Green’s mill

1

Prima pagina del saggio di George Green pubblicato nel 1828

2

George Green (14 July 1793–31 May 1841) was a Britishmathematician and physicist, who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Green, 1828). (Note: This 1828 essay can be found in "Mathematical papers of the late George Green", edited by N. M. Ferrers.) The essay introduced several important concepts, among them a theorem similar to modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions.George Green was the first person to try and explain a mathematical theory of electricity and magnetism which formed the basis for other scientists such as James Clerk Maxwell,William Thomson and others.Green's life story is remarkable in that he was almost entirely self-taught.

3

Green’s functions in one-body quantum problems

2can be used to convert into an integral equation

and to convert volume integral over to surface integrals over boundary S.

1 ( ( ) ) ( ) 02 r V r rε ψ

Ω

− ∇ + − =

ε δ− ∇ + − = −

2 Introduce G satisfying:

with r' source point:

1 ( ( ) ) ( ', ) ( ' )2 r V r G r r r r

ε ψ

ε ψ

− ∇ + − =

⇒ − ∇ + − =

2multiply by

2.

1 ( ( ) ) ( ) 0 G(r',r)2

1G(r',r)( ( ) ) ( ) 02

r

r

V r r

V r r

ε δ ψ

ψ ε ψ δ

− ∇ + − = −

⇒ − ∇ + − = −

2multiply by

2

1( ( ) ) ( ', ) ( ' ) ( )2

1( ) ( ( ) ) ( ', ) ( ) ( ' )2

r

r

V r G r r r r r

r V r G r r r r r

The delta not yet invented at Green’s time

Source

4

ε ψ

ψ ε ψ δ

− ∇ + − =

− ∇ + − = −

2.

2

subtract: V disappears

1G(r',r)( ( ) ) ( ) 02

1( ) ( ( ) ) ( ', ) ( ) ( ' )2

r

r

V r r

r V r G r r r r r

2 2

integrate

1 1( ) ( ) ( ', ) G(r',r)( ) ( ) ( ) ( ' )2 2r rr G r r r r r rψ ψ ψ δ− ∇ + ∇ = −

ψ ψ ψ δ ψΩ Ω

− ∇ + ∇ = − =∫ ∫

2 2

important step: interchange and '

1 [ ( ) ( ', ) G(r',r) ( )] ( ) ( ' ) ( ')2 r r

r r

dr r G r r r dr r r r r

ψ ψ ψΩ

∇⇒ = − ∇∫

2 2 ' '

1( ) '[G(r,r') ( ') ( ') ( , ')]2 r rr dr r r G r r

This is the sought integral equation5

Ω

=∫ ∫

Start from divergence (Gauss or theor' em) .S

divA dr A n dSOstrogradsky s

= Φ∇Ψ −Ψ∇Φ

insert: A

Ω

∇ Φ∇Ψ −Ψ∇Φ = Φ∇Ψ −Ψ∇⇒ Φ∫ ∫

( ) ( ).S

dr n dS

Ω

⇒ Φ∇ Ψ −Ψ∇ Φ = Φ∇Ψ −Ψ∇Φ∫ ∫

2 2( ) ( ).S

dr n dS

Next: convert volume -> surface integral

ψ ψ ψΩ

= ∇ − ∇∫

This lends itself to transform the r.h.s.

2 2

of

' '1( ) '[G(r,r') ( ') ( ') ( , ')]2 r rr dr r r G r r

6

ψ ψ ωω

= = =− =

= +

0 1

Introduce reso, 1

( ) 1Let

lvent:H E E

H GH H H

10 0

1 1 1 1 , or alsoHH H H Hω ω ω ω

= +− − − −

operator identities

0 0 1 1 11 1 1( ) ( ) 1ω ω

ω ω ω− = − − + = +

− − −H H H H H

H H H

10 0

1 1 1 1ω ω ω ω

= +− − − −

HH H H H

Example: H1=δV(r): then, in the coordinate representation

10 0

1 1 1 1' ' 'r r r r r H rH H H Hω ω ω ω

= +− − − −

Green’s functions are a versatile tool:Among their uses the resolvent

7

8

10 0

1 1 1 1' ' '

Insert complete set of position eigenstates:

r r r r r H rH H H Hω ω ω ω

= +− − − −

1''0 0

1

1 1 1 1' ' '' '' '

'' ( '') ''ω ω ω ω

δ

= +− − − −

=

∑r

r r r r r H r r rH H H H

H r V r r

8

9

This gives the Lippmann-Schwinger equation

( ) ( ) ( ) ( ) ( )30 0, ' , ' '' , '' '' '', 'G r r G r r d r G r r V r G r rδ= + ∫

The Lippmann-Schwinger equation is mostconvenient when the perturbation islocalized . Typical examples are impurityproblems, in metals. The alternative isembedding (Inglesfield 1981)

1''0 0

1

1 1 1 1' ' '' '' '

'' ( '') ''ω ω ω ω

δ

= +− − − −

=

∑r

r r r r r H r r rH H H H

H r V r r

9

10

Propagators: Bose field with speed c

( ) ( ) ( )( ) ( )ω ωφε ε

−= + = + † †

'coordinate' of oscillator

2 2k ki t i t

k k k k kk k

t a t a t a e a e

( )

( ) ( )

φ

φ ω φ

=

∂= −

∂

22

2

harmonic oscillator, normal mode of

Bose Field. EOM:

k

k k k

t

t tt

( )

ω

φ

= ⇒

∂+ = ∂

22 2

2 0

k

k

ck

c k tt

( )φ ∂

⇔ − ∇ = ∂

22 2

2 field wave equation , 0.c r tt

( )ω ωπ φω

ε−= += −

†conjuga2

te momentum k kk k

i t i tkk k

k

i a e a e

( )π φ π ω φ ω= + = +∑ ∑

2 2 2 †1 12 2( , ) ( )k k k k k k k k

k kH a a

The Hamiltonian is a collection of oscillators:

10

11

( )

( ) ω

ω δ

φ φ

θ φ φ θ φ φ δ

φε

−

∂+ = − ∂ =

= + − =

= +

22

2

' '

' ' '

Propagator is a Green's function defined by : ( )

Let us show that it is the vacuum average ( ) 0 [ ( ) (0)] 0

( ) 0 ( ) (0) 0 ( ) 0 (0) ( ) 0 ( )

with2

k

k k

kk k k

k k k k kk k

i tk k

k

D t i tt

D t P t

t t t t D t

t a e a( )ω ε ω= † , . Indeed,ki tk k ke

( ) ( )ω ωω θ θ−= + −† †2 ( ) 0 0 0 0k ki t i tk k k k k kD t t a e a t a e a

( ) ( ) ω

ω ωθ θω ω

−−= + − =

||1( )2 2

kk k

i ti t i t

kk k

eD t t e t eω

ω

−

=||

( ) is continuous, however2

ki t

kk

eD t

( ) ( ) ( ) ( )

( ) ( )

