The galitskii-Migdal-koltun sum

The galitskii-Migdal-koltun sum
Arianna Carbone
Monday Morning Meeting,
Dept. of Physics, Univ. of Surrey
26 th March 2012

Outline
2

Outline
What are we talking about?
2

Outline
What are we talking about?
Which tools do we need?
2

Outline
What are we talking about?
Which tools do we need?
the Green functions
2

Outline
What are we talking about?
Which tools do we need?
the Green functions
The GMK sum rule
2

Outline
What are we talking about?
Which tools do we need?
the Green functions
The GMK sum rule
2- and 3-body interactions
2

What are we talking about?
Objective: study the properties of many-body systems..............
≥ than 3 bodies is already many!!!
3

What are we talking about?
Objective: study the properties of many-body systems..............
≥ than 3 bodies is already many!!!
3

What are we talking about?
Calculate the energy of a nuclear many-body system:
Finite systems: nuclei
Infinite systems: neutron stars
September 24, 1997
HUBBLE SEES A NEUTRON STAR ALONE IN SPACE
4

Which tools do we need?
Many-body theory: formalism of Green functions
5

Which tools do we need?
Many-body theory: formalism of Green functions
It’s a very useful mathematical tool to describe the propagation
of an interacting particle with the rest of the system
5

Which tools do we need?
Many-body theory: formalism of Green functions
It’s a very useful mathematical tool to describe the propagation
of an interacting particle with the rest of the system
H|α, t = i ∂ ∂t
|α, t |α,
t 0 ; t = e − i H(t−t 0) |α, t 0
5

Which tools do we need?
Many-body theory: formalism of Green functions
It’s a very useful mathematical tool to describe the propagation
of an interacting particle with the rest of the system
H|α, t = i ∂ ∂t
|α, t |α,
t 0 ; t = e − i H(t−t 0) |α, t 0
ψ(r,t)=r|α, t 0 ; t = r|e − i H(t−t0) |α, t 0
= dr r|e − i H(t−t0) |r r |α, t 0
= i dr G(r, r ; t − t 0 )ψ(r ,t 0 )
5

Which tools do we need?
Many-body theory: formalism of Green functions
It’s a very useful mathematical tool to describe the propagation
of an interacting particle with the rest of the system
H|α, t = i ∂ ∂t
|α, t |α,
t 0 ; t = e − i H(t−t 0) |α, t 0
ψ(r,t)=r|α, t 0 ; t = r|e − i H(t−t0) |α, t 0
= dr r|e − i H(t−t0) |r r |α, t 0
= i dr G(r, r ; t − t 0 )ψ(r ,t 0 )
5
named after George Green (~1830)

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t 0) |r = 0|a r e − i H(t−t 0) a † r |0
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t0) |r = 0|a r e − i H(t−t0) a † r |0
To a many-body system
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t0) |r = 0|a r e − i H(t−t0) a † r |0
To a many-body system
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t0) |r = 0|a r e − i H(t−t0) a † r |0
To a many-body system
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t 0) |r = 0|a r e − i H(t−t 0) a † r |0
To a many-body system
G(α, α ; E) = m
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
Energy formulation for the propagator:
Ψ N 0 |a α |Ψ N+1
m Ψ N+1
m
|a † α |Ψ N 0
E − (E N+1 m − E N 0 )+iη + n
Ψ N 0 |a † α |Ψ N−1
n
E − (E 0 − E N−1 n
Ψ N−1
n |a α |Ψ N 0
) − iη
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t 0) |r = 0|a r e − i H(t−t 0) a † r |0
To a many-body system
G(α, α ; E) = m
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
Energy formulation for the propagator:
Ψ N 0 |a α |Ψ N+1
m Ψ N+1
m
|a † α |Ψ N 0
E − (E N+1 m − E N 0 )+iη + n
Ψ N 0 |a † α |Ψ N−1
n
E − (E 0 − E N−1 n
Ψ N−1
n |a α |Ψ N 0
) − iη
1
E ± iη = P 1 E ∓ iπδ(E)
Plemelj Formula
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t 0) |r = 0|a r e − i H(t−t 0) a † r |0
To a many-body system
G(α, α ; E) = m
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
Energy formulation for the propagator:
Ψ N 0 |a α |Ψ N+1
m Ψ N+1
m
|a † α |Ψ N 0
E − (E N+1 m − E N 0 )+iη + n
Ψ N 0 |a † α |Ψ N−1
n
E − (E 0 − E N−1 n
Ψ N−1
n |a α |Ψ N 0
) − iη
1
E ± iη = P 1 E ∓ iπδ(E)
Plemelj Formula
1
π ImG(α, α; E) = n
|Ψ N−1
n |a α |Ψ N 0 | 2 δ(E − (E N 0 − E N−1
n ))
6

