways so m! term cancels• Terms with simple closed loop contain time ordered product with equal times• These arise from contraction of Hamiltonian where adjoint operator is on left• Terms interpreted as
Many-body Green’s Functions
∑ ∫ ∫ ∞
=
∞
∞−
∞
∞−
+−=
0m connected
om111om1 ])(ψ)(ψ)(tH ...)(tH[dt...dtm!
)(),G( φ φ yxyx T
ii
iG(x, y) = + + + …
{ }
densitychargeginteractin-non )(ρ)(ψ)(ψ
)t',(ψt),(ψ),(G
ooo
oo
lim
'o
xxx
xxxx
−=−=
=
+
++→
φ φ
φ φ δ T it t
7/28/2019 Many_Body_Lecture_3.pdf
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• Rules for generating Feynman diagrams in real space and time (F 97)
• (a) Draw all topologically distinct connected diagrams with m interaction
lines and 2m+1 directed Green’s functions. Fermion lines run continuously
from y to x or close on themselves (Fermion loops)
• (b) Label each vertex with a space-time point x = (r,t)
• (c) Each line represents a Green’s function, Go(x,y), running from y to x
• (d) Each wavy line represents an unretarded Coulomb interaction
• (e) Integrate internal variables over all space and time
• (f) Overall sign determined as (-1)F
where F is the number of Fermion loops• (g) Assign a factor (i)m to each mth order term
• (h) Green’s functions with equal time arguments should be interpreted as
G(r,r’,t,t+) where t+ is infinitesimally ahead of t
• Exercise: Find the 10 second order diagrams using these rules
Many-body Green’s Functions
7/28/2019 Many_Body_Lecture_3.pdf
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• Feynman diagrams in reciprocal space
• For periodic systems it is convenient to work in momentum space
• Choose a translationally invariant system (homogeneous electron gas)
• Green’s function depends on x-y, not x,y
• G(x,y) and the Coulomb potential, V, are written as Fourier transforms
• 4-momentum is conserved at vertices
Many-body Green’s Functions
( )
t-.. ddd
)e',v()'-d()v(
)eG(
2
d),G(
34
4
4
)'.(
).(
ω ω
π
xkxkkk
rrrrq
kk
yx
rrq
yxk
≡≡
=
=
∫
∫ −
−
i-
i
Fourier Transforms
( ) ( )321
43214 2eeed...
qqqxxqxqxq −−=
+∫ δ π
-i-ii
4-momentum Conservation
q1
q2
q3
7/28/2019 Many_Body_Lecture_3.pdf
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• Rules for generating Feynman diagrams in reciprocal space
• (a) Draw all topologically distinct connected diagrams with m interaction
lines and 2m+1 directed Green’s functions. Fermion lines run continuously
from y to x or close on themselves (Fermion loops)
• (b) Assign a direction to each interaction• (c) Assign a directed 4-momentum to each line
• (d) Conserve 4-momentum at each vertex
• (e) Each interaction corresponds to a factor v(q)
• (f) Integrate over the m internal 4-momenta• (g) Affix a factor (i)m/(2π)4m(-1)F
• (h) A closed loop or a line that is linked by a single interaction is assigned a
factor eiεδ Go(k,ε)