1( )2

2

k k k k

k k

i t i t i t i tk k k

k

i t i t

D t i t e i t e t e t et

i t e t e

ω ω ω ω

ω ω

ω θ ω θ δ δω

θ θ

− −

−

∂= − + − + − −

∂−

= − − 11

12

( ) ( ) ω ωθ θ−∂ −= − − ⇒

∂( )

2k ki t i t

kiD t t e t e

t

( ) ( ) ( ) ( ) 2

2 ( )2

k k k ki t i t i t i tk k k

iD t i t e i t e t e t et

ω ω ω ωω θ ω θ δ δ− −∂ −= − − − + +

∂

( ) ( ) ( )2

2 ( )2

k ki t i tkkD t t e t e i t

tω ωω θ θ δ−−∂

= + − −∂

ω

ω

−

=||

Therefore, ( ) obeys2

ki t

kk

eD t

( )2

22 ( )k kD t i t

tω δ

∂+ = − ∂

and is Green’s fuction of wave equation

2 2 ( )k kD iω ω ω − + = − 12

13

( ) ( ) ( )ω ω δ ω ω δω ω∞ − + + −

−∞= +∫ ∫

0

requires convergence factors02 k ki i t i i t

k kD dt e dt e

( ) ( ) ( )ωω ω ω δ ω ω δ

−= +

− + + −

12k

k k k

i iDi i

( ) 2 2kk

iDi

ωω ω δ

−=

− −

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ω

ω ω

φ φ

φε ε

φ −

−

=

= + = +

= + ⇒ =

' '

† †

||†

ummary: Propagator ( ) 0 [ ( ) (0)] 0

2 2For nonrelativistic bosons (phonons,plasmons) one often uses

( ) k

k k

kk k k

i t i tk k k k k

k k

k k ki t

kD t

S D t P t

t a t a t a e a e

a et t a t

( ) ( ) ( )ωω ω δ ω ω δ

−= +

− + + −kk k

i iDi i

ω

ω

−

=||

Fourier transform of the propagator ( )2

ki t

kk

eD t

13

14

Relativistic Bosons, Photons

( )ε = + → − ∇

2 2 4Dispersion With ,k c k m k ic

ψ ψ ∂ −∇ + = − ∂

2 22

2 2

Scalar particle: Klein-Gordon (Relativistic Schrödinger)

1mcc t

( )( )2 2 22 2 4k

k

i iDi c k m c i

ωω ω δ ω δ

− −+

= =− − − −

Relativistic notation ,p k icω =

( )2 2 4( ) iD p

cp m c iδ−

=+ −

14

15

( )2

Massless scalar (if it existed)

( , ) ( ) iD k D pcp i

ωδ

−= =

−

Photon propagator ( ) ( ) ( ), ' 'D x x P A x A xµν µ ν =

( )The Photon has 4 components : ,A A iµ φ=

( )µ

∂− ∇ = ∂

22 2

2 , 0c A r tt

( ) ( )

4 ( ')

, , , ,2 24

Massless vector ( )

( , ) ( , ')(2 )

photonip x xi d p eD k D x x i

cp i cp iµ ν µ ν µ ν µ νω δ δπδ δ

−−= ⇒ = −

− −∫

( )

'

2 4

'

2

( ) 0 [ ( ) (0)

( )

] 0kk k k

iD pcp i

t

m c

D t P

δ

φ φ−

=+ −

=

⇒

φ φ=' 'The previous result ( ) 0 [ ( ) (0)] 0 generalizes tokk k kD t P t

,p k icω =

22 (4)

2 21( , ') ( ) ( ')x xD x x x xc tµν δ∂

= ∇ − = −∂

15

16

the gauge invariance allows introducing arbitrary f:

( )pfppip

ipD νµ

µνµν

δ+

−−

=0

)( 2

( )

4 ( ')

, , 24( , ') ( ')(2 )

ip x xd p eD x x i f x xcp iµ ν µ ν µ νδ

π δ

−

= − + ∂ ∂ −−∫

4( )f p p−= Landau gauge

2 2

1( )0

p pD p i

p i pµ ν

µν µνδ

= − − −

( ) ( ) ( ) ( )there is a gauge invariance , and since 0A x A x x A xµ µ µ µχ→ + ∂ =

( ), , 2( , ) is butiD k OKcp i

µ ν µ νω δδ

−=

−

2 0 and is a divergenceless symmetric tensor.p p p

p D p Dp

µ µ νµ µν ν µν≈ − =

1The Lorentz gauge : . 0 corresponds to choosingA Ac tµ µ

φ∂∂ = ∇ + =

∂

( ) ( ) ( ), ' 'D x x P A x A xµν µ ν =

16

1717

'

(T) †, 'ig (x,t, x',t') = T (x,t) (x',t')

σσ σ σψ ψ

Many-Body Quantum averages in terms of GF

†The average of one-b densitiesody ' '

'

ˆ f(x) = f (x) (x). (x)σσ σ σσσ

ψ ψ∑

( ) ( )σσ σ σ

σσ

σσ σσσσ

ψ ψ

→ + →

∑

∑

†' '

'T

' '' ' '

is done at once: f = f (x)< (x). (x)>=

-i lim lim f (x)g , , ', 't t x x

x t x t

Electron propagator

ψ anticommuting Heisenberg representation operators.The average must be taken over true g.s

(more generally, in equilibrium at temperature T).Difficult, but more feasible than the calculation of wave functions!

Also called time-ordered Green’s function

17

1818

( ) ( ) ( )T

' 'x,t =-i lim lim g , , ', '

t t x xx t x tσ σσρ

→ + →

Important examples: the density

'

(T) †, 'ig (x,t, x',t') = T (x,t) (x',t')

σσ σ σψ ψ

also yields the current density

( ) ( ) ( ) ( )Tx '' '

-1x,t = lim lim g , , ', '2m xt t x x

j x t x tσ σσ→ + →∇ −∇

18

1919

( ) ( )†The average of one-body operators ' '

' A = dx a ( )x x xσσ σ σ

σσ

Ψ Ψ∑ ∫

( ) ( )

( ) ( )' '''

' '''

A = ±i dx lim lim , ' '

=±i lim lim Tr[ , ' ' ]

T

x xt tT

x xt t

a x g xt x t

a x g xt x t

σσ σσ

σσ σσ

+

+

→→

→→

∫

upper sign Bose

lower sign Fermi

'

(T) †, 'ig (x,t, x',t') = T (x,t) (x',t')

σσ σ σψ ψ

More generally

19

Electron non-interacting propagator

( ) ( )ψ ψ

ψ

= −(0) †1 1 2 2 1 1 2 2( ) where

anticommuting Heisenberg representation operators.The average must be taken over noninteracting g.s

(the vacuum or more generally a filled Fermi sphere).