The Green Function:
From a one-body system......
i G(r, r ; t − t 0 )=r|e − i H(t−t 0) |r = 0|a r e − i H(t−t 0) a † r |0
To a many-body system
G(α, α ; E) = m
i G(αt α ,α t α)=Ψ N 0 |T [a α (t α )a † α (t α )]|Ψ N 0
Energy formulation for the propagator:
Ψ N 0 |a α |Ψ N+1
m Ψ N+1
m
|a † α |Ψ N 0
E − (E N+1 m − E N 0 )+iη + n
Ψ N 0 |a † α |Ψ N−1
n
E − (E 0 − E N−1 n
Ψ N−1
n |a α |Ψ N 0
) − iη
1
E ± iη = P 1 E ∓ iπδ(E)
Plemelj Formula
1
π ImG(α, α; E) = n
|Ψ N−1
n |a α |Ψ N 0 | 2 δ(E − (E N 0 − E N−1
n ))
6
Hole spectral function

What exactly is the hole spectral function?
S h (α, E) = 1 π ImG(α, α; E) = n
|Ψ N−1
n |a α |Ψ N 0 | 2 δ(E − (E N 0 − E N−1
n ))
7

the gmK sum rule
1
π ImG(α, α; E)= n
|Ψ N−1
n |a α |Ψ N 0 | 2 δ(E − (E N 0 − E N−1
n ))
8

the gmK sum rule
E
N
0 −EN−1 0
−∞
dEE
1
ImG(α, α; E) = Ψ N 0 |a † α[a α , Ĥ]|ΨN 0
π
8

the gmK sum rule
α(
E
N
0 −EN−1 0
1
dEE
= Ψ N 0 |a † )
ImG(α, α; E)
α[a α , Ĥ]|ΨN 0
−∞
π
8

the gmK sum rule
α(
E
N
0 −EN−1 0
1
dEE
= Ψ N 0 |a † )
ImG(α, α; E)
α[a α , Ĥ]|ΨN 0
−∞
π
Total energy of the many-body ground-state!!!
E N 0
= Ψ N 0 |Ĥ|ΨN 0
8

the gmK sum rule
α(
E
N
0 −EN−1 0
1
dEE
= Ψ N 0 |a † )
ImG(α, α; E)
α[a α , Ĥ]|ΨN 0
−∞
π
The GMK sum rule
Total energy of the many-body ground-state!!!
E N 0
= Ψ N 0 |Ĥ|ΨN 0
8

the gmK sum rule
α(
E
N
0 −EN−1 0
1
dEE
= Ψ N 0 |a † )
α[a α , Ĥ]|ΨN 0
−∞
π ImG(α, α; E) First developed by Galitksii-Migdal (1958),
The GMK sum rule
later applied to finite systems by Koltun (‘70s)
Total energy of the many-body ground-state!!!
E N 0
= Ψ N 0 |Ĥ|ΨN 0
8

the gmK sum rule
Hamiltonian describing the system:
9

the gmK sum rule
Hamiltonian describing the system:
9

the gmK sum rule
Hamiltonian describing the system:
9

the gmK sum rule
Hamiltonian describing the system:
Ψ N 0 |a † α[a α , Ĥ]|ΨN 0 = Ψ N 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
9

the gmK sum rule
Hamiltonian describing the system:
Ψ N 0 |a † α[a α , Ĥ]|ΨN 0 = Ψ N 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E N 0
= Ψ N 0 |Ĥ|ΨN 0 = Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
9

the gmK sum rule
Hamiltonian describing the system:
Ψ N 0 |a † α[a α , Ĥ]|ΨN 0 = Ψ N 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E N 0
= Ψ N 0 |Ĥ|ΨN 0 = Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
9

2-body interactions
Only 2-body forces:
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0
10

2-body interactions
Only 2-body forces:
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0
1
2 [ α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ] =Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0
10

2-body interactions
Only 2-body forces:
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0
1
2 [ α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ] =Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0
Write this in terms of the 1-body
Green function
10

