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Quantum Field Theory in Condensed Matter Physics

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Quantum Field Theory in Condensed Matter Physics QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS Wolfgang Belzig FORMAL MATTERS “Wahlpflichtfach” 4h lecture + 2h exercise Lecture: Mon & Thu, 10-12, P603 Tutorial: Mon, 14-16 (P912) or 16-18 (P712) 50% of exercises needed for exam Language: “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 Matter H. Bruus and K. Flensberg: Many-Body Quantum Theory in Condensed Matter Physics J. Rammer: Quantum Field Theory of Non-equilibium States G. Mahan: Quantum Field Theoretical Methods Fetter & Walecka: Quantum Theory of Many Particle Systems Yu. V. Nazarov & Ya. Blanter: Quantum Transport J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986) CONTENT A. Introduction, the Problem B. Formalities, Definitions C. Diagrammatic Methods D. Disordered Conductors E. Electrons and Phonons F. Superconductivity G. Quasiclassical Methods H. Nonequilibrium and Keldysh formalism I. 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 ( Ψ ( x ) , c k , ... ) 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 G(x, x') = Ψ(x)Ψ†(x′) Physical observable are obtained directly from the Greens functions, e.g. density 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: Exponential decay t / n ~ e− τ ≈ 1− t / τ goes negative Critical temperature −1/ NV not expandable in V kBTc ~ ω ce CONTENT A. Introduction, the Problem B. Formalities, Definitions C. Diagrammatic Methods D. Disordered Conductors E. Electrons and Phonons F. Superconductivity G. Quasiclassical Methods H. Nonequilibrium and Keldysh formalism I. 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 2 ⎛ P j ⎞ Ions in the H = ∑⎜ + U j (Rj )⎟ harmonic potential j ⎝ 2M j ⎠ 2 Electrons in the ⎛ pi ⎞ +∑⎜ + Ui (ri )⎟ periodic potential i ⎝ 2m ⎠ 1 e2 + + Vij Rj ,ri 2 ∑ r r ∑ ( ) i≠ j i − j ij Electron-electron and Elektron-phonon interaction THE HAMILTONIAN Ions in the ⎛ † 1⎞ H = ω qλ aqλaqλ + harmonic ∑ ⎜ ⎟ qλ ⎝ 2⎠ potential † + c c Electrons in the ∑ kσ kσ kσ periodic potential kσ † † + ck +qσ ck '−qσ 'V(q)ck 'σ 'ckσ ∑ kk′qσσ ′ Electron-electron and Electron-phonon † † interaction + gqλck +qσ ckσ a−qλ + aqλ ∑ ( ) kqσλ 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 Problem B. Formal Matters C. Diagrammatic Methods D. Disordered Conductors E. Electrons and Phonons F. Superconductivity G. Quasiclassical Methods H. Nonequilibrium and Keldysh formalism I. 