G x t x t i T x t x t

Also called time-ordered Green’s function; this is needed for many-body perturbation theory

20

Electron non-interacting propagator

( ) ( ) ( ) −= − Φ Φ = 0 00 †

Noninteracting propagator on discrete basis( , , ) 0 , iH t iH t

a b a aG a b t i Tc t c where c t e c e

0

†

det of spin-orbitals labelled with eigenfunctions k of H

k k kc c nσ σ

Φ =

⇒ Φ = Φ

Simple example

21

†

† †

( ) [ ( ) (0)]

( ) ( ) (0) ( ) (0) ( )k k k

k k k k

ig t T c t c

t c t c t c c tσ σ

σ σ σ σθ θ

= Φ Φ

= Φ Φ − − Φ Φ

0 0† †Let us develop ( ) (0) = in detail.iH t iH tk k k kc t c e c e cσ σ σ σ

−Φ Φ Φ Φ

0

0

0 0

†

( )† †

( )†

†

= Fermi sphere (energy E )+ electron if empty

(0) =e

(0) =e

(0) = e

Contribution: ( )(1 )

k

k

k

k

k k

iH t i E tk k

iH t i E tk k

iH t iH t i tk k

i tk

c

e c c

c e c

e c e c

e t n

σ

εσ σ

εσ

εσ σ

ε

ε

θ

Φ

Φ

Φ

− − +

− − +

− −

−

Φ

Φ Φ

Φ Φ

Φ Φ

−

22

23

ωδω δ ω

ω ε η

η

= =− +

+= −

0

0 0 ''

Noninteracting propagator on H eigenfunction basis in

( , ', ) ( , )

0 electrons in empty states0 holes in filled states

kkkk

k k

k

space

G k k G ki

( ) ( )(1 ) ( ) ki tk k kig t e t n t nε θ θ−= − − −

†

† †

† †

Putting all together, ( ) [ ( ) (0)]

( ) ( ) (0) ( ) (0) ( )

( ) (0) (0) ( ) (0) (0)

( )(1 ) ( )

k

k

k k k

k k k k

i tk k k k

i tk k

ig t T c t c

t c t c t c c t

e t c c t c c

e t n t n

σ σ

σ σ σ σ

εσ σ σ σ

ε

θ θ

θ θ

θ θ

−

−

= Φ Φ

= Φ Φ − − Φ Φ

= Φ Φ − − Φ Φ

= − − −

24

Lesser and greater Green’s functions( Heisenberg operators)g<

i,j(t, t’) = <c† i (t’) cj (t)>g>

i,j(t, t’) = <cj (t)c† i (t’)>

g<i,j(t, t’) filled states spectroscopy

g>i,j(t, t) empty states spectroscopy

Other Quantum Green’s functions

=

= Ψ Ψ Ψ = Ψ

0 0 0 0 0

Consider first 0 : averages over,

ground stateO O H E

T

What matters is the order of operators, not of times

GF are versatile and many kinds are needed and used.

24

25

Retarded and advanced Green’s functions; what matters is the order of times:igr

i,j(t, t’) = (g<i,j(t, t’) + g>

i,j(t, t’))θ(t − t’)

-igai,j(t, t’) = (g<

i,j(t, t’) + g>i,j(t, t’))θ(t’ − t)

Other Quantum Green’s functions

gai,j(t, t’)= gr

i,j(t’, t)*

igri,j(t, t’)=< [cj(t’),c† i (t)]+>θ(t − t’)

Equivalently,-iga

i,j(t, t’)=< [cj(t),c† i (t)]+>θ(t’ − t)

25

The different GF have different physical contents. Example: a model also used for transport problems(M.Cini, prb, 1980)

ε=∑ †0 k k k

kH a a =∑ †

1 ' ''

( ) ( )kk k kkk

H t V t a a

σ

∂∂

∂=

∂

(T)k,k'one can see contents by calculating g (t, t').

The derivative of c (t) is obtained by theˆ ˆˆHeisenberg EOM: i [H,A]

k

it

At

†

† †

( ) [ ( ) (0)]

( ) ( ) (0) ( ) (0) ( )k k k

k k k k

ig t T c t c

t c t c t c c tσ σ

σ σ σ σθ θ

= Φ Φ

= Φ Φ − − Φ Φ

26

δ δ ε

δ

δ−

+

∂= − + +

∂+

−

∑∑

(T) (T)k,k' ' k,k'

(T)p,k'

(T)p,k' '

(T)p,k' '

g (t, t') ( ') g (t, t')

(t)g (t, t'),

initial conditions g (t=0, t'=0 )=-i (1 )

g (t=0, t'=0 )=i

kk kk

kpp

kk k

kk k

i t tt

V

f

f

δ(T) (T)kk' k,k' 'g (t, t'=t+0)-g (t, t'=t-0)=i kk

One obtains

27

δ δ ε

δ

∂= − + +

∂+

∑∑

(r) (r)k,k' ' k,k'

(r)p,k'

(r)k,k' '

g (t, t') ( ') g (t, t')

(t)g (t, t'),

initial conditions g (t, t')=-i

kk kk

kpp

kk

i t tt

V

∂=

∂Using the Heisenberg EOM: i [H,A]

one can also obtain

At

Information contents

(T)k,k'g (t, t') knows about the Fermi level

and is a many-body Green's function(r)

k,k' g (t, t') is basically a one-body quantityindependent of filling (in noninteracting models)

28

2929

Simple tight-binding Hamiltonians: retarded GF and DOS

Chain (d=1), square (d=2) , cubic (d=3) lattices

< >

= < >= >∑ †

,, . . 0h i j h

i jH t c c i j n n t

Chain:

( )

1( ) common eigenfuctions of translation an

2 o

d

c sk

i

h

knn e T HN

t kε

ψ

=

=

Bipartite lattice (first neighbors of black sites are red, first neighbors of red sites

are black)

Square lattice

( ) ( )

( )

2

1( ) common eigenfuctions of translation and

2 [cos cos ]

x x y y

k

i k

y

n

h

k

x

nn e TN

t k k

Hψ

ε

+

= +

=

Bipartite lattice

< >

= < >= >∑ †

,, . . 0h i j h

i jH t c c i j n n t

Cubic lattice

( ) ( ) ( )

( )

2

1( ) common eigenfuctions of translatio

2 [

n

cos co

a

os

d

]

n

s c

x x y y z zi k n k n k n

k h x y z

n e T HN

t k k kε

ψ + +

= + +

=

Bipartite lattice

3232

Local Green’s functions for tight-binding Hamiltonians

( )

( ) ( )( )

( )

αα

ε

ρ ω δ ω ε δ ω επ

< >

=

< >= >

=

= − = −

∑

∑

∑ ∫

†

,

2 2

, nearest neighbors 0

d=dimensionality, 2 cos

1| 0 | | 0 |2

h i ji j

hd

k h

dd k kd

k BZ

H t c c

i j t

t k

k d k k

Chain (d=1), square (d=2) , cubic (d=3) lattices

3333

( )2 cosk ht kε =

( ) ( )( ) ( )1dim 0

1 1 , (0, )2 sin 2 sinh h

k kdk k

t k t kπ ω

ωω

δρ ω π

π π−

= = ∈∫

band edges 2 ht± k=0,π

arccos2 h

k ktωω

= =

argument of delta=0 for

Local Green’s functions for tight-binding chain

( ) ( ) ( )πρ ω δ ω ε

π= −∫ 2

0

1 | 0 |2d kdk k

2 2

arccos cos sin 1 1 ,2 2 2 2h h h h

k k kt t t tω ω ωω ω ω ωθ

= ⇒ = ⇒ = ± − −

( )

