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TitleFeynman Diagram Techniques in Condensed Matter Physics
Author
TagsPhysics
LanguageEnglish
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Total Pages416
Table of Contents
                            Cover
Title
Full Title
Copyright Page
Contents
Preface
1 A brief review of quantum mechanics
	1.1 The postulates
		(I) The quantum state
		(II) Observables
		(III) Time evolution
		(IV) Measurements
		(V) Wave function of a system of identical particles
	1.2 The harmonic oscillator
	Further reading
	Problems
2 Single-particle states
	2.1 Introduction
	2.2 Electron gas
	2.3 Bloch states
	2.4 Example: one-dimensional lattice
	2.5 Wannier states
	2.6 Two-dimensional electron gas in a magnetic field
	Further reading
	Problems
3 Second quantization
	3.1 N-particle wave function
	3.2 Properly symmetrized products as a basis set
	3.3 Three examples
	3.4 Creation and annihilation operators
	3.5 One-body operators
	3.6 Examples
	3.7 Two-body operators
	3.8 Translationally invariant system
	3.9 Example: Coulomb interaction
	3.10 Electrons in a periodic potential
	3.11 Field operators
	Further reading
	Problems
4 The electron gas
	4.1 The Hamiltonian in the jellium model
	4.2 High density limit
	4.3 Ground state energy
	Further reading
	Problems
5 A brief review of statistical mechanics
	5.1 The fundamental postulate of statistical mechanics
	5.2 Contact between statistics and thermodynamics
	5.3 Ensembles
	5.4 The statistical operator for a general ensemble
	5.5 Quantum distribution functions
	Further reading
	Problems
6 Real-time Green's and correlation functions
	6.1 A plethora of functions
	6.2 Physical meaning of Green's functions
	6.3 Spin-independent Hamiltonian, translational invariance
	6.4 Spectral representation
	6.5 Example: Green's function of a noninteracting system
	6.6 Linear response theory
	6.7 Noninteracting electron gas in an external potential
	6.8 Dielectric function of a noninteracting electron gas
	6.9 Paramagnetic susceptibility of a noninteracting electron gas
	6.10 Equation of motion
	6.11 Example: noninteracting electron gas
	6.12 Example: an atom adsorbed on graphene
	Further reading
	Problems
7 Applications of real-time Green's functions
	7.1 Single-level quantum dot
	7.2 Quantum dot in contact with a metal: Anderson's model
	7.3 Tunneling in solids
	Further reading
	Problems
8 Imaginary-time Green's and correlation functions
	8.1 Imaginary-time correlation function
	8.2 Imaginary-time Green's function
	8.3 Significance of the imaginary-time Green's function
	8.4 Spectral representation, relation to real-time functions
	8.5 Example: Green's function for noninteracting particles
	8.6 Example: Green's function for 2-DEG in a magnetic field
	8.7 Green's function and the U-operator
	8.8 Wick's theorem
	8.9 Case study: first-order interaction
	8.10 Cancellation of disconnected diagrams
	Further reading
	Problems
9 Diagrammatic techniques
	9.1 Case study: second-order perturbation in a system of fermions
	9.2 Feynman rules in momentum-frequency space
	9.3 An example of how to apply Feynman rules
	9.4 Feynman rules in coordinate space
	9.5 Self energy and Dyson's equation
	9.6 Energy shift and the lifetime of excitations
	9.7 Time-ordered diagrams: a case study
	9.8 Time-ordered diagrams: Dzyaloshinski's rules
	Further reading
	Problems
10 Electron gas: a diagrammatic approach
	10.1 Model Hamiltonian
	10.2 The need to go beyond first-order perturbation theory
	10.3 Second-order perturbation theory: still inadequate
	10.4 Classification of diagrams according to the degree of divergence
	10.5 Self energy in the random phase approximation (RPA)
	10.6 Summation of the ring diagrams
	10.7 Screened Coulomb interaction
	10.8 Collective electronic density fluctuations
	10.9 How do electrons interact?
	10.10 Dielectric function
	10.11 Plasmons and Landau damping
	10.12 Case study: dielectric function of graphene
	Further reading
	Problems
11 Phonons, photons, and electrons
	11.1 Lattice vibrations in one dimension
	11.2 One-dimensional diatomic lattice
	11.3 Phonons in three-dimensional crystals
	11.4 Phonon statistics
	11.5 Electron-phonon interaction: rigid-ion approximation
	11.6 Electron-LO phonon interaction in polar crystals
	11.7 Phonon Green's function
	11.8 Free-phonon Green's function
	11.9 Feynman rules for the electron-phonon interaction
	11.10 Electron self energy
	11.11 The electromagnetic field
	11.12 Electron-photon interaction
	11.13 Light scattering by crystals
	11.14 Raman scattering in insulators
	Further reading
	Problems
12 Superconductivity
	12.1 Properties of superconductors
	12.2 The London equation
	12.3 Effective electron-electron interaction
	12.4 Cooper pairs
	12.5 BCS theory of superconductivity
	12.6 Mean field approach
	12.7 Green's function approach to superconductivity
	12.8 Determination of the transition temperature
	12.9 The Nambu formalism
	12.10 Response to a weak magnetic field
	12.11 Infinite conductivity
	Further reading
	Problems
13 Nonequilibrium Green's function
	13.1 Introduction
	13.2 Schrodinger, Heisenberg, and interaction pictures
	13.3 The malady and the remedy
	13.4 Contour-ordered Green's function
	13.5 Kadanoff-Baym and Keldysh contours
	13.6 Dyson's equation
	13.7 Langreth rules
	13.8 Keldysh equations
	13.9 Steady-state transport
	13.10 Noninteracting quantum dot
	13.11 Coulomb blockade in the Anderson model
	Further reading
	Problems
Appendix A: Second quantized form of operators
	A.1 Fermions
	A.2 Bosons
Appendix B: Completing the proof of Dzyaloshinski's rules
Appendix C: Lattice vibrations in three dimensions
	C.1 Harmonic approximation
	C.2 Classical theory of lattice vibrations
	C.3 Vibrational energy
	C.4 Quantum theory of lattice vibrations
Appendix D: Electron-phonon interaction in polar crystals
	D.1 Polarization
	D.2 Electron-LO phonon interaction
References
Index
                        
