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Many-body Green’s Functions. Propagating electron or hole interacts with other e - /h + Interactions modify ( renormalize ) electron or hole energies Interactions produce finite lifetimes for electrons/holes ( quasi-particles ) - PowerPoint PPT Presentation
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Many-body Green’s Functions
• Propagating electron or hole interacts with other e-/h+
• Interactions modify (renormalize) electron or hole energies• Interactions produce finite lifetimes for electrons/holes (quasi-particles)• Spectral function consists of quasi-particle peaks plus ‘background’• Quasi-particles well defined close to Fermi energy
• MBGF defined by
o
oHHo )t','(ψt),(ψ)t','t,,G(
state, ground Heisenberg exact
over averaged operator field of function ncorrelatio i.e.
rrrr Ti
Many-body Green’s Functions
• Space-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 • Non-interacting Green’s function Go represented by a single line• Interacting Green’s Function G represented by a double or thick single line
time
Add particle Remove particle
t > t’t’
time
Remove particle Add particle
t’ > tt
x
y
y
)t'(t)t',(ψt),(ψ oHHo yx
t)(t't),(ψ)t',(ψ oHHo xy
x
Go(x,y)x,ty,t’
G(x,y)x,ty,t’
Many-body Green’s Functions
• Lehmann Representation (F 72 M 372) physical significance of G
oo
onn
n
oSnnSo
-o-n
oSnnSo
oHnnHooHHo
oHHooHHo
nn
n
o
oHHo
tiEtHitiE-tHi-
)t'tEi(E-
t'Hi-t'HitHi-tHi
ee ee
)'(ψ)(ψ)(
e
)e'(ψe)e(ψe
)t','(ψt),(ψ)t','(ψt),(ψ
t)(t't),(ψ)t','(ψ-)t'(t)t','(ψt),(ψ)t','t,,G(
)t','(ψt),(ψ)t','t,,G(
rr
rr
rrrr
rrrrrr
1
rrrr
i
Ti
formalism number occupation in operator unit
number particleany , state, Heisenberg exact
state, ground Heisenberg exact
Many-body Green’s Functions
• Lehmann Representation (physical significance of G)
oneby in number particle reduces ooS
oSoSSS
on
oSnnSo
on
oSnnSo
-o-noSnnSo
-o-noSnnSo
oHHo
ψ
ψ)1N(ψn )(ψ)(ψ dn
δ)EE(ε
ψψ
δ)EE(ε
ψψ
e)t','t,,)G(t'-d(t),',G(
t)(t')(
e)(ψ)'(ψ
-)t'(t)(
e)'(ψ)(ψ)t','t,,G(
)t','(ψt),(ψ)t','t,,G(
)t'(t
)t'tEi(E
)t'tEi(E-
rrr
rrrr
rr
rrrr
rrrr
ii
ii
i
Ti
i
Many-body Green’s Functions
• Lehmann Representation (physical significance of G)
)1N(E)1N(E)N(E)1N(E
)N(E)1N(E)1N(E)1N(E)N(E)1N(E
ψψψ
)1N(E)1N(E)N(E)1N(E
)N(E)1N(E)1N(E)1N(E)N(E)1N(E
ψψψ
onon
ooonon
2
nSooSnnSo
onon
ooonon
2
oSnoSnnSo
states particle 1N and N connects
states particle 1N and N connects
Many-body Green’s Functions
• Lehmann Representation (physical significance of G)• Poles occur at exact N+1 and N-1 particle energies• Ionisation potentials and electron affinities of the N particle system • Plus excitation energies of N+1 and N-1 particle systems
• Connection to single-particle Green’s function
Fbelow states for as states unoccupied to limited Sum
unoccupied
state ground g)interactin-(non particle-single the is
00c
n 0cc0 )t'(t)e'(ψ)(ψ
)t'(t0)(t'c(t)c0)'(ψ)(ψ
)t'(t0)t','(ψt),(ψ0)t','t,,(G
0
n
mnnmn*
n
unocc
nn
nm*n
nm,m
HHo
)t'-(t-
i
i
rr
rr
rrrr
Many-body Green’s Functions
• Gell-Mann and Low Theorem (F 61, 83)• Expectation value of Heisenberg operator over exact ground state
expressed in terms of evolution operators and the operator in question in interaction picture and ground state of non-interacting system
oIo
oIIIo
oo
oHo
)-,(U
)(t,-U(t)Ot),(U(t)O
o
oo
oHHo
|
)t','(ψt),(ψ)t','t,,G(
rrrr
Ti
Function sGreen'Body -Many
57 F
)(t'IHdt't
0
t
0
t
0
nI2I1I
t
0
n21
n
I e)(tH)...(tH)(tHdt...dtdtn!