2

1dim 2

121

21

2

h

h

h

t

t

t

ωθ

ρ ωπ ω

− =

−

( )1

1One verifies that 1.dωρ ω

−=∫

3434

( )

( )

( )

2 cos

band edges : 2, cos 1

group velocity sin 0 near edges

particles spend more time where they are slo

k 0,

w

k h

k

h

k

t k

ktd kdk

π

εε

ε

=

= ± = ± ⇔

=

=

Symmetry of spectrum is due to bipartite graph

0=dkd kε 0=

dkd kε

band edges 2 ht± k=0,π3 2 1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

k

1( )ρ ω

3535

Chain:

Bipartite lattices, 1d,2d,3d

Chain:

SquareSquare

Changing the sign to all the black orbitals is just a gaugetransformation and cannot change any physical quantity, yet it is equivalentto sending the off-diagonal one-electron matrix element th to −th

However, εk is proportional to th and must change sign as well.

The spectrum is symmetric and the eigenfunctions at εk and− εk get exchanged by the gauge transformation.

36

Off-diagonal matrix elements of the resolvent for the chain

ωω ω

ω ωω ω ω ω

ω+ −

≠

= +− −

= + ⇒ =− −

= + + −− −

+

⇒

=

,

0 , 0 , 1 0, 1

0

Calculation of :1 1take matrix elements of identity 1

1 1 1 10 0 0 0

The chain H is suc ( 1 1 )h that

m n

n n n

h

h

g m n

HH H

n H n n H nH H H H

g g gt

n

H n t n n

0, 00

21 1 0

nn

h h

g g q

q q qt q tω ω

=

= + ⇒ − + =

36

This is a 3-term recurrence relation: try solution

2

12 2h h

qt tω ω

±

⇒ = ± −

| |

0 00( ) ( ) ( ) implies attenuation with distancenng g qω ω ω−=

3737

( )2 2

2 ( ', , ) '2

dz g x x z x xm dx

δ

+ = −

Continuous model, 1d

2 dimensions

( )2

2

2 '

2

2| | '

2( ', , ) , 0, blows at only edge 0

0 exponential damping ( ', , )

mzi x x

mz x x

m eg x x z z zi z

for z g x x z e

−

− −

= > →

<

Square

3838

( )

( ) ( )( )

( )

αα

ε

ρ ω δ ω ε δ ω επ

=

= − = −

∑

∑ ∫2

2 cos , d=dimensionality,

1| 0 |2

d

k h

dd k kd

k BZ

t k

k d k

( )( )

( ) ( )( )2 21 2 [cos cos ]

2 x y h x ydk dk t k kπ π

π πρ ω δ ω

π − −= − +∫ ∫

( ) ( ) ( )( ) ( )( )11 2 cos 2 cos ] 2 cos

2 y h x h y h xdk t k t k t kπ

πδ ω ρ ω

π −− − = −∫

artichocke technique

( ) ( ) ( )( )2 11 2 cos

2 x h xdk t kπ

πρ ω ρ ω

π −= −∫

( )

2

1dim 2

121

21

2

h

h

h

t

t

t

ωθ

ρ ωπ ω

− =

−

3939

This integral can be written in terms of the complete elliptic integral of the first kind. The band now extends from −4thto 4th, and is symmetric (the graph is bipartite).

Singularities become milder when integrated over, however they are still evident here:

Log Van Hove singularity

( ) ( ) ( )( )2 11 2 cos blows up at 0 and jumpsat edges

2 x h xdk t kπ

πρ ω ρ ω ω

π −= − =∫

4040

3 dimensions Simple cubic lattice is bipartite, and LDOS is symetric around O.

( ) ( ) ( )( )3 21 2 cos

2 x h xdk t kπ

πρ ω ρ ω

π −= −∫

Well-known edge singularities+Van HoveE

4141

22 ( ', , ) ( ')

2z g r r z r r

mδ

+ ∇ = −

Continuous 3d case

22 | '|

2( ', , )2 | ' |

mzi r rm eg r r z

r rπ

−

= −−

π π π

−− = →

−− →

2

2 2 2 2 2

2sin( | '|)1 2Im ( ', , )| '|2 2

| '

Well-known singularity(edge singularities depend on dimensional

| 0

ity)

mz r rm m mzg r r zr r

for r r

E

42

Lehmann representationIn non-interacting models, the poles of the Green’s functions close to the realaxis correspond to eigenstates of the Hamiltonian where a particle can existor where one can put a particle. Here we see how to generalize this notion to

the fully interacting case.

Zero-Temperature Fermi Case

M,n

M,n set of many-body eigenstates of H with M electrons M = 0, 1, 2,.... = electron number n = 0, 1, 2,... runs over the M-body eigenstates, such thatH M,n = E M,n , N M,n = M M

,n

†

iHt -iHt

G(N; xt, x't') = -i N, 0 T [ (x, t) (x't')] N, 0

(x, t) = e (x)e

ψ ψ

ψ ψ

average over the N-body interacting ground stateof the Hamiltonian H and the operators are in the Heisenberg picture.

43

inserting the complete set we obtain the very useful Lehmann representation: for t > t’,

1, ,0

†

†

iHt -iHt iHt' † -iHt'

( )( ') †

1, 1,

( , ', ') N, 0 (x, t) (x't') N, 0

N, 0 (x, t) (x't') N, 0

N, 0 e (x)e 1, 1, e (x')e N, 0

,0 ( ) 1, 1, ( ') ,0N n N

n

ni E E t t

n

N n

iG x x t t

N n N n

e N x N n N n x

n

N

N

ψ ψ

ψ ψ

ψ ψ

ψ ψ+− − −

− =

=

= + +

+ +

+ +

=

∑

∑

∑

1, ,0

†

( )( ') †

( , ', ') N, 0 (x't') (x, t) N, 0

,0 ( ') 1, 1, ( ) ,0N n Ni E E t t

n

iG x x t t

e N x N n N n x N

ψ ψ

ψ ψ−− − −

− = −

= − − −∑

for t < t’,

ω δ ω δ

ω δ

δ

δ ω∞ − +

−∞

−= =

+ −∫ ∫0

0

Fourier transform using (with infinitesimal >0 ):

i t t i t ti idt e t ei i

d

44

†

1, ,0

†

1, ,0

,0 ( ) 1, 1, ( ') ,0( , ', )

( )

,0 ( ') 1, 1, ( ) ,0( )

n N n N

n N n N

N x N n N n x NG x x

E E i

N x N n N n x NE E i

ψ ψω

ω δ

ψ ψω δ

+

−

+ +=

− − +

− −+

+ − −

∑

∑

quasiparticle excitations, including IP and E.A.

