Transcript Polarisation Propagator

Polarisation Propagator
•
•
•
•
Collective excitations (M227, F 171, F 558)
Poles of G are single-particle excitations (creation of particles or holes)
Poles of P are collective excitations
Density and Density Fluctuation operators


ψ̂  (r, t)ψ̂(r, t)  ˆ (r, t) Density operator  c.f. number operator  ci ci  n̂ 
i


~ (r, t)  ˆ (r, t) - ˆ (r ) Density fluctuatio n operator
~ (r, t) ~ (r ' , t' )  ˆ (r, t) - ˆ (r ) ˆ (r ' , t' ) - ˆ (r ' ) 
Measures correlatio ns in density fluctuatio ns
ˆ (r, t) -
ˆ (r ) ˆ (r ' , t' ) - ˆ (r ' )

 ˆ (r, t) ˆ (r ' , t' )  ˆ (r, t) ˆ (r ' )  ˆ (r ) ˆ (r ' , t' )  ˆ (r ) ˆ (r ' )
 ˆ (r, t) ˆ (r ' , t' )  ˆ (r ) ˆ (r ' )
Polarisation Propagator
• Dielectric Function
• The dielectric function for a material relates the electrostatic potential
due to all charges to an external electrostatic potential
V tot (r, t)   dr dt' ε 1 (r, r ' , t  t' )φ ext (r ' , t' )
Total potential  external potential  induced potential
1
ext
φ (r ' , t' ) (r - r ' ) (t - t' ) 
 (r ' , t' )
r  r'
 (r ' , t' )   dr ' ' dt' ' P R (r ' , r ' ' , t' t' ' )φ ext (r ' ' , t' ' )


1
R

ε (r, r ' , t  t' )   dr ' ' dt' '   (r - r ' ) (t - t' ) 
P (r ' ' , r ' , t' ' t' ) 
r  r' '


1
Polarisation Propagator
•
•
•
•
Space-time interpretation of Polarisation Propagator
Drawn as a pair of directed lines
Non-interacting propagator Po represented by single directed lines
Interacting propagator P represented by filled loop
Create electron-hole pair
r’,t’
Destroy electron-hole pair
t > t’
Po
r’,t’
r,t
P
time
iP R (r, r ' , t  t' ) 
-
o ψ̂ H (r, t)ψ̂ H (r, t)ψ̂ H (r ' , t' )ψ̂ H (r ' , t' ) o
o o
o ψ̂ H (r )ψ̂ H (r ) o
o ψ̂ H (r ' )ψ̂ H (r ' ) o
o o
o o
 (t  t' )
r,t
Polarisation Propagator
• Lehmann Representation (F 172 M 375) physical significance of P
n̂, ψ̂  ψ̂
n̂, ψ̂   ψ̂

n̂ψ̂  ψ̂n̂  ψ̂  ψ̂(n̂  1)

n̂ψ̂   ψ̂  n̂  ψ̂   ψ̂  (n̂  1)
n̂ψ̂  ψ̂  ψ̂  (n̂  1)ψ̂  ψ̂  ψ̂  ψ̂  n̂ψ̂  ψ̂  ψ̂  ψ̂  ψ̂(n̂  1)  ψ̂  ψ̂ n̂
n̂ψ̂  ψ̂ n  ψ̂  ψ̂ n̂ n  Nψ̂  ψ̂ n
i.e. number of particles is unchanged
iP  (r, r ' , t - t' )  o ψ̂ H (r, t)ψ̂ H (r, t)ψ̂ H (r ' , t' )ψ̂ H (r ' , t' ) o  ( t  t ' )
insert complete set of states and make time dependence explicit
iP  (r, r ' , t - t' )  o e  iĤt ψ̂S (r )ψ̂S (r )e -iĤt n n e  iĤt'ψ̂S (r ' )ψ̂S (r ' )e -iĤt' o  ( t  t ' )
 o ψ̂S (r )ψ̂S (r ) n n ψ̂S (r ' )ψ̂S (r ' ) o e
-i(E n - E o )(t - t')
 (t  t' )
Polarisation Propagator
• Lehmann Representation (F 172 M 375) physical significance of P
-i(E - E )(t - t')
iP  (r, r ' , t - t' )  o ψ̂S (r )ψ̂S (r ) n n ψ̂S (r ' )ψ̂S (r ' ) o e n o
 (t  t' )