(t,0)U
i
TTi
Many-body Green’s Functions
• Perturbative Expansion of Green’s Function (F 83)
• Expansion of the numerator and denominator carried out separately• Each is evaluated using Wick’s Theorem• Denominator is a factor of the numerator• Only certain classes of (connected) contractions of the numerator survive• Overall sign of contraction determined by number of neighbour permutations• n = 0 term is just Go(x,y)• x, y are compound space and time coordinates i.e. x ≡ (x, y, z, tx)
o
- -
nI2I1I
-
on210n
n
oIo
o
- -
nI2I1I
-
on210n
n
oIo
)(tH)...(tH)(tHdt...dtdtn!
,U
)(ψ)(ψ)(tH)...(tH)(tHdt...dtdtn!,U
1),G(
Ti
Ti
i yxyx
Many-body Green’s Functions
• Fetter and Walecka notation for field operators (F 88)
bb- t t ),(G
t t 0)(ψ)(ψ
t t 0
aa t t ),(G)(ψ)(ψ
ba)(ψ)(ψ)(ψ
ba)(ψ)(ψ)(ψ
yxo
yx)()(
yx
yxo)()(
(-))(
(-))(
yx
yx
yxyx
xxx
xxx
i
i
0ψψ 0ψψ 0bbabbaaa
bbabbaaa
ba ba
ψψψψψψ (-))()()(
similarly
Many-body Green’s Functions
• Nonzero contractions in numerator of MBGF
(-1)3 (i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y)
(-1)4(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y)
(-1)5(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y)
(-1)4(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y)
(-1)6(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y)
(-1)7(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)(6) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(5) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(4) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(3) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(2) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
(1) )(ψ)(ψ)(ψ)'(ψ)'(ψ)(ψ
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
yxrrrr
Many-body Green’s Functions
• Nonzero contractions
-(i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y) (1)
+(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y) (2)
-(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y) (3)
+(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y) (4)
+(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y) (5)
-(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y) (6)
y
x
r r’
y
x
r r’
x
y
r r’
y
r r’
x
y
r’ r
xx
y
r’ r
(1) (2)
(3) (4)
(5) (6)
• Nonzero contractions in denominator of MBGF• Disconnected diagrams are common factor in numerator and denominator
Many-body Green’s Functions
(8) )(ψ)'(ψ)'(ψ)(ψ
(7) )(ψ)'(ψ)'(ψ)(ψ
rrrr
rrrr
(-1)3(i)2v(r,r’)Go(r’,r) Go(r,r’)
(-1)4(i)2v(r,r’)Go(r,r) Go(r’,r’)
r r’(7)
r r’(8)
Denominator = 1 + + + …
Numerator = [ 1 + + + … ] x [ + + + … ]
• Expansion in connected diagrams
• Some diagrams differ in interchange of dummy variables• These appear m! ways so m! term cancels• Terms with simple closed loop contain time ordered product with equal times• These arise from contraction of Hamiltonian where adjoint operator is on left• Terms interpreted as
Many-body Green’s Functions
0m connected
om111om1 ])(ψ)(ψ)(tH ... )(tH[dt...dtm!