Document Text Contents
Page 1

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Page 208

192 Diagrammatic techniques

Figure 9.9 The two first-order connected, topologically distinct diagrams arising
from the correction to g(kσ, ωn ) in first order of the interaction (n = 1). (a) is a
direct interaction diagram and (b) is an exchange interaction diagram.

For n = 2,

C(2) = 5!− 2!
1!1!

2!C(1)− 2!
2!0!

4!C(0) = 80, CT D(2) =
80

222!
= 10.

For n = 3, we find CT D(3) = 74. Clearly, the number of connected, topologically
distinct diagrams grows rapidly with increasing perturbation order.

9.3 An example of how to apply Feynman rules

Consider the correction to g(kσ, ωn) in first order of the interaction (n = 1). There
is one interaction line and three fermion lines, two of which are external lines
with coordinates (kσ, ωn). There are two connected, topologically distinct dia-
grams, shown in Figure 9.9. Using the Feynman rules, we can readily write the
contributions of these two diagrams:

δg(1)a (kσ, ωn) = −
(−1

βh̄2

)
v0

V
[g0(kσ, ωn)]2


k′σ ′


n′

g0(k′σ ′, ωn′)eiωn′0
+

(9.12)

δg
(1)
b (kσ, ωn) =

(−1
βh̄2

)
[g0(kσ, ωn)]2


qm

vq

V
g0(k− qσ, ωn − ωm)ei(ωn−ωm)0

+

δg(1)(kσ, ωn) = δg(1)a (kσ, ωn)+ δg(1)b (kσ, ωn). (9.13)

The first minus sign in δg(1)a results from the existence of a closed fermion loop in
diagram (a). The convergence factor eiωn′0

+
is inserted because the fermion line with

coordinates (k′σ ′, ωn′) closes in on itself. On the other hand, the convergence factor
ei(ωn−ωm)0

+
in δg(1)b arises because the fermion line with coordinates (k− qσ, ωn −

ωm) is joined by an interaction line. These convergence factors are important;
without them, the summation over frequencies would diverge (see Problem 9.3).

Page 209

9.4 Feynman rules in coordinate space 193

Frequency sums, as in the above expression for δg(1)a (kσ, ωn) and δg
(1)
b (kσ, ωn),

often arise in applications of the finite temperature Green’s function. Here we
record the following formula:

∞∑
n=−∞

eiωn0
+

iωn − �̄/h̄
=
{
−βh̄n�̄ bosons
βh̄f�̄ fermions

(9.14)

(see Problem 9.3). In the above equation, n�̄ and f�̄ are the Bose–Einstein and
Fermi–Dirac distribution functions, respectively.

9.4 Feynman rules in coordinate space

The perturbation expansion for Green’s function, derived in the previous chapter,
applies to both momentum and coordinate space; hence

g(rστ, r′σ ′τ ′) = −
∞∑

n=0

1
n!

(−1/h̄)n
∫ βh̄

0
dτ1 . . .