 d(t  t' )e
i (E n E o - i )(t  t') iE(t  t')
0
iP  (r, r ' , E) 
e

i
E - (E n  E o )  i
o ψ̂S (r )ψ̂S (r ) n n ψ̂S (r ' )ψ̂S (r ' ) o
E - (E n  E o )  i
The poles of the retarded polarisati on propagator occur at the exact excitation
energies of the N particle system
Polarisation Propagator
• Time-Ordered (Causal) Polarisation Propagator


o T ψ̂  (r, t)ψ̂(r, t)ψ̂  (r ' , t' )ψ̂(r, t' ) o
connected
ψ̂ (r, t)ψ̂(r, t)ψ̂ (r' , t' )ψ̂(r' , t' )
(i)2Go(r,r) Go(r’,r’)
ψ̂ (r, t)ψ̂(r, t)ψ̂ (r' , t' )ψ̂(r' , t' )
(i)2Go(r,r’) Go(r’,r)
 o ψ̂  (r)ψ̂(r) o o ψ̂  (r' )ψ̂(r' ) o
-(i)2Go(r,r) Go(r’,r’)
Polarisation Propagator
• Single-particle polarisation propagator Po in coordinate form


iP o (1,2)  G o (1,2)G o (2,1)  G o (1,2)  G -o (1,2) G o (2,1)  G -o (2,1)

 G o (1,2)G -o (2,1)  G o (1,2)G o (2,1)
occ
G (1,2)  i  ψ m (1)ψ*m (2)e

o
i ( m  i )(t 1  t 2 )
 (t 2  t1 )
m
unocc
G (1,2)  i  ψ n (1)ψ*n (2)e

o
i ( n - i )(t 1  t 2 )
 (t 1  t 2 )
n
occ
G (2,1)  i  ψ m (2)ψ*m (1)e

o
i ( m  i )(t 2  t1 )
 (t 1  t 2 )
m
unocc
G (2,1)  i  ψ n (2)ψ*n (1)e

o
i ( n - i )(t 2  t1 )
 (t 2  t1 )
G o (2,1)  â(2)â  (1)  (t 2 - t1 )
n
1,t1
Po1,2
2,t2
G o (1,2)  b̂(2)b̂ (1)  (t 2 - t1 )
Polarisation Propagator
• Single-particle polarisation propagator Po in coordinate form

o
G (1,2)G (2,1) 
occ,unocc
 ψ n (1)ψ*n (2)ψ m (2)ψ*m (1)e
o
i ( n - m )(t 1  t 2 )
 (t 1  t 2 )
m, n

o

o
G (1,2)G (2,1) 
occ,unocc
*
*
ψ
(
1
)
ψ
(
2
)ψ
(
2
)ψ
 m m n n (1)e
i ( m - n )(t 1  t 2 )
 (t 2  t1 )
m, n

o
G (1,2)G (2,1)  i
occ,unocc
o

m, n
G o (1,2)G o (2,1)  i
occ,unocc

m, n
P o (1,2,  ) 
occ,unocc

m, n
ψ n (1)ψ*n (2)ψ m (2)ψ*m (1)
  ( n -  m ) - i
poles at   ( n -  m )  i
ψ m (1)ψ*m (2)ψ n (2)ψ*n (1)
poles at   ( n -  m )  i
  ( n -  m )  i
ψ m (1)ψ*m (2)ψ n (2)ψ*n (1) ψ n (1)ψ*n (2)ψ m (2)ψ*m (1)

  ( n -  m )  i
  ( n -  m ) - i
Poles of the single - particle, time - ordered polarisati on propagator occur at
single particle excitation energies of the N particle system
Polarisation Propagator
• Poles of time-ordered P or Po in the complex energy plane
Im()
Advanced
xxx
xxx
xx xx xxx x x xxx
Retarded
Re()
xx xx xxx x x xxx
Polarisation Propagator
• Leading terms in expansion of Polarisation Propagator
r’
ψ̂ (r, t)ψ̂(r, t)ψ̂ (r' , t' )ψ̂(r' , t' )
(i)2Go(r,r’) Go(r’,r)
r’
r
 d1d2
1 
ψ̂ (1, t' ' )ψ̂  (2, t' ' )ψ̂(2, t' ' )ψ̂(1, t' ' )ψ̂  (r, t)ψ̂(r, t)ψ̂  (r' , t' )ψ̂(r' , t' ) 1
1- 2
2
r
1 



d
1
d
2
ψ̂
(
1
,
t'
'
)
ψ̂
(
2
,
t'
'
)
ψ̂
(
2
,
t'
'
)
ψ̂
(
1
,
t'
'
)
ψ̂
(
r
,
t)
ψ̂
(
r
,
t)
ψ̂
(r' , t' )ψ̂(r' , t' )