)(),G( yxyx T
ii
iG(x, y) = + + + …
density charge ginteractin-non )(ρ)(ψ)(ψ
)t',(ψt),(ψ),(G
ooo
oolim
'o
xxx
xxxx
Ti tt
• Rules for generating Feynman diagrams in real space and time (F 97)
• (a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops)
• (b) Label each vertex with a space-time point x = (r,t)
• (c) Each line represents a Green’s function, Go(x,y), running from y to x
• (d) Each wavy line represents an unretarded Coulomb interaction• (e) Integrate internal variables over all space and time• (f) Overall sign determined as (-1)F where F is the number of Fermion loops• (g) Assign a factor (i)m to each mth order term• (h) Green’s functions with equal time arguments should be interpreted as
G(r,r’,t,t+) where t+ is infinitesimally ahead of t
• Exercise: Find the 10 second order diagrams using these rules
Many-body Green’s Functions
• Feynman diagrams in reciprocal space
• For periodic systems it is convenient to work in momentum space• Choose a translationally invariant system (homogeneous electron gas)• Green’s function depends on x-y, not x,y • G(x,y) and the Coulomb potential, V, are written as Fourier transforms• 4-momentum is conserved at vertices
Many-body Green’s Functions
t-.. ddd
)e',v()'-d()v(
)eG(2
d),G(
34
4
4
)'.(
).(
xkxkkk
rrrrq
kk
yx
rrq
yxk
i-
i
Fourier Transforms
32143214 2eeed
...qqqx
xqxqxq
-i-ii
4-momentum Conservation
q1
q2
q3
• Rules for generating Feynman diagrams in reciprocal space
• (a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops)
• (b) Assign a direction to each interaction• (c) Assign a directed 4-momentum to each line• (d) Conserve 4-momentum at each vertex• (e) Each interaction corresponds to a factor v(q)• (f) Integrate over the m internal 4-momenta• (g) Affix a factor (i)m/(2)4m(-1)F
• (h) A closed loop or a line that is linked by a single interaction is assigned a
factor ei Go(k,)
Many-body Green’s Functions
)(ψ)(ψ1
)(ψ)(ψddH
)(ψ)(ψ1
)(ψd H, ψψt
)(ψ)(h)(ψdH
)(ψ)(h H, ψψt
1H2H21
2H1H21H
H2H2
2H2HHH
1H11H1H
HHHH
2
1rr
rrrrrr
rrrr
rr
rrrr
rr
for
for
i
i
Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators (from Lecture 2)
oo
oHHo )t','(ψt),(ψ)t','t,,G(
rrrr
Ti
Equation of Motion for the Green’s Function
• Equation of Motion for Field Operators
t),(ψt),(ψ1
t),(ψd t),(ψt),(ht
t),(ψt),(ψ1
t),(ψd t),(ψt),(h
tHe )(ψ)(ψ1
)(ψd tHe tHe )(ψ)(h tHe
tHeH,ψtHet),(Ht),,(ψt),(ψt
H2H2
2H2H
H2H2
2H2H
22
22
SSHHH
rrrr
rrrr
rrrr
rrrr
rrrr
rrrr
rrr
i
iiii
iii
Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument
)t'-(t)-(|)t'-(t)t',(ψt),,(ψ
)t'-(t)t',(ψt),,(ψ
(t'-t)t
t),(ψ)t',(ψ-)t'-(t)t',(ψ
t
t),(ψ
)t'-(tt),(ψ)t',(ψ--)t'-(t)t',(ψt),(ψ
(t'-t)t
t),(ψ)t',(ψ-)t'-(t)t',(ψ
t
t),(ψ
(t'-t)t),(ψt
)t',(ψ-)t'-(t)t',(ψt),(ψt
)t',t,,G(t
)t',(ψt),,(ψ)t',t,,G(
oooHHo
oHHo
oH
HHH
o
oHHHHo
oH
HHH
o
oHHHHo
oHHo
yxyx
yx
xyy
x
xyyx
xyy
x
xyyxyx
yxyx
i
Ti
Equation of Motion for the Green’s Function
• Differentiate G wrt first time argument
)t'-(t)-(
)t',(ψt),(ψt),(ψt),(ψ1
d)t',t,,G(ht
)t'-(t)-(
