Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 12 Proof# 1 - 29/07/09

364 12 Many Body Interactions: Green’s Function Method

This gives that 62

(t) − Ψ

)=−i





)Ψ



). (12.22)

This is an integral equation for Ψ

(t) that we can try to solve by iteration. Let 63

us write 64

(t) ≡ Ψ

(0)

(t)+Ψ

(1)

(t)+···+ Ψ

(n)

(t)+··· . (12.23)

Here

(0)

(t)=Ψ

(1)

(t)=−i



)Ψ

(0)



(2)

(t)=−i



)Ψ

(1)



(n)

(t)=−i



)Ψ

(n−1)



(12.24)

This result can be expressed as

(t)=S(t, t

)Ψ

), (12.25)

where S(t, t

) is the so-called S matrix is given by 68

S(t, t

)=1− i

)+(−i)

)+···



∞

n=0

(−i)

···

n−1



) ···H

)



(12.26)

Let us look at the third term involving integration over t

and t



. (12.27)

In the second

, let us reverse the order of integration (see Fig. 12.1). We 70

ﬁrst integrated over t

from t

to t

,thenovert

from t

to t. Inverting the 71

order gives 72



⇒



Therefore, we have



(12.28)

But t

and t

are dummy integration variables and we can interchange the 74

names to get 75



). (12.29)

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BookID 160928 ChapID 12 Proof# 1 - 29/07/09

12.1 Formulation 365

Fig. 12.1. Order of integration I

appearing in (12.27)

Adding this term to the

that was left in its original form gives 76



). (12.30)

We are integrating over a square of edge Δt = t − t

in the t

-plane. The 77

second term, with t

, is just an integral on the lower triangle shown 78

in Fig. 12.1. The ﬁrst term, where t

, is an integral on the upper tri- 79

angle. Therefore, we can combine the time integrals and write the limits of 80

integration from t

to t. 81



)θ(t

− t

)+H

)θ(t

− t

)] .

(12.31)

The thing we have to be careful about here, however, is that H

)andH

) 82

do not necessarily commute. We can get around this diﬃculty by using the 83

time ordering operator T. The product of functions H

) that follows the 84

operator T must have the largest t values on the left. In the ﬁrst term of 85

(12.31), t

, so we can write the integrand as 86

)=T{H

)}.

In the second term, with t

,wemaywrite 87

)=T{H

)}.

Equation (12.31) can, thus, be rewritten as



T {H

)}. (12.32)

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

366 12 Many Body Interactions: Green’s Function Method

For the general term we have 89

···

n−1

) ···H

)

···dt

) ···H

)witht ≥ t

≥ t

≥···≥t

and it is not diﬃcult to see that the same technique can be applied to give



···



T{H

) ···H

)}. (12.33)

Note that

T{H

) ···H

)} = H

) ···H

)ift

≥ t

≥···≥t

Making use of (12.33), the S matrix can be written in the compact form

S(t, t

)=T



−i



)dt





, (12.34)

where it is understood that in the nth term in the expansion of the exponential, 94

(12.33) holds. We note that, at t = 0, the wave functions Ψ

, Ψ

coincide, 95

(0) = Ψ

(0) = S(0,t

)Ψ

)=S(0,t

wherewehaveusedΨ

(t)=e

(t)andΨ

(t)=S(t, t

)Ψ

). 96

12.2 Adiabatic Approximation 97

Suppose that we multiply H

by e

−β|t|

where β ≥ 0, and treat the resulting 98

interaction as one that vanishes at t = ±∞. Then, the interaction is slowly 99

turned on from t = −∞ up to t = 0 and slowly turned oﬀ from t = 0 till 100

t =+∞.WecanwriteH(t = −∞)=H

, the noninteracting Hamiltonian, 101

and 102

(t = −∞)=Ψ

(t = −∞)=Φ

. (12.35)

Here, Φ

is the Heisenberg state vector of the noninteracting system. We know 103

that eigenstates of the interacting system in the Heisenberg, Schr¨odinger, and 104

interaction representation are related by 105

(t)=e

iHt

(t)and Ψ

(t)=e

(t). (12.36)

Therefore, at time t = 0,

106

(t =0)=Ψ

. (12.37)

Henceforth, we will use Ψ

to denote the state vector of the fully interacting 107

system in the Heisenberg representation. We can express Ψ

as 108

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BookID 160928 ChapID 12 Proof# 1 - 29/07/09

12.2 Adiabatic Approximation 367

= S(0, −∞)Φ

, (12.38)

where S is the S matrix deﬁned in (12.34). Because Ψ

(t =0)=Ψ

we can 109

write 110

(t)=S(t, 0)Ψ

= S(t, −∞)Φ

. (12.39)