Singularities:

( )

( )

''

, ', '( , ', ) ', sign( ' )*

'1, ', ' Im( )sign(

Spectral function

) 0

:x x

G x x di

x x G

ωω

ρ ωω ω δ ω µ δ

ω ω δ

ρ ω ωπ

ρ

µ

= = −− +

= − − >

∫

Re z

Im z IP

EA

μ=chemical potential

( )

( )

†1, ,0

†1, ,0

( , ', ) ,0 ( ) 1, 1, ( ') ,0 ( )

,0 ( ') 1, 1, ( ) ,0 ( )

N n Nn

N n Nn

x x N x N n N n x N E E

N x N n N n x N E E

ρ ω ψ ψ δ ω

ψ ψ δ ω

+

−

= + + − −

+ − − + −

∑

∑

particle addition

particle subtraction

Similar to G we met in Fano theory, except that Im G changes sign at Fermi level

( )

( )

''

, ', '( , ', ) ,

spectral f

( ' )*'

1, ', ' I unctio( ( nm ) ) 0

x xG x x sign

i

x x G sign

ωω

ρ ωω δ ω µ δ

ω ω δ

ρ ω ω µπ

= = −− +

= − − >

∫

45

Finite Temperatures, Fermi and BoseInstead of the ground state average we make a grand-canonical one,:

1 Fermione can extend to Bose by defining

1 Boses = −

( ) ( )†( ') ',mn n x m m xR nx x ψ ψ=

1 1ˆ ˆ( ), = , , grand partition functionnKK

nB

A Tr A e K H N Z eZ K T

ββρ ρ β µ −−= = = − =∑

( ) †

( ') ( ')

( , ', ') Tr [ ( , ), ( , )] ( ')= ( ( , ') ( ', ) )n mn mn

r

K i t t i t tmn mn

mn

g x x t t i x t x ti t t e R x x e R x x e

Zβ ω ω

ρ ψ ψθ

+

− − −

− = −− −

+∑

( ) ( , ')1( , ', ')= ( ) .n mK Kr mn

mn mn

R x xg x x t t e seZ i

β β

ω ω δ− −− +

+ +∑

( ) ( ) ( )

( ) ( ) ( )

, ', '', ',2 '

, ', '', ',2 '

r

a

x xdg x xi

x xdg x xi

ρ ωωωπ ω ω δ

ρ ωωωπ ω ω δ

= − +

= − −

∫

∫

( ) ( )( )

( ) ( )( )

( ) ( )( )

( )

T-dependent Spectral function2 , ' 1

2 , ' 1

2 , '

n

n n m

n m

Kmn mn

mn

K K Kmn mn

mn

K Kmn mn

mn

e R x x seZ

e R x x seZ

R x x e seZ

β βω

β β

β β

πρ δ ω ω

π δ ω ω

π δ ω ω

−

− −

− −

= + +

= + +

= + +

∑

∑

∑

The general analytic structure is the same for all systemsand similar formulas hold for Fermi and Bose particles.

In a similar way,

1Time-ordered ( , ', ) ( , ')n mK K

mnmn mn mn

e seG x x R x xZ i i

β β

ωω ω δ ω ω δ

− − = − + + + −

∑

( ) ( )†( ') ',mn n x m m xR nx x ψ ψ=

48

Fluctuation-Dissipation TheoremFor the time-ordered G

' Im ( , ', ') 'Re ( , ', ) tanh Bose' 2

' Im ( , ', ') 'Re ( , ', ) coth Fermi' 2

d G x xG x x

d G x xG x x

ω ω βωωπ ω ωω ω βωωπ ω ω

= − − = − −

∫

∫

The above results imply a number of relationships involving ρ and the realand imaginary parts of the various Green’s functions. Recall

( ) ( )† ' .( , ')mn n x m mx nR x xψ ψ=

One shows that assuming Rmn(x, x) is real (it must be positive forx = x’ and it can be taken real anyhow in the absence of magnetic fields)

H. B. Callen, T. A. Welton (1951)

49

' Im ( , ', ') 'Re ( , ', ) tanh Bose' 2

' Im ( , ', ') 'Re ( , ', ) coth Fermi' 2

d G x xG x x

d G x xG x x

ω ω βωωπ ω ωω ω βωωπ ω ω

= − − = − −

∫

∫

At 0 Kelvin they become the well known Hilbert transforms.

Albert Einstein (1905) found that i Brownian motion the mean-free-path isproportional to the viscosity, Harry Nyquist 1928 found that the noise in a conductor is proportional to the resistivity. Spontaneous fluctuationsproduce the response of a system to an external perturbation.

50

Interacting system

Adiabatically switchedweak perturbation,the

cause

)(ˆ)(' tFAtH −=

Kubo formulae for linear response theory

B=effect of cause A with <B>=0 in equilibrium

0S

S

H

eq eq HeTr B

Tre

β

βρ ρ−

−= =

ˆ ˆ( ) we need density matrix B t Tr Bρ ρ= ⇒

'SH H H= +

χ

φ φ∞

∆ ∫Recall the familiar relation P = E of electric field to polarization, the linear response equation is

t

BA BA-B(t) = dt' (t - t')F(t'), (t )=response function

50

Typically, one excites the system with a probe A F(t) and measures some induced quantity B

51

Hence, the first order density matrixobeys

( ) ( ) ˆ, ( ) ,S eqdi t H t F t Adt

ρ ρ ρ− −

− ∆ = ∆ + −

( ) ( )ρ ρ−

− = + '( ),Sdi t H H t tdt

( ) ( )

[ ]

ρ ρ ρ

ρ ρ ρ

−

−− −

− ∆ = − + ∆

= + − + ∆ +

( ),

ˆ, ( ) , , ...

S eq

S eq eq S

di t H AF t tdt

H F t A H

Linearized equation of motion

51

Heisenberg EOM

( ) ( )ρ ρ ρ ρ≈ + ∆S ; commuet tes with Heq eq St t

)(ˆ)(' tFAtH −=

check: just differentiate

( ') ( ')

( ') ( '' (

'

' )( (,

,

,

'

)'S S

S

S S

S

t iH t t iH t t

eq

iH t t iH t t

eqt

t iH t t iH t t

eq

t

d d Fe A e dtdt i

F te A ei

de A e Fdtt

d

d

tρ ρ

ρρ− −

−

−∞

− −−

− −

−∞

=

−

+

∆ =

−

−

+

∫

∫

Compact solution formula

( )ρ ρ−

− −

−∞ ∆ = − ∫

( ') ( ')(it looks anti-Heisenberg)

1 , ( ') '.S Si it H t t H t t

eqt e A e F t dti

( ) ( ) ˆ, ( ) ,S eqdi t H t F t Adt

ρ ρ ρ− −

∆ = ∆ −

,( )( ) eqS

SFAi t Htid

it

Hd ρ ρρρ∆ = − +∆−

∆+

( ) ( ) , ( ), . .S S eqdi H t t H A F t O Kdt

ρ ρ ρ ρ ⇒ ∆ = ∆ −∆ + −

52

Comment: if probe A commutes with system Hamiltonian, no B.