1- 2
r’
1
r
2
Polarisation Propagator
• Further Diagrams in the Polarisation Propagator
• Zeroth Order
• First Order
• Second Order
• Third Order
Polarisation Propagator
• Classification of Diagrams in the Polarisation Propagator
• Proper
• Improper
• Ladder
• Ring (Bubble)
Vertex Part
Polarisation Propagator
• Effective Interparticle Interaction (M 189, F 111, 154)
• Total inter-particle/hole interaction is sum of direct (instantaneous
Coulomb) interaction plus (retarded) reaction from medium
V
=
+
=
+
=
v
+
+…+
+…+
P
+
vPv
V effective (dressed) interaction
v Coulomb (bare) interaction
P Polarisation propagator (polarisation insertion)
+
Polarisation Propagator
• Effective Interparticle Interaction
• Summation in terms of proper polarisation insertion P*
P* =
+
+
+…
=
v
+
v P* V
=
=
v
v
+
+
v P* v + v P* V)
v P* v + v P* v P* v + v P*)V
P
=
P*
+
P* v P
V
=
v
+
vPv
V
V
=
=
v
v
+
+
v (P* + P* v P) v
v P*v + v P* v (P* + P* v P) v
V
Polarisation Propagator
• Effective Interparticle Interaction: Dielectric Function
• V = v + v P* V
• (1 - v P*) V = v
• V = (1 - P*v)-1 v
• -1 = (1 - v P*)-1
•
 = (1 - v P*)
•
P* = Po yields the Random Phase Approximation to 
•
RPA = (1 - v Po)
Polarisation Propagator
• Selective Summation of Ring Diagrams
• Restricting P* to Po in -1 sums ring diagrams to infinite order
•
•
-1RPA v = (1 - v Po)1 v = (1 + v Po + v Po v Po + v Po v Po v Po + … ) v
= v + v Po v + v Po v Po v + v P o v Po v Po v + …
=
+
+
+…
Polarisation Propagator
• Model of the Density Response Function
EExt
p1
e2
α
I Atom polarisability
2
2
m(ωo - ω )
 p1   α1 0   T12 .p 2  E1Ext 

   
.
Ext 
 p 2   0 α 2   T21.p1  E 2 
p2

p i  α i .Eiloc  α i . Tij.p j  EiExt
 α1 0 


 0 α2 
 α1-1 - T12   p1   E1Ext 

.    Ext 
-1  
- T


 21 α 2   p 2   E 2 
 α1-1 (0)  
- T12   p1   0 

.    
-1
 -T
α 2 (0)     p 2   0 
21

1

 p1   T12 .p 2  E1Ext 

.   
Ext 
 p 2   T21.p1  E 2 
mω 2
 2
e
Polarisation Propagator
• Model of the Density Response Function
α
-1
1


(0)   α -12 (0)    T12 .T21  0
2
2
 mω

mωo2 mω
2
 2     T12
 2   T12
2
e
e
 e

e 2 T12
2
2
 i  ωo 
m
 α1-1 (0)  
- T12   p1   E1Ext 

.    Ext 
-1
 -T
α 2 (0)     p 2   E 2 
21

inverse of A - I may be written as
2
o
A -  I 
1
p i p iT
 2
2
i i  ω
 p1 
p i p iT  E1Ext 

    2
.
2  Ext 
 p 2  i i  ω  E2 
p i p iT
i  2  ω2 is a dipole - dipole correlatio n function
i
Polarisation Propagator
• Model of the Density Response Function
1
 α1-1 - T12 
1
1
1 1






rewrite
as
A
B

A
1
BA
-1 
- T
 21 α 2 
- T12 
 α1-1 0 
 α1 0 
 0
 0
1
1




 A  
A
B  
BA  
-1 

0 
 - T21
 0 α2 
 0 α2 
 - T21.α1
1 - BA 
1 1
- T12 .α 2 

0 
 1  BA 1  BA 1BA 1  ...
A - B 1  A 1  A 1BA 1  A 1BA 1BA 1  ...
 α1
 
0
1
Eext
0 
0

α 2   - α 2 .T21.α1
2
- α1 .T12 .α 2   α1 .T12 .α 2 .T21.α1

 
0
0
 
1
Eext
-T12
2
1
Eext

0
 .
α 2 .T21.α1 .T12 .α 2 
-T12
-T21
2
Polarisation Propagator
• Model of the Density Response Function
Si
 1

m 2 

  (0) 2 I + Tq . p q ( ) = 0
2e 


Herrendörfer and Patterson J Phys Chem Solids 58, 207 (1997)
Si
Polarisation Propagator
• Model of the Density Response Function
p(q,  )   (q,  ).EExt (q,  )
2
x xT *
2e
nq nq
2
 (q,  )  
=
e
Xq
2
2
n m ( nq - i )  
p pT *
2
nq nq
2
2
n m ( nq - i )  