In the last step, we have used S(t

, −∞)=S(t

)S(t

, −∞). If we write 111

|Ψ

(t) = S(t, 0)|Ψ

 and Ψ

(t)| = Ψ

−1

(t, 0), then for some operator F 112

Ψ

(t)|F

|Ψ

(t) = Ψ

−1

(t, 0)F

S(t, 0)|Ψ

 = Ψ

(t)|F

|Ψ

(t). (12.40)

But this must equal Ψ

|Ψ

. Therefore, we have 113

= S

−1

(t, 0)F

S(t, 0). (12.41)

Now, look at the expectation value in the exact Heisenberg interacting state

114

of the time ordered product of Heisenberg operators 115

Ψ

|T{A

) ···Z

)}|Ψ

.

If we assume that the t

’s have been arranged in the order t

≥ t

≥ 116

···≥t

,thenwecanwrite 117

Ψ

|T{A

)···Z

)}|Ψ



Ψ

|Ψ



Φ

|S(∞,0)S

−1

,0)A

)S(t

,0)S

−1

,0)B

)S(t

,0)S

−1

,0)···S

−1

,0)Z

)S(t

,0)S(0,−∞)|Φ



Φ

|S(∞,0)S(0,−∞)|Φ



118

(12.42)

But from S(t

, 0) = S(t

)S(t

, 0) we can see that 119

S(t

, 0)S

−1

, 0) = S(t

). (12.43)

Using this in (12.42) gives

120

Ψ

|T{A

)···Z

)}|Ψ



Ψ

|Ψ



Φ

|S(∞,t

)S(t

)···Z

)S(t

,−∞)|Φ



Φ

|S(∞,−∞)|Φ



(12.44)

We note that, in (12.44), the operators are in time-ordered form, i.e. t

≥−∞, 121

≥ t

, ∞≥t

, so the operators 122

S(∞,t

)S(t

) ···Z

)S(t

, −∞)

are chronologically ordered, and hence we can rewrite (12.44) as

123

Ψ

|T{A

)···Z

)}|Ψ



Ψ

|Ψ



Φ

|T{S(∞,−∞)A

)···Z

)}|Φ



Φ

|S(∞,−∞)|Φ



. (12.45)

124

125

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

368 12 Many Body Interactions: Green’s Function Method

12.3 Green’s Function 126

We deﬁne the Green’s function G

αβ

(x, x



), where x = {r,t} and α, β are spin 127

indices, by 128

αβ

(x, x



)=−i

Ψ

|T{ψ

(x)ψ

H†



)}|Ψ



Ψ

|Ψ



. (12.46)

129

Here, ψ

(x) is an operator (particle ﬁeld operator) in the Heisenberg repre- 130

sentation. 131

By using Eq.(12.45) in Eq.(12.46), we obtain 132

αβ

(x, x



)=−i

Φ

|T{S(∞, −∞)ψ

(x)ψ

I†



)}|Φ



Φ

|S(∞, −∞)|Φ



. (12.47)

The operator ψ

(x) is now in the interaction representation. If we write out 133

the expansion for S(∞, −∞) in the numerator and are careful to keep the 134

time ordering, we obtain 135

αβ

(r,t,r



)=−

S(∞,−∞)



∞

n=0

(−i)

∞

−∞

···dt

×Φ

|T{ψ

(r,t)ψ

I†



) ···H

)}|Φ

.

(12.48)

12.3.1 Averages of Time-Ordered Products of Operators

136

If F

(t)andF



) are Fermion operators, then by T{F

(t)F



)} we mean 137

T{F

(t)F



)} = F

(t)F



)ift>t



= −F



(t)ift<t



(12.49)

In other words, we need a minus sign for every permutation of one Fermion

138

operator past another. For Bosons no minus sign is needed. 139

In G

αβ

we ﬁnd the ground state average of products of time-ordered oper- 140

ators like T{ABC ···}. Here, A, B, ... are ﬁeld operators (or products of ﬁeld 141

operators). When the entire time ordered product is expressed as a product of 142

†

’s and ψ’s, it is useful to put the product in what is called normal form, in 143

which all annihilation operators appear to the right of all creation operators. 144

For example, the normal product of ψ

†

(1)ψ(2) can be written 145

N{ψ

†

(1)ψ(2)} = ψ

†

(1)ψ(2) while N{ψ(1)ψ

†

(2)} = −ψ

†

(2)ψ(1). (12.50)

The diﬀerence between a T product and an N product is called a pairing or

146

a contraction. For example, the diﬀerence in the T ordered product and the 147

N product of AB is given by 148

T(AB) − N(AB)=A

. (12.51)