Many-body Green’s Functions
7/28/2019 Many_Body_Lecture_3.pdf
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[ ]
[ ]
)(ψ)(ψ1
)(ψ)(ψddH
)(ψ)(ψ1)(ψdH,ψψt
)(ψ)(h)(ψdH
)(ψ)(h H,ψψt
1H2H
21
2H1H21H
H2H
2
2H2HHH
1H11H1H
HHHH
2
1rr
rrrrrr
rrrr
rr
rrrr
rr
−=
−==∂∂
=
==∂∂
++
+
+
∫ ∫
∫
∫
for
for
i
i
Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators (from Lecture 2)
{ }
oo
oHHo )t','(ψt),(ψ)t','t,,G(
ΨΨ
ΨΨ=
+ rrrr
T
i
7/28/2019 Many_Body_Lecture_3.pdf
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Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators
[ ] [ ]
t),(ψˆ
t),(ψˆ
1
t),(ψˆ
dt),(ψˆ
t),(h
ˆ
t
t),(ψt),(ψ1
t),(ψdt),(ψt),(h
tH
e)(ψ)(ψ
1
)(ψd
tH
e
tH
e)(ψ)(hˆ
tH
e
tHeH,ψtHet),(Ht),,(ψt),(ψt
H2H2
2H2H
H2H
2
2H2H
22
22
SSHHH
rrrrrrrr
rrrr
rrrr
rrrrrrrr
rrr
−=
−∂
∂
−+=
−−
++
−+=
−+==∂∂
+
+
+
∫
∫ ∫
i
iiii
iii
7/28/2019 Many_Body_Lecture_3.pdf
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Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument
{ }
[ ]
[ ] )t'-(t)-(|)t'-(t)t',(ψt),,(ψ
)t'-(t)t',(ψt),,(ψ
(t'-t)t
t),(ψ)t',(ψ-)t'-(t)t',(ψt
t),(ψ
)t'-(tt),(ψ)t',(ψ--)t'-(t)t',(ψt),(ψ
(t'-t)t
t),(ψ)t',(ψ-)t'-(t)t',(ψ
t
t),(ψ
(t'-t)t),(ψt
)t',(ψ-)t'-(t)t',(ψt),(ψt
)t',t,,G(t
)t',(ψt),,(ψ)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
7/28/2019 Many_Body_Lecture_3.pdf
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Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument
[ ]
[ ]
)t'-(t)-(
)t',(ψt),(ψt),(ψt),(ψ1
d)t',t,,G(ht
)t'-(t)-(
)t',(ψt),(ψt),(ψt),(ψ
1
d),G(hˆ
)t'-(t)-(
(t'-t)t),(ψt),(ψt),(ψ)t',(ψ-1
d
)t'-(t)t',(ψt),(ψt),(ψt),(ψ1
d
(t'-t)t),(ψ)t',(ψ-)t'-(t)t',(ψt),(ψh)t',t,,G(t
oHH1H1Ho
1
1
oHH1H1Ho1
1
oH1H1HHo
1
1
oHH1H1Ho
1
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
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Equation of Motion for the Green’s Function
• Evaluate the T product using Wick’s Theorem
• Lowest order terms
• Diagram (9) is the Hartree-Fock exchange potential x G o(r1,y)
• Diagram (10) is the Hartree potential x G o(x,y)
• Diagram (9) is conventionally the first term in the self-energy
• Diagram (10) is included in Ho in condensed matter physics
[ ]connectedoHH1H1Ho
1
1 )t',(ψt),(ψt),(ψt),(ψ1
d ΨΨ−
++∫ yxrrrx
r T
)t',(ψt),(ψt),(ψt),(ψ HH1H1H yxrr ++
)t',(ψt),(ψt),(ψt),(ψ HH1H1H yxrr ++
(i)2v(x,r1)Go(x,r1) Go(r1,y)
(i)2v(x,r1)Go(r1,r1) Go(x,y)
x
y
r1
(10)
(9)y
r1
x
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Equation of Motion for the Green’s Function
• One of the next order terms in the T product
• The full expansion of the T product can be written exactly as
(i)3v(1,2) v(x,r1)Go(1,x) Go(r1,2) Go(2,r1) Go(1,y)
)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ-
1
-
1ddd HH1H1HHHHH
1
1 yxrr1221rx21
r21 ++++∫
(11)
Go(1,y)
y
1
x
Σ(x,1)
2
r1
diagramsproperiteratingbygeneratedarelatterThe diagrams.