)t',(ψt),(ψt),(ψt),(ψ1
d ),G( h
)t'-(t)-(
(t'-t)t),(ψt),(ψt),(ψ)t',(ψ-1
d
)t'-(t)t',(ψt),(ψt),(ψt),(ψ1
d
(t'-t)t),(ψ)t',(ψ-)t'-(t)t',(ψt),(ψh)t',t,,G(t
oHH1H1Ho1
1
oHH1H1Ho1
1
oH1H1HHo1
1
oHH1H1Ho1
1
oHHHHo
yx
yxrrrx
ryx
yx
yxrrrr
ryx
yx
xrryrx
r
yxrrrx
r
xyyxyx
Tii
Tiii
i
i
ii
Equation of Motion for the Green’s Function
• Evaluate the T product using Wick’s Theorem
• Lowest order terms
• Diagram (9) is the Hartree-Fock exchange potential x Go(r1,y)• Diagram (10) is the Hartree potential x Go(x,y)• Diagram (9) is conventionally the first term in the self-energy• Diagram (10) is included in Ho in condensed matter physics
connectedoHH1H1Ho
11 )t',(ψt),(ψt),(ψt),(ψ
1d
yxrr
rxr T
)t',(ψt),(ψt),(ψt),(ψ HH1H1H yxrr
)t',(ψt),(ψt),(ψt),(ψ HH1H1H yxrr
(i)2v(x,r1)Go(x,r1) Go(r1,y)
(i)2v(x,r1)Go(r1,r1) Go(x,y)
x
y
r1
(10)
(9)y
r1
x
Equation of Motion for the Green’s Function
• One of the next order terms in the T product
• The full expansion of the T product can be written exactly as
(i)3v(1,2) v(x,r1)Go(1,x) Go(r1,2) Go(2,r1) Go(1,y)
)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ)(ψ-
1
-
1ddd HH1H1HHHHH
11 yxrr1221
rx21r21
(11)
Go(1,y)y
1
x
(x,1)
2
r1
diagrams proper iteratingby generated are latter The diagrams.
and into diagrams order higher divides ndistinctio This
line G single a cuttingby two into cut be cannot diagrams Unique
unique are others and repeated are diagrams some orders higher At
diagram) this in ( variabledummy a is
energy-self the is
o
improper
proper
1x
yxxxx
'
),'()G',('d o
Equation of Motion for the Green’s Function
• The proper self-energy * (F 105, M 181)• The self-energy has two arguments and hence two ‘external ends’• All other arguments are integrated out• Proper self-energy terms cannot be cut in two by cutting a single Go
• First order proper self-energy terms *(1)
• Hartree-Fock exchange term Hartree (Coulomb) term
Exercise: Find all proper self-energy terms at second order *(2)
r1
x
x’ (10)(9)x’
x
Equation of Motion for the Green’s Function
• Equation of Motion for G and the Self Energy
potential ncorrelatio-exchange the is
here suppressed dependence time
indirect put to is physics matter condensed in Convention
direct
exchangedirect
)',(
, ,
)-(),'(G)',('d),G(Vht
)',(V)',()',(
H )(
)',(V),(G)'('
1d)',)((
)()(
),'(G)',('d)(ψ)(ψ)(ψ)(ψ1
d
1
oH
H
o)1(
H11o1
1)1(
)1()1()1(
ooHH1H1Ho1
1
xx
ryx
yxyxxxxyx
xxxxxx
xxrrxxrx
rxx
yxxxxyxrrrx
r
ii
iTi
Equation of Motion for the Green’s Function
• Dyson’s Equation and the Self Energy
),''(G)'','()',(G''d'd),(G),G(
VH H
)-(),(GVht
)-(),'(G)',('d),G(Vht
ooo
Ho
oH
oH
Equation sDyson'
) incl. ( system ginteractin-non for G for Motion of Equation
system ginteractin for G for Motion of Equation
o
yxxxxxxxyxyx
yxyx
yxyxxxxyx
i
ii
Equation of Motion for the Green’s Function
• Integral Equation for the Self Energy
equation sDyson' inbyreplacemay weHence
and using
and Compare
energy self proper the iteratingby generated
energy self the in terms (repeated) improper i.e.