P(q,  )   o  (q,  ).EExt (q,  )
p pT *
2
nq nq
 (q,  )  
2
2
n  o mV ( nq - i )  
Expt - - - Model ____
Polarisation Propagator
• Model of the Density Response Function
• Reflectance Anisotropy of stepped silicon surfaces
 i r r i
R
  xs  b -  xs  b 
 4k z d Im
 xs   s   b
2

R



b


b  bulk s  surface d  selvedge width k z  2/
Hogan and Patterson, Phys. Rev. B 57, 14843 (1998)
Polarisation Propagator
• Model of the Density Response Function
• Expansion of the polarisability Po in s (occ) and p (unocc) ETB Bloch orbitals
occ unocc *
P o (r, r ' ,  ) = 2    (r ) (r )* (r ' ) (r ' ) x
ik
jk '
jk '
ik
ik jk '


1
1

x

  - E  E + i   + E - E - i 
jk '
ik
jk ' ik


 2 
1k (r) =   
 
3/4
exp[- (r - R  ) 2 ] exp[ik. R  ]
s Bloch state
5/4
2



2k (r ) =  2 1/2   (x - R
) exp[-  (r - R ) 2 ] exp[i k.R  ]
, x

 

px Bloch state
Nicastro, Galamic-Mulaomerovic and Patterson,
J. Phys. Condens. Matt. 13, 1215 (2001)
Polarisation Propagator
• Model of the Density Response Function
• Polarisability and Coulomb expansion coefficients
P o (r, r ' ,  ) =  P o ij (q,  ) (r )* (r ' )
iq
jq
i jq
P oij (q,  ) =  d3r d3r ' i* (r ) P o (r, r ' ,  )j (r' )
V
 0 0
 5/2

P o ( ) = a ( )
3/2
(2 )
0 I 
R1q
e2
=  v (q) (r )* (r ' )
i
j
r - r ' i j q ij
R2q
v (q) =   dr dr '
ij

i* (r ) exp[i q.R  ]j (r ' )
R  + r' - r

r ' - r .R 
1
1


 ..
R  + r' - r
R
R
Tq
Polarisation Propagator
• Model of the Density Response Function
    3/2
   + R
3
/
2
2
1q

2 
e   2 
v(q) =
5/2 4 

1/ 2 *
o

R 2q


W =  1v = v + v P v
e2  0

 R 2q 


3/2
1  
  + Tq 
3  2 

1/ 2
-1


R
X
 1 0
q
2
q

.
  (q, ) = 
Tq Xq 
 0 I  4o  0
-1
W(q, ) =
2 3 / 2
 5/2
1/2

1/ 2

R

R
e 
1q
2q

4   1 / 2R*
T
o 
2
q
q

2

P = 1 - P o v  P o = P o - v
-1

1
p pT *
2
nq nq
2
2
n m ( nq - i )  
e 2 Xq  
 R X R *

1/ 2

2

R
X
T

 e
2q q 2q
2q q q 



1
/
2
*

 4o   T X R
T X T
q
q
2
q
q
q
q



The GW Approximation
• Self Energy
• Expressed generally as S = G W G M 211
4
1
3
2
4
S(1,2) = G(1,3) W(1,4) G(3,4,2)
3
2
• G dressed Green’s function
• W is the screened interaction W = v + v P v
• G is the vertex part (vertex correction)
2
4
=
+
3
1
+
+…
The GW Approximation
• Self Energy: GoWo COHSEX approximation
• Wo= -1RPA v
Wo is the screened interaction in RPA
• G = (2 - 3) (2 - 4)
• S(1,2) = Go(1,3) Wo(1,4)(2 - 3)(2 - 4) = Go(1,2) Wo(1,2)
2


d ' -i '
dq
G o (k - q,  -  ' ) Wo (q,  ' )
e
S(k ,  ) = i 
3 
 2   2
e-i '  integral converges in lower half plane
G o (k - q,  -  ' )  
n
W(q, ’)
Go(k - q,  - ’)
ψ n,k -q (r )ψ*n,k -q (r ' )
 -  ' -  n,k -q  i sgn(  n,k -q -  )
1
model for screened interactio n, W(q,  ), in translatio nally invariant system
2



p
-1
 v( q)

