many body lecture 3 (1)

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


2/35


• $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’


3/35


• 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


4/35


• 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


5/35


• 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


6/35


• 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


7/35


• 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


8/35


• 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


9/35


• 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


10/35


•.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

+++

+++

+++

+++

+++

+++


11/35


• .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"


12/35

• .on!ero contractions in denominator of M(GF• ?isconnected diagrams are common factor in numerator and denominator


(+) )(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


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


∑ ∫ ∫ ∞

=

∞

∞−

∞

∞−

+−=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


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



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


( )

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


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"



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


18/35


• ;#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


19/35


• ?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


20/35


• ?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


21/35


• ;,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


22/35


• @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∫


23/35


• 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


24/35


• ;#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


25/35


• ?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


26/35


• Integral ;#uation for the $elf ;nergy

e#uations?yson


27/35

•?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


∫ ∫ ∫ ∫

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 + +


28/35

• @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


29/35

•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


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


30/35

• Integral may be e,aluated in either half of comple plane


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

+−+−=

−−++−=

++−=−−++−

++

++

qq

qq

22

atpolefor residue


31/35

• From 0esidue 8heorem

• ; ercise K @btain this result by closing the contour in the lower half plane


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


32/35

• 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


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

−+−−

++−=−

++∫ ∫

qq

q2

2

d2

2

d),(o

44 k qk + Boiπ

),(G),(G2d

oo β α β π β ++∫ q ii

44 k qk >+<


33/35

( ) ( )

( ) ( )

( ) ( )

( ) ( )

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

qq

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


44 k qk >+<

β) ℓω−α) k-q

α) q

α) q

α+β) ℓ+q


34/35

( ) ( )

( ) ( )

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

qq

qq

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


444 , , k qk k qk >−>+<

444 , , k qk k qk


35/35

many body lecture 3 (1)

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