andintodiagramsorderhigherdividesndistinctioThis
lineGsingleacuttingbytwointocutbecannotdiagramsUnique
uniqueareothersandrepeatedarediagramssomeordershigherAt
diagram)thisin (variabledummyais
energy-selftheis
o
improper
proper
1x
yxxxx
'
),'()G',('d o ΣΣ∫
7/28/2019 Many_Body_Lecture_3.pdf
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Equation of Motion for the Green’s Function
• The proper self-energy Σ* (F 105, M 181)• The self-energy has two arguments and hence two ‘external ends’• All other arguments are integrated out
• Proper self-energy terms cannot be cut in two by cutting a single Go
• First order proper self-energy terms Σ*(1)
• Hartree-Fock exchange term Hartree (Coulomb) term
Exercise: Find all proper self-energy terms at second order Σ*(2)
r1
x
x’ (10)(9)x’
x
i f i f h i
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Equation of Motion for the Green’s Function
• Equation of Motion for G and the Self Energy
[ ]
potentialncorrelatio-exchangetheis
heresuppresseddependencetime
indirectputtoisphysicsmattercondensedinConvention
direct
exchangedirect
)',(
,,
)-(),'(G)',('d),G(Vht
)',(V)',()',(
H )(
)',(V),(G)'('1d)',)((
)()(
),'(G)',('d)(ψ)(ψ)(ψ)(ψ1
d
1
oH
H
o)1(
H11o
1
1)1(
)1()1()1(
ooHH1H1Ho
1
1
xx
ryx
yxyxxxxyx
xxxxxx
xxrrxxrx
rxx
yxxxxyxrrrx
r
∑
=∑+
−−
∂∂
−∑→∑
∑
=−−=∑
∑+∑=∑
∑=ΨΨ−
∫
∫
∫ ∫ ++
δ
δ
ii
iT i
i f i f h i
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Equation of Motion for the Green’s Function
• Dyson’s Equation and the Self Energy
),''(G)'','()',(G''d'd),(G),G(
VH H
)-(),(GVht
)-(),'(G)',('d),G(Vht
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EquationsDyson'
)incl.( systemginteractin-nonforGforMotionofEquation
systemginteractinforGforMotionofEquation
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∫ ∫
∫
∑+=
=
=
−−
∂∂
=∑+
−−
∂∂
δ
δ
i
ii
E i f M i f h G ’ F i
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Equation of Motion for the Green’s Function
• Integral Equation for the Self Energy
equationsDyson'inbyreplacemay weHence
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...GGG
)','''()''',''()G'',('''d''d)',()',(
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ΣΣ
+ΣΣΣ+ΣΣ+Σ=Σ
+ΣΣΣ+ΣΣ+Σ=Σ
ΣΣ+Σ=ΣΣ+=ΣΣ
+ΣΣΣ+ΣΣ+Σ=Σ
ΣΣ+Σ=Σ
ΣΣ
∫ ∫
∫∫
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• Dyson’s Equation (F 106)
• In general, Σ∗ is energy-dependent and non-Hermitian• Both first order terms in Σ are energy-independent
• Quantum Chemistry: first order self energy terms included in Ho
• Condensed matter physics: only ‘direct’ first order term is in Ho
• Single-particle band gap in solids strongly dependent on ‘exchange’ term
Equation of Motion for the Green’s Function
∫ ∫ ∫ ∫
Σ+=
Σ+=
),''()G'','()',(G''d'd),(G),G(
),'')G('','()',(G''d'd),(G),G(
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G(x,y) = = + + + …
Σ(x’,x’’)= + + …
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• One of the 10 second order diagrams for the self energy• The first energy dependent term in the self-energy
• Evaluate for homogeneous electron gas (M 170)
Evaluation of the Single Loop Bubble
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),(
),(G),(G2
d
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d(-1).2.x
x))V((),(G2d
2d
iiii
i
i
ii
ii
=−=−
−=−=
++
−−−=
∫ ∫
∫ ∫
π
π
α π
β α β π
β
π
α ω π α
π
TheoremsWick'
