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QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS
Wolfgang Belzig
FORMAL MATTERS
“Wahlpflichtfach” 4h lecture + 2h exerciseLecture: Mon & Thu, 10-12, P603Tutorial: Mon, 14-16 (P912) or 16-18 (P712)50% of exercises needed for examLanguage: “English” (German questions allowed)No lecture on 21. April, 25. April, 9. May, 2. June, 13. June, 16. June, 23. June, 14. July
EXERCISE GROUPS
• Two groups @ 14h (P912) and 16h (P712)• Distribution: see list• Exercise sheets usually 5-8 days before exercise
(see webpage for preview)• Question on exercise to one of the tutors
(preferably the one who is responsible)• Plan: 18.4. Milena Filipovic; 2.5. Martin Bruderer; 9.5. Fei Xu
16.5. Cecilia Holmqvist; 23.5. Peter Machon
LITERATURE
G. Rickayzen: Green’s functions and Condensed MatterH. Bruus and K. Flensberg: Many-Body Quantum Theory in Condensed Matter PhysicsJ. Rammer: Quantum Field Theory of Non-equilibium StatesG. Mahan: Quantum Field Theoretical MethodsFetter & Walecka: Quantum Theory of Many Particle SystemsYu. V. Nazarov & Ya. Blanter: Quantum TransportJ. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986)
CONTENT
A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise
PHYSICAL OVERVIEW
Solid state physics: Many electrons and phonons interacting with the lattice potential and among each other (band structure, disorder, electron-electron interaction, electron-phonon interaction)
We need to explain:
Why some materials are conductors, semiconductors, insulators, superconductors or ferromagnets
Thermodynamic quantities, transport coefficients, electric and magnetic susceptibilities
All phenomena follow from the same Hamiltonian, but with different microscopic parameters (number of electrons per atom, lattice structure, nucleus masses...)
THE THEORETICAL PROBLEM
Due to the huge number of degrees of freedom (M states, N particles: MN), a wave function treatment is not feasible.
Quantum fields ( ) are more suitable, since they can represent a large number of identical particles
The central objects are expectation values of quantum field operators - Greens function
Physical observable are obtained directly from the Greens functions, e.g. density
Ψ(x),ck†,...
G(x, x ') = Ψ(x)Ψ†( ′x )
n(x) = G(x, x)
THE TECHNICAL PROBLEM
Often approximations are based on some perturbation expansion up to some finite order (first or second)
In many practical problems we need non-pertubative solutions:
n ~ e− t /τ ≈ 1− t / τ
kBTc ~ ω ce−1/NV not expandable in V
goes negative
Critical temperature
Exponential decay
CONTENT
A. Introduction, the ProblemB. Formalities, DefinitionsC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise
A. INTRODUCTION
1. Greens functions (general definition, Poisson equation, Schrödinger equation, retarded, advanced and causal)Linear response (general theory of response functions, Kubo formula, Greens function)
2. Statement of the problem (physical problem, 2nd quantization, field operators, electron-phonon interaction, single particle potentials)
THE HAMILTONIAN
H =P2j2M j
+Uj (Rj )
⎛
⎝⎜⎞
⎠⎟j∑
+pi2
2m+Ui (
ri )⎛⎝⎜
⎞⎠⎟i
∑
+12
e2ri −rji≠ j
∑ + VijRj ,ri( )
ij∑
Ions in the harmonic potential
Electrons in the periodic potential
Electron-electron and Elektron-phonon interaction
THE HAMILTONIAN
H = ω qλ a†qλa qλ +12
⎛⎝⎜
⎞⎠⎟qλ
∑+ kσc
†kσc kσ
kσ∑
+ c†k + qσc†k '− qσ 'V (
q)c k 'σ 'c kσk′k qσ ′σ∑
+ gqλc†k + qσc kσ a†− qλ + a qλ( )
kqσλ∑
Ions in the harmonic potential
Electrons in the periodic potential
Electron-electron and Elektron-phonon interaction
SUMMARY OF A.