Wo (q,  )   (q,  )v( q)  1 
2
2
  - i    
q


e2
 q plasmon energy
 
 o mV
2
p
Hybertsen and Louie Phys. Rev. B 34, 5390 (1986) (861 citations)
The GW Approximation
• Self Energy: GoWo COHSEX approximation
• Poles of Go: Screened Exchange (SEX) contribution to S
• Poles of Wo: Coulomb hole (COH) contribution to S
2
2






p
p
-1



 v( q)
Wo (q,  ' )   (q,  ' )v( q)  1 
v( q)  1 
2
2
  '  i  ' - i  
  - i    ' 
q
q
q




poles of Wo at  '    q - i   '   q - i in lower half plane
 p2
residues of poles of Wo :
ψ n,k -q (r )ψ*n,k -q (r ' ) v( q)
2 q  -  q -  n,k -q  i 
G o (k - q,  -  ' )  
n
ψ n,k -q (r )ψ*n,k -q (r ' )
 -  ' -  n,k -q  i sgn(  n,k -q -  )
poles of G o at  '   -  n,k -q  i
 n,k -q  
 '   -  n,k -q  i
 n,k -q  


 p2
*

ψ
residues of poles of G o : 1  2
(
r
)ψ
n,k -q (r ' ) v( q )
2  n,k -q
    - 
q
n,k -q  i 


n, k - q limited to occupied states for this term!
The GW Approximation
• Poles of time-ordered 1 x and Go x in the complex energy plane
Im()
Advanced

xxx
xxx
xx xx xxx x x xxx
xxx xx xx xxx
x x xxx
Retarded
Re()
x
xx xx xxx x x xxx
xx xx xx x x xxx
The GW Approximation
• Self Energy: GoWo COHSEX approximation
• Poles of Go: Screened Exchange (SEX) contribution to S
• Poles of Wo: Coulomb hole (COH) contribution to S
2



p
*

ψ
residues of poles of G o : 1  2
(
r
)ψ
n,
k
q
n,k -q (r ' ) v( q )
2 
    - 
q
n,k -q  i 


 p2
residues of poles of Wo :
ψ n,k -q (r)ψ*n,k -q (r' ) v( q)
2 q  -  q -  n,k -q  i 
Sr, r ' ,    
ψ nk S
COH
dq
iq.( r -r ')


S
q
,

e
(2 ) 3
r, r' ,   ψ n'k
 ψ nk

e



ψ nk SSEX r, r ' ,   ψ n'k  ψ nk e
i q  G .r
i q  G .r
ψ n,k -q ψ n,k -q

e



ψ n,k -q ψ n,k -q e
Compare Eq 34 in Hybertsen and Louie
i q  G ' .r '
i q  G ' .r '
ψ n',k
ψ n',k
 p2
2 q  -  q -  n,k -q 
2



p
1 

2
2
    - 

q
n,k -q  

The GW Approximation
• Model of the Density Response Function
• Self Energy: GoWo approximation

d ' -i '
S(r, r' ,  ) = i 
e
G o (r, r' ,  -  ' ) Wo (r, r' ,  ' )
 2
1


2   - c + i
G o (k ,  ) =

0


Wo (q,  ) =
2 3 / 2
 5/2
2



1

I
 - v - i 
0

1/ 2

R

R
e 
1q
2q


4o   1 / 2 R*
T
2
q
q

2
1
 R X R*

1/ 2


R
X
T
2
 e 
2q q 2q
2q q q 



1
/
2
*
4


o 

T X R
T X T
q q 2q
q q q



Nicastro, Galamic-Mulaomerovic and Patterson,
J. Phys. Condens. Matt. 13, 1215 (2001)
The GW Approximation
• Model of the Density Response Function
• Self Energy: GoWo approximation
2
1
0

9
2  0

2
e
1/2
on - site


=
HF
5 4o  0
3/2
3
32  

 1
1/ 2 * 
Tr
T

R 2q 

q
2 e
inter
site

Σq
=

5 4o

1
/
2
HF

3
R 2q
R1q I 


9
Σ(q,  ) =
1
2

nq 
2
2
2


1
2  e 
x
5  4o 

-   sgn(  -  )  3 

i
f
i
nq 
9


  R X R*

1/ 2 *
 1
1/ 2
* T


R
X
T
Tr
T
X
T

T
X
R
q
q
q


2
q
2
q
2
q

q q q
q q 2q
1 






  1/ 2

1
*
T
4

T
X
R
T
X
T
  1/ 2 R* X T

*
q q 2q
q q q 
 

R
X
R
I

q
q
q
2
q
2
q
2
q
i=1 
 
i= 2-4 