53

ˆEffect: ( ) ( )B t Tr t Bρ = ∆

cyclic property

( ') ( ')ˆ( ) , ( ') '

S St iH t t iH t t

eqiB t Tr e A e B F t dtρ

− − −

−∞

∆ = ∫

( ') ( ')ˆ ˆ( ) , ( ') ', that is,

S St iH t t iH t t

eqiB t Tr A e Be F t dtρ

− − −

−∞

∆ = ∫

ˆ( ) , ( ') ( ') 't

eq HiB t Tr A B t t F t dtρ

−∞

∆ = − ∫

Heisenberg picture operators with HS

Obtain first-order response

53

( )ρ ρ−

− −

−∞ ∆ = − ∫

( ') ( ')(it looks anti-Heisenberg)

1From , ( ') '.S Si it H t t H t t

eqt e A e F t dti

54

( ) , ( ') ( ') 't

eq HiB t Tr A B t t F t dtρ

−∞

∆ = − ∫

This is the Response equation

Hence, ( ) ' ( ') ( ') with the response functiont

BAB t dt t t F tφ−∞

∆ = −∫

( ) ( ) ( ) (ugly!)BA eq eqi t Tr A B t AB tφ ρ ρ− = −

cyclic property

[ ]( ) ( ) ( ) ( ),BA eq eqi t Tr B t A AB t B t Aφ ρ ρ−

− = − =

This is Kubo formula in its simplest form. The response function is a correlation function.

54

φ∞

∆ ∫t

BA-B(t) = dt' (t - t')F(t'),

55

[ ]( ) ( ) ( ) ( ),BA eq eqi t Tr B t A AB t B t Aφ ρ ρ−

− = − =

Let us derive a new form restarting from

( ) ( ) ( ) (ugly!)BA eq eqi t Tr A B t AB tφ ρ ρ− = −

55

Comment: if probe A commutes with system Hamiltonian, no B.

Comment: if probe A commutes with B, no B.

Since , the formulaSH

eqe ugly

Z

β

ρ−

=

( )yields: ( ) , ( )SHBAi t Tr A e B tβφ − − =

56

( )yields: ( ) , ( )SHBAi t Tr A e B tβφ − − =

We get another Kubo formula by a chain of transformations, that we prove below:

( )( ) ( ),1S

BAHAi t Tr B te

Zβφ − − =

0

identity: , ( )H H H HA e e e HA AH e dβ

β β τ τ τ− − − = − ∫

0

, [ ]H Ht i

dAA e i e ddt

ββ β

τ τ− −=− = − ∫

0

( )( ) | ( )BA t idA tt B t d

dt

β

τφ τ=−= ∫ 56

( ) ( ) ( )BA eq eqi t Tr A B t AB tφ ρ ρ− = −

57

( ),( ) ( )SBA

HA ei t Tr B tβφ − − =

0

, ( )H H H HA e e e HA AH e dβ

β β τ τ τ− − − = − ∫

is obtained by the identity

( ) ( )

0

by integratin

)

g (

,

:

(

, )

H H H H H H H H

H

H H H H

H

d de A e e Ae A He Ae e AHed d

e HA AH e

e A e e HA AH e dβ

β β τ τ

β β β β β β β β

β β

β β

τ− −

− − − −

−

= − = −

= −

= − ∫57

Proofs

58

0

, ( )H H H HA e e e HA AH e dβ

β β τ τ τ− − − = − ∫

0

, [ ]H Ht i

dAA e i e ddt

ββ β

τ τ− −=− = − ∫

58

One can eliminate the commutator in favor of a time derivative using

[ , ( )] [ ] [ , ] [, ]Ht H

H Ht i

ti idA dAi H e Ae idt

H A t H Ad

et

e τ ττ

−− =−

−= −= = ⇒−

0

, [ ] ,H Ht i

dAA e i e ddt

ββ β

τ τ− −=−−

⇒ = − ∫

0 0

( ) ( )( ) | ( ) | ( )HBA t i t i

dA t dA tt Tr e d B t B t ddt dt

β ββ

τ τφ τ τ−=− =−= =∫ ∫

0

, [ ]H Ht i

dAA e i e ddt

ββ β

τ τ− −=− = − ∫

,( ) ( )SBA

HA ei t Tr B tβφ − − =

Finally, inserting

one obtains

59

into

Example: conductivity tensor

β τ τ

β βτ τ

σ τ

τ τ τ

−−

−−

=

= = +

∫

∫ ∫

0

0 0(0) ( )

Ht Hti iH Hij i j

Ht Hti iH Hi j i j

d e J e e J e

d J e e J e e d J J t i

depends on the autocorrelation function of current in equilibrium

60

σ ( , )ij t T

= =(cause) (effect)ii

dxA x J nedt

σ=i ij jJ E= −∇

,E V V x

β βσ τ τ −= =∫0 ( ) ( ) , Hiij j

dxd i J t X Tre X

dt

Re time

Im time

iβ

t

( ) ( )

( ) ( ) ( ) ( ) ( )

λ λλ

α β α β γ δ δ γα β γ δ

= +

= Ψ Ψ

= Ψ Ψ − Ψ Ψ

∑ ∫

∑ ∑ ∫

0†

0 0

† †' ' ' ' ' ' ' '

' ' ' '

Many-body theory : the full Hamiltonian reads

( )

12

H H VH dy y h y y

V dydz y z v y z z y

Many-body problem and Green’s functions

ψ ψ ψ ψ = − † †

2 1 2 3 4 1 2 3 4

Two-particle

( , , , ) ( ) ( ) ( ) ( )

(Heisenberg operators)

GF

G x x x x T x x x x

σσ σ σψ ψ

'

(T) †, 'Along with ig (x,t, x',t') = T (x,t) (x',t') ,

which describes the propagation of a Fermion, we also need to propagate 2.

2x

1x 3x

4x

This arises from the equation of motion for g (see below). 61

β α β β β α β

αβ β αβ αδ δ δ δ

Ψ Ψ Ψ Ψ +Ψ Ψ Ψ

= − Ψ = − Ψ

† †( ) : ( ) ( ) ( ) ( )Multiply b ( ) ( )

( ) (

y

) (

) (x)

y x y y y x y

x y y x y

( ) ( )λ λλ

λ

α α

α λ λ α

− −

−

ΨΨ = Ψ

= Ψ Ψ Ψ Ψ

Ψ

=

∑ ∫∑ ∫

†0

0

0

0

0

†

c( )

( ) (x)

ommutator with H[ ( ),H ] [ ( ), ]

[ ( ), ( ) ( )] ( )

dy y h y y

dyh y h

x x

x y y x

The field operators anticommute:α β

α β β α αβδ δ+Ψ Ψ =

Ψ Ψ +Ψ Ψ = −

†

† †

[ ( ), ( )]

( ) ( ) ( ) ( ) ( )

x y

x y y x x y

Equations of motion technique [ , ]di A A Hdt

=

( ) ( ) ( ) ( )αα β αβ γ δ δ γ

β γ δ

α

∂Ψ= Ψ + Ψ − Ψ Ψ

∂Ψ

∑ ∑ ∫ †0 ' ' ' ' ' '

' ' '

First,we show that( , )

( ) ( , ) .