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 functions 5.Wicks theorem 1. DEFINITIONS Two-time Greens functions for two operators A and B Retarded Greens function R i ˆ ˆ G (t,t′) = − ⎡AH (t), BH (t′)⎤ θ(t − t′) ⎣ ⎦ 1 ˆ ˆ = Tr(ρˆ) ρˆ = e−β(H − µN ) Z ⎪⎧ −1 Bose operators ⎡Aˆ, Bˆ ⎤ = AˆBˆ + BˆAˆ = ⎨ ⎣ ⎦ +1 Fermi operators ⎩⎪ † † ˆ ˆ ˆ Bose: aˆk , aˆk , ρˆ(r ) = ψˆ (r )ψˆ (r ), j(r ), H, N † † Fermi: cˆk , cˆk , ψˆ (r ), ψˆ (r ) 1. DEFINITIONS Temperature Greens function t → −iτ Wick rotation ˆ ˆ Hτ ˆ − Hτ AH (t) → A(−iτ ) = A(τ ) = e Ae 1 G(τ,τ′) = − T Aˆ(−iτ )Bˆ(−iτ′) ( ) Time-ordering operator T (A(τ )B(τ′)) = A(τ )B(τ′)θ(τ − τ′) − B(τ′)A(τ )θ(τ '− τ ) “Later times to the left” Higher-order Greens function G(τ1,τ 2 ,τ 3,…) ~ T (A(τ1 )B(τ 2 )C(τ 3 )) 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 functions 5.Wicks theorem MATSUBARA GREEN’S FUNCTION Definition of temperature GF ⎧ ⎪ − A(τ )B τ > 0 A(τ ) = Aˆ(−iτ ) = eHτ Ae− Hτ G(τ ) = ⎨ ⎪ BA(τ ) τ < 0 ⎩ For 0 < τ ≤ β we find the symmetry relation G(τ − β) = −G(τ ) Definition of Matsubara GF for −β < τ ≤ β β 1 −iω τ iωντ ν G (ω ) = dτe G(τ ) G(τ ) = e G(ων ) ν ∫ β ∑ 0 ν Symmetry implies π ⎪⎧ 2ν = −1 for bosons ων = ⎨ ν = 0,±1,±2,… 2ν + 1 = +1 for electrons β ⎩⎪ ( ) SPECTRAL REPRESENTATION ∞ A(x) G (ω ) = dx Matsubara GF can be represented as ν ∫ −βω iω − x 1 e −∞ ν + −βEn SAME function as before A(x) = ∑e Anm Bmnδ(x − Em + En ) Z nm Matsubara GF can be found from G as G(ων ) = G(iων ) Inverse question: Can we determine G from MGF? No, since G '(ω) = G(ω) + (1+ eβω ) f (ω) has the same MGF f ( ) G(ων ) = G(iων ) = G '(iων ) for an arbitrary analytical function ω Unique definition through condition lim G(ω) ~ 1 / ω ω →∞ THE SPECTRAL FUNCTION 1+ e−βω ∞ A(x) A(x) = e−βEn A B δ(x − E + E ) → G(ω) = dx ∑ nm mn n m ∫ Z nm −∞ ω − x ∞ ∞ R A(x) A(x) G (E) = G(E + iδ) = ∫ dx G A (E) = G(E − iδ) = dx −∞ E − x + iδ ∫ −∞ E − x − iδ ∞ A(x) G (ω ) = G(iω ) = dx ν ν ∫ −∞ iων − x CONSEQUENCES OF ANALYTICAL PROPERTIES Kramers-Kronig relations (from analycity of G in upper half-plane) 1 ∞ ImG(E′) 1 ∞ ReG(E′) ReG R (E) = P ∫ dE ' ImG R (E) = − P ∫ dE ' π −∞ E′ − E π −∞ E′ − E Real and imaginary parts of response functions are related Fluctuation-Dissipation relation (for Bose operators and A=B) π ∞ ImG R (E) = (e−βE − 1) ∫ dteiEt B(t)B† 2 −∞ Dissipation (c.f. exercise) Fluctuations (spectral density) B.3 SINGLE PARTICLE GREEN’S FUNCTION Definition: G(r,τ;r′,τ′) = − T (Ψ (r,τ )Ψ (r ',τ ')) H † H Ψ (r,τ ) = eHτ Ψ(r )e− Hτ Ψ (r,τ ) = e τ Ψ (r )e− τ Spectral representation β ∞ A(r,r′, x) G (r,r′;ω ) = dτeiωντ G(r,r′;τ ) = dx ν ∫ ∫ 0 −∞ iων − x 1 e−βω + −βEm † A(r,r′; x) = ∑e Ψnm (r )Ψ mn (r ')δ(x − Em + En ) Z nm MOMENTUM REPRESENTATION G(k,τ ) = − T c (τ )c ( k k ) Free particles β ∞ A(k, x) 1 iωντ G(k,ων ) = dτe G(k,τ ) = dx G(k,ων ) = ∫ ∫ i 0 −∞ iων − x ων − k −βx 1+ e −βE 2 A(k, x) e m c (x E E ) = ∑ knm δ − m + n Z nm Density of states 1 R N(E,k) = − ImG (k,E) = A(k,E) N(E,k) = δ( − E) π k 1 N(E) = − Im G R (k,E) = A(k,E) ∑ ∑ π k k WICK’S THEOREM Recursion relation for GF of noninteracting particles (n) G0 (1,2,…,n;1',2',…,n') = n i−1 (n−1) ∑(−) G0 (1,i ')G0 (1,2,,n;1',, i',,n') i=1 (2) Example: G0 (1,2;1',2') = G0 (1,1')G0 (2,2') −G0 (1,2')G0 (1',2) WICK’S THEOREM II For Fermions (n) G0 (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.
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