q
q
qqkq
α+β, ℓ+qβ, ℓ
α+β, ℓ+qβ, ℓω−α, k-q
α, q
α, q
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•Polarisation bubble: frequency integral over β
• Integrand has poles at β = ε ℓ - iδ and β = -α + ε ℓ+q + iδ
• The polarisation bubble depends on q and α
• There are four possibilities for ℓ and q
Evaluation of the Single Loop Bubble
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),(G2
d
oo
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FF
FF
FF
FF
kqk
kqk
kqk
kqk
>+<
<+<
>+><+>
x
y
δ ε α β i++−= +q
δ ε β i−=
FF kqk <+>
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•Integral may be evaluated in either half of complex plane
Evaluation of the Single Loop Bubble
x
y
δ ε α β i++−= +q
δ ε β i−=
FF kqk <+>
( )0
1
ee2
ed
2
d
2
lim
=∝∝
=+=
∫ ∫
∫ ∑∫ ∫
∞→
−
∞
∞
−
rr
i
r
ir
i
ii
i
rφ φ
φ
π π
β
π planehalfupperincirclesemi
-planehalfUpper
clockwiseAnti residues
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1
bzaz
1f(z)
−=→
−−=
azatf(z)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
+−+−=
−−++−=
++−=
−−++−
++
++
q
q
22
atpoleforresidue
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• From Residue Theorem
• Exercise: Obtain this result by closing the contour in the lower half plane
Evaluation of the Single Loop Bubble
δ ε ε α
δ ε ε α π
π β α β
π
β
i
i
i
iii
−+−
=
+−+−−
=++
+
+∫
q
q
q
1
2
2),(G),(G
2
doo
l i h i l bbl
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• Polarisation bubble: continued
• For
• Both poles in same half plane
• Close contour in other half plane to obtain zero in each case
• Exercise: For
• Show that
• And that
Evaluation of the Single Loop Bubble
FF kqk >+<− A oiπ
FF
FF
kqk
kqk
<+<
>+>
δ ε ε α β α β π
β
i
i
ii ++−−
=++ +∫
qq ),(G),(G2
doo
( ) ( ) δ ε ε α π δ ε ε α π α π
i
i
i
ii
−+−−
++−=−
++∫ ∫
q2
2
d2
2
d),(
3
3
3
3
o
FF kqk <+>+ Boiπ
),(G),(G2
doo β α β
π
β ++∫ q ii
FF kqk >+<
l i f h i l bbl
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2
ε)V(
2
d
2
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2
d
),(ε
)V(2
d
2
d
),(),())V((ε
2
d
2
d
),(G),(G2
d
2
d))V((),(G
2
d
2
d-2
kqk
qqq
qqqq
qqqkq
qqkqk
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qkqk
qkqk
>−
−−=±−=
++−±−−=
−±−−
=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
• Self Energy
Evaluation of the Single Loop Bubble
FF kqk >+<
β, ℓω−α, k-q
α, q
α, q
α+β, ℓ+q
E l i f h Si l L B bbl
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( ) ( )
( ) ( )
dependentvector waveandenergyisenergySelf
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−
>>+<+−−+
−=
+−→
++−+−−
−−
−+
−+
−+−
∫ ∫
∫ ∫
qkq
qkq
qkq
qk
qqk
kq-kkqk
εεε
1)V(
2
d
2
d2
εεε
1)V(
2
d
2
d2
,,
εεε
2
εεε
2
ε
2
3
3
3
3B
2
3
3
3
3A
FFF
• Self Energy: continued
Evaluation of the Single Loop Bubble
FFF , , kqkkqk >−>+<
FFF , , kqkkqk <−<+>
E l i f h Si l L B bbl
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• Real and Imaginary Parts
• Quasiparticle lifetime τ diverges as energies approach the Fermi surface
( ) ( )
( ) ( )( )
( ) 2A1
2
3
3
3
3A
2
3
3
3
3A
ε)Im(
εεε)V(2
d
2
d 2)Im(
εεε
1)V(
2
d
2
d 2)Re( P
Σ
−−+−=Σ
−−+=Σ
−+
−+
∫ ∫
∫ ∫
ωτ
ω δ π π
π
ω π π
qkq
qkq
Evaluation of the Single Loop Bubble
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aa
1Im
a
1
a
a
a
1Re
a
a
a
1
22
lim
02
2
2
ε
π ε δ δ π
δ
δ
δ
δ δ
δ
δ
δ
ε +=−=
+−=
+
=+
=
+
+−
=+
→i
Pi
i
i