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