2-body interactions
Only 2-body forces:
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0
1
2 [ α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ] =Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0
Write this in terms of the 1-body
Green function
The 1-body propagator provides the expectation value of any one-body operator:
Ψ N 0 |Ô|ΨN 0 = αα α|O|α Ψ N 0 |a † αa α |Ψ N 0 = αα α|O|α n αα
10

2-body interactions
Only 2-body forces:
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0
1
2 [ α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ] =Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0
Write this in terms of the 1-body
Green function
The 1-body propagator provides the expectation value of any one-body operator:
Ψ N 0 |Ô|ΨN 0 = αα α|O|α Ψ N 0 |a † αa α |Ψ N 0 = αα α|O|α n αα
One-body
density matrix
10

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
11

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
= n
Ψ N 0 |a † α |Ψ N−1
n
Ψ N−1
n |a α |Ψ N 0 = Ψ N 0 |a † α
a α |Ψ N 0 = n αα
11

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
= n
Ψ N 0 |a † α |Ψ N−1
n
Ψ N−1
n |a α |Ψ N 0 = Ψ N 0 |a † α
a α |Ψ N 0 = n αα
Problem solved!
11

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
= n
Ψ N 0 |a † α |Ψ N−1
n
Ψ N−1
n |a α |Ψ N 0 = Ψ N 0 |a † α
a α |Ψ N 0 = n αα
Problem solved!
E N 0
= Ψ N 0 |Ĥ|ΨN 0 = 1 2 [ α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ]
11

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
= n
Ψ N 0 |a † α |Ψ N−1
n
Ψ N−1
n |a α |Ψ N 0 = Ψ N 0 |a † α
a α |Ψ N 0 = n αα
Problem solved!
E N 0
= Ψ N 0 |Ĥ|ΨN 0 = 1 2 [ α
= 1
2π
α
E
N
0 −EN−1 0
−∞
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ]
dE (Eδ α,α
+ α|T |α )Im G(α, α ; E)
11

2-body interactions
E
N
0 −EN−1 0
−∞
dE 1 π ImG(α, α ; E) =
E
N
0 −EN−1 0
−∞
dE n
Ψ N 0 |a † α
|Ψ N−1
n Ψ N−1
n |a α |Ψ N 0 δ(E − (E0 N − En N−1 )
= n
Ψ N 0 |a † α |Ψ N−1
n
Ψ N−1
n |a α |Ψ N 0 = Ψ N 0 |a † α
a α |Ψ N 0 = n αα
Problem solved!
E N 0
= Ψ N 0 |Ĥ|ΨN 0 = 1 2 [ α
= 1
2π
α
E
N
0 −EN−1 0
−∞
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E)+ΨN 0 | ˆT |Ψ N 0 ]
dE (Eδ α,α
+ α|T |α )Im G(α, α ; E)
The Koltun sum rule
11

2- and 3-body interactions
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
12

2- and 3-body interactions
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E
N
0 −EN−1 0
−∞
1
dEE
π Im G(α, α; E)+2 3 ΨN 0 | ˆT |Ψ N 0 + 1
3 ΨN 0 | ˆV |Ψ N 0
= Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
12

2- and 3-body interactions
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E
N
0 −EN−1 0
−∞
1
dEE
π Im G(α, α; E)+2 3 ΨN 0 | ˆT |Ψ N 0 + 1
3 ΨN 0 | ˆV |Ψ N 0
= Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
12

2- and 3-body interactions
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E
N
0 −EN−1 0
−∞
1
dEE
π Im G(α, α; E)+2 3 ΨN 0 | ˆT |Ψ N 0 + 1
3 ΨN 0 | ˆV |Ψ N 0
= Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
This quantity depends on the 2-body Green
function, G II , which is complicated to calculate....
12

2- and 3-body interactions
α
E
N
0 −EN−1 0
−∞
dEE 1 π Im G(α, α; E) =ΨN 0 | ˆT |Ψ N 0 +2Ψ N 0 | ˆV |Ψ N 0 +3Ψ N 0 |Ŵ |ΨN 0
α
E
N
0 −EN−1 0
−∞
1
dEE
π Im G(α, α; E)+2 3 ΨN 0 | ˆT |Ψ N 0 + 1
3 ΨN 0 | ˆV |Ψ N 0
= Ψ N 0 | ˆT |Ψ N 0 + Ψ N 0 | ˆV |Ψ N 0 + Ψ N 0 |Ŵ |ΨN 0
This quantity depends on the 2-body Green
function, G II , which is complicated to calculate....
Open problem: how should we transform this formula in order to
continue benefiting of the power of the GMK sum rule?
12

Thank you for your attention!
13