149

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

12.3 Green’s Function 369

We note that the contraction of a pair of operators is the anticommutator we 150

omit when we formally reorder a T product of a pair of operators to get an N 151

product. The contractions are c-numbers for the operators we are interested 152

in. 153

12.3.2 Wick’s Theorem 154

The Wick’s theorem states that T product of operators ABC ··· can be 155

expressed as the sum of all possible N products with all possible pairings. 156

By this we mean that 157

T(ABCD ···XY Z)

=N(ABCD ···XY Z)

+N(A

CD ···XY Z)+N(A

D ···XYZ)+N(A

BCD

···XY Z)

+ ···+N(ABCD ···XY

)

+N(A

···XYZ)+···+N(ABCD ···W

)

+N(A

···Y

)+N(A

···Y

)+ all other pairings.

(12.52)

158

In evaluating the ground state expectation value of (12.52) only the term in 159

which every operator is paired with some other operator is nonvanishing since 160

the normal products that contain unpaired operators must vanish (they anni- 161

hilate excitations that are not present in the ground state). In the second and 162

third lines on the right, in each term we bring two operators together by anti- 163

commuting, but neglecting the anticommutators, then replace the pair by its 164

contraction, and ﬁnally take the N product of the remaining n −2 operators. 165

We do this with all possible pairings so we obtain

n(n−1)

terms, each term 166

containing an N product of the n −2 remaining operators. In the fourth line 167

on the right, we choose two pairs in all possible ways, replace them by their 168

contractions, and leave in each term an N products of the n − 4 remaining 169

operators. We repeat the same procedure, and in the last line on the right, 170

every operator is paired with some other operator in all possible ways leaving 171

no unpaired operators. Only the completely contracted terms (last line on 172

the right of (12.52)) give ﬁnite contributions in the ground-state expectation 173

value. That is, we have 174

T(ABCD ···XY Z)

= T(AB)T(CD)···T(YZ)±T(AC)T(BD)···T(YZ)

± All other pairings.

(12.53)

Here,wehaveusedA

=T(AB) −N(AB) and noted that N(AB) =0,so 175

the ground state expectation value of A

 = T(AB).Nowletusreturn 176

to the expansion of the Green’s function. The ﬁrst term in the sum over n in 177

(12.48) is 178

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BookID 160928 ChapID 12 Proof# 1 - 29/07/09

370 12 Many Body Interactions: Green’s Function Method

(0)

(r,t,r



)=−

S(∞, −∞)

Φ

∗

| T{ψ

(r,t)ψ

†



)}|Φ



, (12.54)

where, now, the operators ψ(r,t)andψ

†

(r,t) are in the interaction repre- 179

sentation. G

(0)

is the noninteracting Green’s function (i.e., it is the Green 180

function when H



= 0). Here we shall take the interaction to be given, in 181

second-quantized form, by 182





†

)ψ

†

)U(r

− r

)ψ(r

). (12.55)

Now, introduce a function V (x

−x

) ≡ U(r

−r

)δ(t

−t

) to write the ﬁrst 183

correction due to interaction as (let x = r,t) 184

δG

(1)

(x, x



)=−

2S(∞,−∞)

V (x

− x

)

×T{ψ(x)ψ(x



)ψ

†

)ψ

†

)ψ(x

)}

(12.56)

The time-ordered product of the six operators (3 ψ’s and 3ψ

†

’s) can be written 185

out by using (12.53) 186

T{ψ(x)ψ

†



)ψ

†

)ψ

†

)ψ(x

)}

= T(ψ(x)ψ

†

))T(ψ

†

)ψ(x

))T(ψ(x

)ψ

†



))

−T(ψ(x)ψ

†

))T(ψ

†

)ψ(x

))T(ψ(x

)ψ

†



))± all other pairings.

(12.57)

But T(ψ(x

)ψ

†

)) is proportional to G

(0)

). Therefore, the ﬁrst term 187

on the right hand side of (12.57) is proportional to 188

(0)

(x, x

(0)



). (12.58)

It is simpler to draw Feynman diagram for each of the possible pairings. There

189

are six of them in δG

(1)

(x, x



) because there are six ways to pair one ψ

†

with 190

one ψ. The diagrams are shown in Fig. 12.2. Note that x

and x

are always 191

connected by an interaction line V (x

− x

). An electron propagates in from 192

x and out to x



.Ateachx

and x

theremustbeoneG

(0)

entering and one 193

leaving. 194

In a standard book on many body theory, such as Fetter–Walecka(1971), 195

Mahan(1990), and Abrikosov–Gorkov–Dzyaloshinskii(1963), one can ﬁnd 196

1. Rules for constructing the Feynman diagrams for the n

order correction 197

and 198

2. Rules for writing down the analytic expression for δG

(n)

associated with 199

each diagram. 200

Let us give one simple example of constructing diagrams. For the n

order cor- 201

rections, there are n interaction lines and (2n + 1) directed Green’s functions, 202