We need to commute ( , ) with H.

x ti h x x t dy y v x y y x

tx t

62

( ) ( ) ( ) ( ) ( )α β α β γ δ δ γα β γ δ

= Ψ Ψ − Ψ Ψ∑ ∑ ∫ † †' ' ' ' ' ' ' '

' ' ' '2 .V dydz y z v y z z y

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

α α β α α β α β α

αα α α β α β αδ δ−Ψ Ψ Ψ = Ψ Ψ Ψ −Ψ Ψ Ψ =

= − −Ψ Ψ Ψ −Ψ Ψ Ψ

† † † † † †' ' ' ' ' '

† † † †' ' ' ' '

Using the anticommutation relations, [ , ]

[ ( ) ]

x y z x y z y z x

x y y x z y z x

( ) ( ) ( ) ( ) ( ) ( ) ( )αα β α α β α β αδ δ= − Ψ −Ψ Ψ Ψ −Ψ Ψ Ψ† † † † †' ' ' ' ' '( )x y z y x z y z x

( ) ( ) ( ) ( ) ( ) ( ) ( )αα β α αβ β α α β αδ δ δ δ= − Ψ −Ψ − −Ψ Ψ −Ψ Ψ Ψ† † † † †' ' ' ' ' ' '( ) [ ( ) ]x y z y x z z x y z x

( ) ( )αα β αβ αδ δ δ δ= − Ψ − − Ψ† †' ' ' '( ) ( ) . Therefore,x y z x z y

( ) ( ) ( ) ( ) ( ) ( )α α α β α β γ δ δ γα β γ δ

− −Ψ = Ψ Ψ Ψ − Ψ Ψ∑ ∑ ∫ † †' ' ' ' ' ' ' '

' ' ' '[ (x),2 ] [ , ]V dydz x y z v y z z y

( ) ( ) ( ) ( ) ( )αα β αβ α α β γ δ δ γα β γ δ

δ δ δ δ= − Ψ − − Ψ − Ψ Ψ∑ ∑ ∫ † †' ' ' ' ' ' ' ' ' '

' ' ' '[ ( ) ( ) ]dydz x y z x z y v y z z y

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )β αβ γ δ δ γ α α αγ δ δ γβ γ δ α γ δ

= Ψ − Ψ Ψ − Ψ − Ψ Ψ∑ ∑ ∑ ∑ ∫† †' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' '] .dz z v x z z y dy y v y x x y

63

( ) ( ) ( ) ( )α α

α β αβ γ δ δ γβ γ δ

−

−

Ψ = Ψ

Ψ = Ψ − Ψ Ψ∑ ∑ ∫0 0

†' ' ' ' ' '

' ' '

Summarizing : [ ( ), ] ( ) ( )[ ( ), ]

x H h x xx V dz z v x z z x

( ) ( ) ( ) ( )αα β αβ γ δ δ γ

β γ δ

∂Ψ= Ψ + Ψ − Ψ Ψ

∂ ∑ ∑ ∫ †0 ' ' ' ' ' '

' ' '

( , )( ) ( , )

x ti h x x t dy y v x y y x

t

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )β αβ γ δ δ γ α α αγ δ δ γβ γ δ α γ δ

= Ψ − Ψ Ψ − Ψ − Ψ Ψ∑ ∑ ∑ ∑ ∫† †' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' '] .dz z v x z z y dy y v y x x y

( ) ( ) ( ) ( )δ γ γ δ α β γ δ β α δ γΨ Ψ → −Ψ Ψ

→

→' ' ' ' and ( ' ' ' ' )Now in the first term we rename z y wh

( ' ' ' ' ).The second contribution is id

ile in the secon

entical to the firs

d

t.

x y y x

( ) ( ) ( ) ( )αα β αβ γ δ δ γ

β γ δ

∂Ψ= Ψ + Ψ − Ψ Ψ

∂ ∑ ∑ ∫ †0 ' ' ' ' ' '

' ' '

write for the time associated with x( , )

( ) ( , ) .

x

xx

x

I tx t

i h x x t dy y v x y y xt

64

( ) ( ) ( ) ( )αα β αβ γ δ δ γ

β γ δ

∂Ψ= Ψ + Ψ − Ψ Ψ

∂ ∑ ∑ ∫ †0 ' ' ' ' ' '

' ' '

( , )( ) ( , ) .x

xx

x ti h x x t dy y v x y y x

t

( ) ( ) ( ) ( )α α

α

αα

β αβ γ δ δ γβ γ δ

∂ΨΨ = Ψ Ψ +

∂Ψ Ψ − Ψ Ψ∑ ∑ ∫

† †0

† †' ' ' ' ' '

' ' '

( , )( , ) ( ) ( , ) ( , )

( , )

x ti z t h x z t x t

tz t dy y v x y y x

αΨ†Now left multiply by ( , )z t

( ) ( ) ( ) ( )

† †0

† †' ' ' ' ' '

' ' '

( , ) ( , ) ( ) ( , ) ( , )

( , ) , ,

z x z xx

z y y

i z t x t h x z t x tt

dyv x y z t y t y t x

α α

α

α α

αβ γ δ β δ γβ γ δ

∂Ψ Ψ = Ψ Ψ +

∂

− Ψ Ψ Ψ Ψ∑ ∑ ∫

Ground-state average, tz>tx

65

( ) ( )

( ) ( )β δδ

+

+

⇒

Ψ Ψ

>

∂− =

=

−

+

∂ ∑ ∫†

' '

0 2

the eom,for ,involves a special ordering of operators:

( ) ( , ) ( , ) , , ;

we need to write in order to ge , ,t

,

.( , )y

z x

y

x

y

t t

i h x iG

y t

x z dyv x y G x

y

y y

y t

t

t

z

y

( )δ δ +

≤

∂− = − − − ∂

∫0 2

Extending to the case ,we obtain the EOM

( ) ( , ) ( ) ( ) ( , ) , , , .

z x

x zx

t t

i h x G x z x z t t i dyv x y G x y y zt

( ) ( ) ( )α β δ γΨ Ψ Ψ Ψ† †

' ' 'New object: ( , ) , ,z y yz t y t y t x

ψ ψ ψ ψ = − † †

2 1 2 3 4 1 2 3 4

introduce the two-particle

( , , , ) ( ) ( ) ( ) ( )

(Heisenberg operators)

GF

G x x x x T x x x x

σσ σ σψ ψ

'

(T) †, 'Recalling the definition ig (x,t, x',t') = T (x,t) (x',t') ,

2x

1x 3x

4x

x z =∂ − ∂

0( )i h xt y +y

zx+

66

67

( )