1. Greens functions obey differential equations with generalized delta-perturbation
2. Analytical properties of GF are related to causality
3. Linear response response of a quantum system (Kubo formula) related to a retarded function
4. Many-body Hamiltonian in second quantization: electrons, phonons, potentials, interaction
CONTENT
A. Introduction, the ProblemB. Formal MattersC. Diagrammatic MethodsD. Disordered ConductorsE. Electrons and PhononsF. SuperconductivityG. Quasiclassical MethodsH. Nonequilibrium and Keldysh formalismI. Quantum Transport and Quantum Noise
B. FORMAL MATTERS
1.Definitions of double-time GF (retarded, advanced, causal, temperature), higher order GF
2.Analytical properties (spectral representation, spectral function, Matsubara frequencies)
3. Single particle Greens functions, spectral function, density of states, quasiparticles
4.Equation of motion for Greens functions5.Wicks theorem
1. DEFINITIONS
Two-time Greens functions for two operators A and BRetarded Greens function
GR (t, ′t ) = −i
AH (t), BH ( ′t )⎡⎣ ⎤⎦ θ(t − ′t )
A, B⎡⎣ ⎤⎦ = AB + BA
= Tr ρ( ) ρ =1Ze−β ( H −µN )
=−1 Bose operators+1 Fermi operators
⎧⎨⎪
⎩⎪Bose: ak , ak
†, ρ(r ) = ψ †(r )ψ (r ),j (r ), H , N
Fermi: ck , ck†, ψ †(r ), ψ (r )
1. DEFINITIONS
Temperature Greens function
G(τ , ′τ ) = −1T A(−iτ )B(−i ′τ )( )
t→ −iτ Wick rotation
AH (t)→ A(−iτ ) = A(τ ) = eHτ Ae−Hτ
Higher-order Greens function
G(τ1,τ 2 ,τ 3,…) ~ T A(τ1) B(τ 2 ) C(τ 3)( )
T A(τ ) B( ′τ )( ) = A(τ ) B( ′τ )θ(τ − ′τ ) − B( ′τ ) A(τ )θ(τ '− τ )
Time-ordering operator
“Later times to the left”
B. FORMAL MATTERS
1.Definitions of double-time GF (retarded, advanced, causal, temperature), higher order GF
2.Analytical properties (spectral representation, spectral function, Matsubara frequencies)
3. Single particle Greens functions, spectral function, density of states, quasiparticles
4.Equation of motion for Greens functions5.Wicks theorem
MATSUBARA GREEN’S FUNCTION
G(τ ) =− A(τ ) B τ > 0
B A(τ ) τ < 0
⎧⎨⎪
⎩⎪A(τ ) = A(−iτ ) = eHτAe−Hτ
Definition of temperature GF
0 < τ ≤ β G(τ − β) = −G(τ )Definition of Matsubara GF for
For we find the symmetry relation
G(ων ) = dτeiωντG(τ )0
β
∫Symmetry implies
ων =πβ
2ν = −1 for bosons 2ν +1( ) = +1 for electrons
⎧⎨⎪
⎩⎪
G(τ ) = 1β
e− iωντ G(ων )ν∑
ν = 0,±1,±2,…
−β < τ ≤ β
SPECTRAL REPRESENTATION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − Em + En )
nm∑
G(ων ) = dx A(x)iων − x−∞
∞
∫Matsubara GF can be represented as
SAME function as before
G(ων ) = G(iων )Matsubara GF can be found from G as
f (ω )
SPECTRAL REPRESENTATION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − Em + En )
nm∑
G(ων ) = dx A(x)iων − x−∞
∞
∫Matsubara GF can be represented as
SAME function as before
G(ων ) = G(iων )Matsubara GF can be found from G as
f (ω )
Inverse question: Can we determine G from MGF?
