022 - pr 08 - one particle Green function aka density matrix: For N non-interacting spin one-half fermions, an ideal Fermi gas, the ground state is obtained from the vacuum state according to, pF 2 † D.C . ; G0 [ , p p ap , ] 0 (1.1) 0 [ , p p ap, ]; F F F 2m
All the states below the Fermi energy are occupied in accordance with Pauli’s exclusion principle, and all states above are empty for the case of the ground state. The Fermi sphere: Pictorially, the ground state is that of a filled sphere in momentum space, the Fermi sea, with the Fermi surface separating occupied and unoccupied states.
Show that the one- particle particle Green’s function or density matrix becomes , , 3n sin k F x x kF x x cos k F x x G G (x, x) G (x x) G0 | † (x) (x ) | G0 3 2 k x x F
3n sinc k F x x cos k F x x
k F x x
2
2
; n
N V
3 F 2
k
3
(1.2)
;
Here, n is the density of the fermions, and k F is the Fermi-wavevector, and in the considered three dimensions 3 2 you have k F 3 n . The field operators appear as, ( x)
3
d p
(2
)
3/ 2
e
ipx /
3
ap D.C .
d p
(2
)
3/ 2
e
i p x /
ap (x) ; †
†
In which we have the commutation-relations, {ap , ap†} p | p (p p); {ap , ap} 0 0 † {ap†, ap†};
(1.3)
(1.4)
And the action on the vacuum ket, for fermions, ap† 0 0 1 0,0.. ,0...p,.. ,...0,0,. ,0,.... 1 p p ;
(1.5)
ap 0,0.. ,0...p,... ,...0 0,0,.. ,0,... 1 0,0.. ,0...0,.. ,...0,0,.. ,0,... 0 ; Then, G 0
(2 1 )
3
G
Then we,
1 3 (2 )
p1 p F
ap1
e
(2 )
i ( p x p x ) /
p1 p F p1 p F
e
1
i p( x x )/
3/ 2
e ip x / ap† d 3 p
1 3/ 2
(2 ) 0 ap ap† ap ap† 0 d 3 pd 3 p G pp pp pp 1
1
0 ap ap† ap ap† 0 d 3 p
1
1 3 (2 )
e
ip ( x x ) /
eipx / ap d 3 p
d 3 p;
1
p1 pF
ap†1 0
(1.6)
2 1 p F
G
e
1 (2 )3
i p x x cos /
pF
p dp d (cos )d 2
0 1 0
2
2 (2 )3
e
i p x x /
0 x F
3
(2 )3 x x
2 x dx sinc x
2
3
0
x p x x / k x x
3 k F 3 sin xF xF cos x F 2 3 2
e i p x x / 2 p dp i p x x /
( k F x x ) 3
3n sinc x F cos x F
(1.7)
xF 2
2
Interpretation of the result: The considered amplitude specifies the overlap between the state (x) G0 where a particle with spin σ at position x has been removed from the ground state and the state G0 † (x) where a
particle with spin σ at position x has been removed from the ground state. G
x x' n
0.5
0.4
0.3
0.2
0.1
10
5
5
x k F x x'
10
(1.8)
Equivalently: it specifies the amplitude for transition to the ground state of the state where a particle with spin σ at position x has been removed from the ground state and subsequently a particle with spin σ has been added at position x .
At small spatial separation, you have, G G0 | † (x) ( x) | G0
3n sinc x F cos xF 2
2 F
x
3n
2
1 3
301 x F 2 O4 ( xF )
1 2
n 1 101 xF 2 ;
(1.9)
Interpretation of leading order term: At x x , you have x F kF x x k F 0 0 identically, the Green function counts the density of fermions per spin at the position in question. Show that the pair correlation function is related to the one-particle density matrix according to, G G (x, x) G0 | † † | G0 G0 | † ( x) † ( x) ( x) ( x) | G0
( 12 n) 2 G (x x) 2 ; Starting with (1.3) interpreted as (1.5), we have,
(1.10)
G
1 (2 )6
(2 1 )
1
G
1
6
6
d
0
a d p1
p1 p F
d 3
3
x /
pe ip
ap†
d
3
pe ip x / ap†
)x /
pd 3 pd 3 pd 3 p ei(p p )x / ei (pp 3
p F
0
pF
0
p1 p F ; p1 pF
† p
† † p p p p1
0 ap1 a a
a
d
3
p p 1
pd p
2 6 (4 )
3
a a
peip x / ap
d
3
peip x / ap
a p1 pF
† p1
0 ap1 ap† ap† ap ap ap†1 0
F ; p1 pF
0 pp1 G pp1
d 3 pd 3 p 0 ap ap† ap† ap ap ap† 0
2 6 (4 )
1
pF
0
p2 dp 0 ap† ap 0
Interpret the result and note in particular the anti-bunching of non-interacting fermions: the avoidance of identical fermions to be at the same position in space, a repulsion solely due to the exchange symmetry, the exclusion principle at work in real space.
0
(1.11)