ψ ψ ψ ψ

δ

ψ ψ ψ ψ

+

∂− = − ∂

= −

= = −

∫

† †2 2

† †2 1 2 3 4 1 2

0 21

3 4

(1,2|3, 4) (1,2,3, 4) (1) (2) (3) (4) ; the EOM

( , , , ) ( ) ( ) ( ) ( ) often denote

(1) (1,1') (1,1') 2 (1,2) 1; 2

d

|2 ;1' or

as:

Popular shorthand notations :

i h G i d v G

G x x x x T

t

G

x x x

G T

x

( )δ + ∇∂+ − = − ∂

∫21

21

(1) (1,1') (1,1') 2 (1,2) 1; 2|2 ;1

al

'

s

.

o

2i U G i d v G

t m

( )δ + ∂ ∇+ − = − ∂

∫22

22

One can also differentiate with respect to 2

obtaining th

(2) (1,2) (1,2) 1 1; 1|1 ; 2 (1,2)2

e adjoint equation:

i U G i d G vt m

( )δ δ + ∂− = − − − ∂

∫0 2( ) ( , ) ( ) ( )

eom reads

( ,

:

, ) , ,x zi h x G x z x z t t i dyv x y G x y y zt

67

Hierarchy of Green’s functions: eom for G2 yieldsG3 objects (several kinds).......

In principle for finite systems one can find the exact solution by solving a set of coupled equations.

In reality one must truncate the hierarchy in some way; one can improve a non-interacting approximation for G by including a non-interacting approximation for G2, and presumably one can do betterby truncating at higher order.

68

696969

Recall: Quantum averages in terms of GF

†The average of one-b densitiesody ' '

'

ˆ f(x) = f (x) (x). (x)σσ σ σσσ

ψ ψ∑( ) ( )σσ σ σ σσ σσ

σσ σσ

ψ ψ→ + →∑ ∑ T†

' ' ' '' '' 'f = f (x)< (x). (x)>=-i lim lim f (x)g , , ', '

t t x xx t x t

'

(T) †, 'ig (x,t, x',t') = T (x,t) (x',t')

σσ σ σψ ψ

( ) ( )†The average of one-body operators ' '

' A = dx a ( )x x xσσ σ σ

σσ

Ψ Ψ∑ ∫

( ) ( )σσ σσ+ →→∫

' ''' A = -i dx lim lim , ' 'T

x xt ta x g xt x t

But we can gain info on interaction energy, too

And this is crucial

( ) ( ) ( )T

' 'x,t =-i lim lim g , , ', '

t t x xx t x tσ σσρ

→ + →

( )δ δ

ψ ψ ψ ψ

δ

+

+

+

∂− = − − − ∂

= − = − ≠ = +

∂− = − ∂

∫

∫

0 2

† †2 1 2 3 4 1 2 3 4

0 2

eom : ( ) ( , ) ( ) ( ) ( , ) , , ,

where ( , , , ) ( ) ( ) ( ) ( ) .

Setting , with 0 but ,

( ) ( , ) ( , ) ,

x zx

x z z x

x

i h x G x z x z t t i dyv x y G x y y zt

G x x x x T x x x x

z x t t t t

i h x G x x i dyv x y G xt ( )+ +, ,y y x

Ground-state energy from 1-body GF(is the g.s. magnetic, superconducting, CDW,.... Questions that require precise gs energy calculations)

recall: T orders earlier times on the right; each exchange brings (-)

( )ψψ ψ ψ

ψψ ψψ

ψ

ψ

ψ+ +

+ +

+

+

+

+ +

= =

∂− = − = ∂

−

+− =

−

∫

∫∫

0

†

†

2

†

††

†

( ) ( , ) ( , ) , , , (by definition)

( , ) (exchange( ) ( ) ( ) ( )

(

with )

( , ) ( ) (bring ( ) to t

( )

( )

( ) (

) ( ) he left

)

) ( )

x

y y

y

i h x G x x i dyv x y G x y y xt

i dyv x y

i dyv x y

T x y x y

T xy xx

ψ ψ ψ ψ

ψ ψ ψ ψ

ψ

+

+

+

+ =

= −

= + =

∫∫

†

† †

†( , ) (bring to the right )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ,, )

)

(

(

y x

T x y y x

T

i dyv x xy

i dyv x y tx y x ty 70

71

ψ ψ ψ ψ+ +∂ − = + ∂

∫ † †0( ) ( , ) ( , ) ( ) ( ) ( ) ( )i h x G x x i dyv x y x y y x

t

ψ ψ ψ ψ+ +∂ − = = ∂

∫ ∫ † †0( ) ( , ) ( , ) ( ) (So, ) ( ) ( ) 2dx i h x G x x i dxdy v x y x y y x i V

t

01 ( ) ( , )2

V dx i h x G x xi t

∂ = − ∂ ∫

x z =∂ − ∂

0( )i h xt y +y

zx

x

=∂ − ∂

0( )i h xt

y +y

x +x

The previous result

becomes

)𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅: −𝑖𝑖𝑖𝑖(𝑥𝑥t, 𝑥𝑥𝑡𝑡 + 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑅𝑅 𝑑𝑑𝑅𝑅𝑑𝑑𝑖𝑖𝑖𝑖𝑡𝑡𝑑𝑑

The propagator ‘knows’ the interaction. 71

( ) ( )

( ) ( )λ λ

λ

λ λλ

+

= Ψ Ψ

= Ψ Ψ = −

∑ ∫∑∫ ∫

†0 0

†0 0 0

The average of the one-body part is( )

( ) ( ) ( , )

H dy y h y y

H dy h y y y i dy h y G y y

→ →

∂= + = − + ∂

∫0 ' , ' 0

The knowledge of is enough to get exact .A reliable calculation of E is of crucial importance since important many-body techniques (DFT etc.) are var

lim ( ) ( , ' ')2

iati

t t x x

G E

iE H V dx i h x G xt x tt

onal.

πωπ ε ω

∞

≠

= − +∑ ∫2

200

Recall results of part 11 1 2ˆ [ Im ]

2

:

( )

0

q,q

neV dq

The propagator also ‘knows’ the kinetic+potential term

72

73

Recall: we have already met the Coulomb interactionin terms of density fluctuations

2† †

20

†

Using

ˆ ˆ1 4ˆˆ ˆ ˆ ( ).2q k k q q q k

k qk

ec c Vq

nσ σσ

πρ ρ ρ ρρ+ −≠

= = = −∑ ∑

𝑡𝑡𝑥𝑥Relation to ground state potential energy

22

20

ˆInteraction in ground state : 0 0 inserting complete set,

2 ˆ( | 0 | ). qq n

V

e n nqπ ρ

≠

=

−∑ ∑

( ) [ ]2 2

0 02

2

200

1 4Comparing with Im 0 ( ) ( ) ,q,

1 1 2ˆ0 0 [ Im ].2 (q, )

q n nn

q

e nq

neV dq

π ρ δ ω ω δ ω ωε ω

πωπ ε ω

∞

≠

= + − −

= − +

∑

∑ ∫