(0)

’s. The rules for the nth-order corrections are as follows: 203

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

12.3 Green’s Function 371

Fig. 12.2. Feynman diagrams in the ﬁrst-order perturbation calculation

1. Form all connected, topologically nonequivalent diagrams containing 2n ver- 204

tices and two external points. Two solid lines and one wavy line meet at 205

each vertex. 206

2. With each solid line associate a Green’s function G

(0)

(x, x



)wherex and 207



are the coordinates of the initial and ﬁnal points of the line. 208

3. With each wavy line associate V (x − x



)=U(r − r



)δ(t − t



)forawavy 209

line connecting x and x



. 210

4. Integrate over the internal variables d

for all vertex coordi- 211

nates (and sum over all internal spin variables if spin is included). 212

5. Multiply by i

(−)

,whereF is the number of closed Fermion loops. 213

6. Understand equal time G

(0)

’s to mean, as δ → 0

, 214

(0)

t, r

t) → G

(0)

t, r

t + δ).

The allowed diagrams contributing to δG

(2)

(x, x



) are shown in Fig. 12.3. 215

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

372 12 Many Body Interactions: Green’s Function Method

Fig. 12.3. Feynman diagrams in the second order perturbation calculation

12.3.3 Linked Clusters 216

In writing down the rules, we have only considered linked (or connected ) dia- 217

grams, but diagrams (e) and (f) in Fig. 12.2 are unlinked diagrams. By this we 218

mean that they fall into two separate pieces, one of which contains the coordi- 219

nates x and x



of G(x, x



). It can be shown (see a standard many body text like 220

Abrikosov–Gorkov–Dzyaloshinskii (1963).) that when the contributions from 221

unlinked diagrams are included, they simply multiply the contribution from 222

linked diagrams by a factor S(∞, −∞). Since this factor appears in denom- 223

inator of G

αβ

(x, x



) in (12.48), it simply cancels out. Furthermore, diagrams 224

(a) and (c) in Fig. 12.2 are identical except for interchange of the dummy 225

variables x

and x

, and so too are (b) and (d). The rules for constructing 226

diagrams for δG

(n)

(x, x



) take this into account correctly and one can ﬁnd the 227

proof in standard many body texts mentioned above. 228

12.4 Dyson’s Equations 229

If we look at the corrections to G

(0)

(x, x



) we notice that for our linked cluster 230

diagrams the corrections always begin with a G

(0)

(x, x

), and this is followed 231

by something called a self energy part. Look, for example, at the ﬁgures labeled 232

(a) or (b) in Fig. 12.2 or (j) in Fig. 12.3. The ﬁnal part of the diagram has 233

another G

(0)



). Suppose we represent the general self energy by Σ. Then 234

we can write 235

G = G

(0)

+ G

(0)

ΣG. (12.59)

236

Uncorrected Proof

BookID 160928 ChapID 12 Proof# 1 - 29/07/09

12.4 Dyson’s Equations 373

Fig. 12.4. Diagrammatic expressions of (a) Dyson equation G = G

(0)

+ G

(0)

ΣG,

(b)selfenergyΣ

,(c) Dyson equation W = V + V ΠW ,(d) polarization part

= G

(0)

This equation says that G is the sum of G

(0)

and G

(0)

followed by Σ which 237

in turn can be followed by the exact G we are trying to determine. We can 238

express (12.59) in diagrammatic terms as is shown in Fig. 12.4a. The simplest 239

self-energy part that is of importance in the problem of electron interactions 240

in a degenerate electron gas is Σ

,where 241

= G

(0)

W. (12.60)

242

In diagrammatic terms this is expressed as shown in Fig. 12.4b, where the dou- 243

ble wavy line is a screened interactionandwecanwriteaDysonequationfor 244

it by 245

W = V + V ΠW. (12.61)

246

The Π is called a polarization part ; the simplest polarization part is 247

= G

(0)

, (12.62)

248

the diagrammatic expression of which is given in Fig. 12.4d. Of course, in 249

(12.60) and (12.62) we could replace G

(0)

by the exact G to have a result 250

that includes many terms of higher order. Approximating the self energy by 251

the product of a Green’s function G and an eﬀective interaction W is often 252

referred to as the GW approximation to the self-energy. The simplest GW 253

approximation is the random phase approximation (RPA). In the RPA, the 254

G is replaced by G

(0)

and W is the solution to (12.61) with (12.62) used for 255