No, since has the same MGFG '(ω ) = G(ω ) + (1+ eβω ) f (ω )G(ων ) = G(iων ) = G '(iων ) for an arbitrary analytical function
Unique definition through condition limω →∞
G(ω ) ~ 1 /ω
THE SPECTRAL FUNCTION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − En + Em )
nm∑ → G(ω ) = dx A(x)
ω − x−∞
∞
∫
THE SPECTRAL FUNCTION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − En + Em )
nm∑ → G(ω ) = dx A(x)
ω − x−∞
∞
∫
G(ων ) = G(iων ) = dx A(x)iων − x−∞
∞
∫
THE SPECTRAL FUNCTION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − En + Em )
nm∑ → G(ω ) = dx A(x)
ω − x−∞
∞
∫
G(ων ) = G(iων ) = dx A(x)iων − x−∞
∞
∫
GR (E) = G(E + iδ ) = dx A(x)E − x + iδ−∞
∞
∫
THE SPECTRAL FUNCTION
A(x) = 1+ e−βω
Ze−βEn AnmBmnδ (x − En + Em )
nm∑ → G(ω ) = dx A(x)
ω − x−∞
∞
∫
G(ων ) = G(iων ) = dx A(x)iων − x−∞
∞
∫
GR (E) = G(E + iδ ) = dx A(x)E − x + iδ−∞
∞
∫ GA (E) = G(E − iδ ) = dx A(x)E − x − iδ−∞
∞
∫
CONSEQUENCES OF ANALYTICAL PROPERTIES
Kramers-Kronig relations (from analycity of G in upper half-plane)
ReGR (E) = 1πP dE ' ImG( ′E )
′E − E−∞
∞
∫ ImGR (E) = −1πP dE ' ReG( ′E )
′E − E−∞
∞
∫Real and imaginary parts of response functions are related
CONSEQUENCES OF ANALYTICAL PROPERTIES
Kramers-Kronig relations (from analycity of G in upper half-plane)
ReGR (E) = 1πP dE ' ImG( ′E )
′E − E−∞
∞
∫ ImGR (E) = −1πP dE ' ReG( ′E )
′E − E−∞
∞
∫Real and imaginary parts of response functions are related
Fluctuation-Dissipation relation (for Bose operators and A=B)
ImGR (E) = π2e−βE −1( ) dteiEt B(t)B†
−∞
∞
∫
CONSEQUENCES OF ANALYTICAL PROPERTIES
Kramers-Kronig relations (from analycity of G in upper half-plane)
ReGR (E) = 1πP dE ' ImG( ′E )
′E − E−∞
∞
∫ ImGR (E) = −1πP dE ' ReG( ′E )
′E − E−∞
∞
∫Real and imaginary parts of response functions are related
Dissipation (c.f. exercise)
Fluctuation-Dissipation relation (for Bose operators and A=B)
ImGR (E) = π2e−βE −1( ) dteiEt B(t)B†
−∞
∞
∫
CONSEQUENCES OF ANALYTICAL PROPERTIES
Kramers-Kronig relations (from analycity of G in upper half-plane)
ReGR (E) = 1πP dE ' ImG( ′E )
′E − E−∞
∞
∫ ImGR (E) = −1πP dE ' ReG( ′E )
′E − E−∞
∞
∫Real and imaginary parts of response functions are related
Dissipation (c.f. exercise)
Fluctuation-Dissipation relation (for Bose operators and A=B)
ImGR (E) = π2e−βE −1( ) dteiEt B(t)B†
−∞
∞
∫Fluctuations (spectral density)
B.3 SINGLE PARTICLE GREEN’S FUNCTION
Definition: G(r ,τ; ′r , ′τ ) = − T Ψ(r ,τ ) Ψ(r ',τ ')( )Ψ(r ,τ ) = eHτΨ(r )e−Hτ Ψ(r ,τ ) = eHτΨ†(r )e−Hτ
Spectral representation
A(r , ′r ; x) = 1+ e−βω
Ze−βEmΨnm (
r )Ψmn† (r ')δ (x − Em + En )
nm∑
G(r , ′r ;ων ) = dτeiωντG(r , ′r ;τ )0
β
∫ = dx A(r , ′r , x)
iων − x−∞
∞
∫
MOMENTUM REPRESENTATION
G(k ,τ ) = − T ck (τ )ck( )
A(k , x) = 1+ e
−βx
Ze−βEm cknm
2δ (x − Em + En )
nm∑
G(k ,ων ) = dτeiωντG(
k ,τ )
0
β
∫ = dx A(k , x)
iων − x−∞
∞
∫
Density of states
N(E,k ) = −
1πImGR (
k ,E) = A(
k ,E)
N(E) = −1πIm GR (
k ,E)
k∑ = A(
k ,E)
k∑
G(k ,ων ) =
1iων − k
Free particles
N(E,k ) = δ (k − E)
WICK’S THEOREM
G0(n) (1,2,…,n;1',2 ',…,n ') =
−( )i−1G0 (1,i ')G0(n−1)(1,2,,n;1',, i ' ,,n ')
i=1
n
∑
G0(2)(1,2;1',2 ') = G0 (1,1')G0 (2,2 ')
−G0 (1,2 ')G0 (1',2)
Recursion relation for GF of noninteracting particles
Example:
WICK’S THEOREM II
For Fermions
G0(n) (1,2,…,n;1',2 ',…,n ') =
G0 (1,1') G0 (1,2 ') ... G0 (1,n ')G0 (2,1') G0 (2,2 ')
G0 (n,1') ... ... G0 (n,n ')
B. FORMAL MATTERS - SUMMARY
1. Definitions of different GFs: retarded, advanced, causal, temperature (Wick rotation, imaginary time)
2. Analytical properties, all GF determined by the same spectral function A(x), Matsubara GF and frequencies
3. Single particle Green’s function, density of states, relation to observables, quasiparticles as poles of the 1P-GF