by related areenergy self proper the and energy -self The
G G
GGGGGGG
GGGGGGG
G GGGG
),')G(',('d ),'()G',('d
...GGG
)','''()''',''()G'',('''d''d)',()',(
*o
o*
o*
o*
o*
o*
o**
o*
o*
o*
o*
o*
o*
o
o**
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*o
**o
**
o**
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• Dyson’s Equation (F 106)
• In general, is energy-dependent and non-Hermitian• Both first order terms in are energy-independent • Quantum Chemistry: first order self energy terms included in Ho
• Condensed matter physics: only ‘direct’ first order term is in Ho
• Single-particle band gap in solids strongly dependent on ‘exchange’ term
Equation of Motion for the Green’s Function
),''()G'','()',(G''d'd ),(G),G(
),'')G('','()',(G''d'd ),(G),G(
ooo
*oo
yxxxxxxxyxyx
yxxxxxxxyxyx
G(x,y) = = + + + …
(x’,x’’)= + + …
• One of the 10 second order diagrams for the self energy• The first energy dependent term in the self-energy• Evaluate for homogeneous electron gas (M 170)
Evaluation of the Single Loop Bubble
oooo2
o
ooo
o
oo3
3
2o3
3
GGGG
GG
),(
),(G ),(G 2
d
2
d(-1).2.x
x))V((),(G 2
d
2
d
iiii
i
i
ii
ii
Theorem sWick'
q
q
qqkq
, ℓ+q, ℓ
, ℓ+q, ℓ, k-q
, q
, q
• Polarisation bubble: frequency integral over
• Integrand has poles at = ℓ - i and = - + ℓ+q + i• The polarisation bubble depends on q and • There are four possibilities for ℓ and q
Evaluation of the Single Loop Bubble
i
ii
i
ii
ii
q
q
q
),(G ),(G
),(G ),(G 2
d
oo
oo
FF
FF
FF
FF
kqk
kqk
kqk
kqk
x
y
i q
i
FF kqk
• Integral may be evaluated in either half of complex plane
Evaluation of the Single Loop Bubble
x
y
i q
i
FF kqk
0
1
ee2
ed
2
d
2
lim
rr
i
r
ir
i
ii
i
r
plane half upper in circlesemi
-plane half Upper
clockwiseAnti residues
ba
1
bzaz
1f(z)
az at f(z) residue
i
i
ii
i
ii
i
i
i
22
at pole for residue
• From Residue Theorem
• Exercise: Obtain this result by closing the contour in the lower half plane
Evaluation of the Single Loop Bubble
i
i
i
iii
q
q
q
1
2
2),(G ),(G
2
doo
• Polarisation bubble: continued
• For
• Both poles in same half plane• Close contour in other half plane to obtain zero in each case
• Exercise: For
• Show that
• And that
Evaluation of the Single Loop Bubble
FF kqk Aoi
FF
FF
kqk
kqk
i
iii
q
q ),(G ),(G 2
doo
i
i
i
ii
q2
2
d2
2
d),( 3
3
3
3
o
FF kqk Boi
),(G ),(G 2
doo
q ii
FF kqk
plane half lower in poles both otherwise be must
and at poles
F
2
3
3
3
3
Ao
2
3
3A
Bo
Ao
23
3
oo3
32
o3
3
εεε
εε
2
ε)V(
2
d
2
d
2
d
),(ε
)V( 2
d
2
d
),(),())V((ε
2
d
2
d
),(G ),(G 2
d
2
d))V((),(G
2
d
2
d-2
kqk
qqq
qqqq
qqqkq
qqkqk
qqkqk
qkqk
qkqk
ii
i
i
i
i
ii
i
iiii
i
iiii
• Self Energy
Evaluation of the Single Loop Bubble
FF kqk
, ℓ, k-q
, q
, q
, ℓ+q
dependent vector waveandenergy isenergy Self
at residue
iii
iii
i
ii
i
i
i
qkq
qkq
qkq
qkqqk
kq-kkqk
εεε
1)V(
2
d
2
d2
εεε
1)V(
2
d
2
d2
,,εεε
2
ε εε
2
ε
2
3
3
3
3B
2
3
3
3
3A
FFF
• Self Energy: continued
Evaluation of the Single Loop Bubble
FFF , , kqkkqk
FFF , , kqkkqk
• Real and Imaginary Parts
• Quasiparticle lifetime diverges as energies approach the Fermi surface
2A1
2
3
3
3
3A
2
3
3
3
3A
ε )Im(
εεε)V(2
d
2
d 2)Im(
εεε
1)V(
2
d
2
d 2)Re(
F
P
qkq
qkq
Evaluation of the Single Loop Bubble
1 lecture from x
/)x( )a(
aa
1Im
a
1
a
a
a
1Re
a
a
a
1
22lim
02
2
2
i
Pi
i
i