∫ βh̄
0
dτn

〈T �σ (rτ )�†σ ′(r′τ ′)V (τ1) . . . V (τn)〉0,c. (9.15)
All operators are interaction picture operators. For two-particle interactions (see
Eq. [3.53]),

V (τ ) = 1
2


λλ′


μμ′


d

3r


d3r ′�†λ(rτ )�


μ(r

′τ )vλμ,λ′μ′(r, r′)�μ′(r′τ )�λ′(rτ )

where λ, λ′, μ, and μ′ are spin projection indices and vλμ,λ′μ′(r, r′) =
〈λμ|v(rσ, r′σ ′)|λ′μ′〉. If v is spin-independent, vλμ,λ′μ′(r, r′) = v(r, r′)δλλ′δμμ′ . Let
U (rστ, r′σ ′τ ′) = v(rσ, r′σ ′)δ(τ − τ ′), where 0 < τ, τ ′ < βh̄. V (τ ) may be written
as

V (τ ) = 1
2


λλ′μμ′


d

3r


d3r ′

∫ βh̄
0

dτ ′�†λ(rτ )�

μ(r

′τ ′)Uλμ,λ′μ′(rτ, r′τ ′)

×�μ′(r′τ ′)�λ′(rτ ). (9.16)
The interaction is depicted in Figure 9.10. The two vertices of the interaction line
are assigned coordinates (rτ ) and (r′τ ′). The first-order correction is given by

δg(1)(rστ, r′σ ′τ ′) = 1
2h̄


d3r1

∫ βh̄
0

dτ1


d3r ′1

∫ βh̄
0

dτ ′1

λλ′


μμ′
〈T �σ (rτ )�†σ ′(r′τ ′)

ψ

λ(r1τ1)ψ


μ(r




1)Uλμ,λ′μ′(r1τ1, r




1)ψμ′(r




1)ψλ′(r1τ1)〉0,c. (9.17)

Page 415

Index 399

three electrons, 40
two electrons, 41, 42

spectral density function, 101, 103, 106, 152,
197

noninteracting particles, 107, 154
spectral representation, 98

advanced Green’s function, 101
meaning, 98
retarded correlation function, 103–104
retarded Green’s function, 101
single-particle correlation function, 101–102

spin density, 109
spin waves, 64
spin-density correlation function, 118
spin-density operator, 118, 119
statistical operator, 84, 331, 352

definition, 85
general ensemble, 85
grand canonical ensemble, 84
Heisenberg picture, 335
interaction picture, 336
properties, 86
Schrödinger picture, 334
time evolution, 86

steady-state transport, 352–360
Anderson’s impurity model, 354
bias voltage, 353
current formula, 358
Landauer formula, 363
level-width function, 358
Meir–Wingreen formula, 360
mixed lesser function, 355
model Hamiltonian, 353
proportional coupling, 360
tunneling, 354

step function, 6, 71, 121, 131, 159, 198, 230, 232,
262, 320, 337, 350

Stirling’s formula, 89
superconductivity, 284

BCS theory, 299–304
Green’s function approach, 309–316
high-TC , 287
infinite conductivity, 325
mean field approach, 304–309
pair fluctuation, 327
properties of superconductors, 284–289
response to a weak magnetic field, 319–325
two-band model, 328

superconductors, 284
copper oxide family, 287
critical magnetic field, 284
critical temperature, 284
electronic specific heat, 286
flux expulsion, 286
iron-based superconductors, 288
isotope effect, 287
Meissner effect, 285
perfect diamagnetism, 285

resistivity, 284
tunneling experiments, 287
type-I, 286
type-II, 286

lower critical field, 286
upper critical field, 286

switching the interaction on and off adiabatically,
338

thermodynamic limit, 65
thermodynamic potential, 210

and self energy, 211
interacting electrons, 211

Thomas–Fermi model, 233–234
charge impurity, 234
dielectric function, 233
induced charge density, 234
screened Coulomb interaction, 233
wave number, 233
wave number in two dimensions, 246

tight binding method, 61
time evolution operator, 3, 16, 331
time-ordered diagrams, 199–210

example: a ring diagram, 199
internal frequency coordinates, 203
section, 203

time-ordered product, 162, 163, 173
equal time arguments, 164

time-ordering operator, 92, 144, 159, 262, 333
transition rate, 16, 17, 105, 275
translation operator, 23

eigenvalues, 23
translationally invariant system, 51, 126, 176
triplet, 14, 297
tunneling, 135

current, 137
elastic, 136
inelastic, 136
linear response theory, 138
model Hamiltonian, 136
Ohm’s law, 141
retarded correlation function, 140
steady state, 139

two-particle interaction, 179, 190, 193, 194, 196, 205,
212

spin-independent, 186, 195

uncertainty principle, 292
uniform positive background, 19, 21, 53, 65, 66, 213

vector space, 1
spatial, 8
spin, 8

Wannier states, 29–31
wave function, 1

bosons, 10
fermions, 10

Page 416

400 Index

Wick’s theorem, 162, 168, 180, 187, 240, 265,
311, 313, 314, 318, 321, 339, 346, 347,
356

an example, 163
bosons, 177

fermions, 162
pictorially, 171
proof, 167
remarks, 168
statement, 163

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