4. Equation of motion: 1P-GF obeys differential equation like ordinary GF, interaction couples 1P-GF, 2P-GF,..., NP-GF
5. Wicks theorem: NP-GF for non-interacting particles can be decomposed into products of 2P-GF
C. DIAGRAMMATIC METHODS
Systematic pertubation theory in external
potential, two-particle interaction or/and electron-
phonon interaction
Symbolic language in terms of Feynman diagrams
enables efficient algebraic manipulations
C. DIAGRAMMATIC METHODS
1.Single-particle potential
2. Interacting particles
3.Translational invariant problems
4. Selfenergy and Correlations
5. Screening and random phase approximation (RPA)
C. 1 SINGLE-PARTICLE POTENTIAL
x= +
xDyson equation:
G(1,2) G0 (1,2) U(1)
G(1,2) = G0 (1,2) + d∫ 3G0 (1, 3)U(3)G(3,2)
Dictionary:
C. 1 SINGLE-PARTICLE POTENTIAL
x= +
xDyson equation:
G(1,2) G0 (1,2) U(1)
G(1,2) = G0 (1,2) + d∫ 3G0 (1, 3)U(3)G(3,2)Rules for order-n contribution:
• Draw all topologically distinct and connected diagrams with 2 external and n internal vertices
• Associate a lines with G0 and each internal vertex with U and integrate over all internal coordinates
Dictionary:
C. 2 TWO-PARTICLE INTERACTION
H = H0 +VS0 (τ ) = T exp − H0 (τ ')0
τ
∫⎡⎣⎢
⎤⎦⎥
VI (τ ) = S0−1(τ )VS0 (τ )
SI (τ ) = T exp − VI (τ ')0
τ
∫⎡⎣⎢
⎤⎦⎥
ready for expansion....
C. 2 TWO-PARTICLE INTERACTION
H = H0 +VS0 (τ ) = T exp − H0 (τ ')0
τ
∫⎡⎣⎢
⎤⎦⎥
VI (τ ) = S0−1(τ )VS0 (τ )
SI (τ ) = T exp − VI (τ ')0
τ
∫⎡⎣⎢
⎤⎦⎥
G(1,2) =T SI (β) ψ I (1) ψ I (2)⎡⎣ ⎤⎦ 0
T SI (β)[ ] 0
Expression for 1P-GF in interaction picture
ready for expansion....
C. 2 TWO-PARTICLE INTERACTION
G(1,2) G0 (1,2)Dictionary: −V (1,2)
++=
C. 2 TWO-PARTICLE INTERACTION
G(1,2) G0 (1,2)
Rules for order-n contribution:• Draw all topologically distinct and connected diagrams
with 2 external and 2n internal vertices• Connect all vertices with direct sold lines and all internat
vertices with dashed lines• Integrate over all internal coordinates• ...
Dictionary: −V (1,2)
++=
C. 3 TRANSLATIONAL INVARIANCE
++=
G(k,ων ) G0 (k,ων )−V (q)
C. 3 TRANSLATIONAL INVARIANCE
Feynman rules for order-n contribution im momentum and frequency space:
• Draw the same diagrams as in real space• Associate lines with Fourier-transformed GF and V• Momentum conservation at vertices• Sum over internal momenta and frequencies• ...
++=
G(k,ων ) G0 (k,ων )−V (q)
C. 4 SELFENERGY AND CORRELATIONS
G = G0 +G0ΣG
Summing all diagram which can be cut by cutting a single line
G =1
G0−1 − ΣPhysical implications
• real part of selfenergy implies energy shifts• imaginary part leads to finite lifetimeHartree-Fock approximations• leads to (diverging) real self energy• neglects correlations in 2P-GF
= +
Dyson equation
C. 5 SCREENING AND RANDOM PHASE APPROXIMATION (RPA)
Dressing the interaction
Physical implications• screening of the interaction by creation of electron-hole pairs• equivalent diagram occurs in charge response to external potential -> relation to dielectric functionConsequences: Finite life time & Thomas-Fermi screening
= +
Polarization bubble
Π(k,ων )V (q,ων ) =
V0 (q)1− Π(q,ων )V0 (q)
C. DIAGRAMMATIC METHODS
Systematic pertubation theory for Greens functions in external potential, two-particle interaction or/and electron-phonon interactionSymbolic language in terms of Feynman diagrams enables efficient algebraic manipulationsSelfenergy leads to an improved approximation by summing infinite series of certain diagram classesHartree-Fock diagrams provide a first rough approximationScreening of the interaction and finite lifetime in the RPA
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