QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS. Wolfgang Belzig

Transcription

1 QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS Wolfgang Belzig

2 FORMAL MATTERS Wahlpflichtfach 4h lecture + 2h exercise Lecture: Mon & Thu, 10-12, P603 Tutorial: Mon, (P912) or (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

3 EXERCISE GROUPS Two 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: Milena Filipovic; 2.5. Martin Bruderer; 9.5. Fei Xu Cecilia Holmqvist; Peter Machon

4 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)

5 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

6 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...)

7 THE THEORETICAL PROBLEM Due to the huge number of degrees of freedom (M states, N particles: M N ), 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 Physical observable are obtained directly from the Greens functions, e.g. density G(x, x') = Ψ(x)Ψ ( x ) n(x) = G(x, x)

8 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 n ~ e t /τ 1 t / τ goes negative Critical temperature k B T c ~ ω c e 1/ NV not expandable in V

9 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

10 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, 2 nd quantization, field operators, electron-phonon interaction, single particle potentials)

11 THE HAMILTONIAN H = j P 2 j + U ( R ) 2M j j j Ions in the harmonic potential + i p 2 i 2m + U i( r ) i Electrons in the periodic potential e 2 ( ) r r + V ij Rj, r i i j i j ij Electron-electron and Elektron-phonon interaction

12 THE HAMILTONIAN H = ω a a + 1 qλ qλ qλ 2 qλ + kσ c kσ c kσ kσ + c k + qσ c k ' qσ ' V( q)c k 'σ ' c kσ kk qσ σ ( ) + g qλ c k + qσ c kσ a qλ + a qλ k qσλ Ions in the harmonic potential Electrons in the periodic potential Electron-electron and Elektron-phonon interaction

13 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

14 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

15 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

16 1. DEFINITIONS Two-time Greens functions for two operators A and B Retarded Greens function G R (t, t ) = i  (t), ˆB ( t ) H H θ(t t ) = Tr( ˆρ ) ˆρ = 1 Z e β (Ĥ µ ˆN ) Â, ˆB =  ˆB + ˆB = Bose: â k, â k, ˆρ( r ) = ˆ ψ ( r ) ˆ ψ ( r ), Fermi: ĉ k, ĉ k, ˆ ψ ( r ), ˆ ψ ( r ) 1 Bose operators +1 Fermi operators ĵ( r ), Ĥ, ˆN

17 1. DEFINITIONS Temperature Greens function t iτ Wick rotation  H (t) Â( iτ ) = A(τ ) = e Hτ Âe Hτ Time-ordering operator Higher-order Greens function ( ) G(τ, τ ) = 1 T Â( iτ ) ˆB( i τ ) T ( A(τ ) B( τ )) = A(τ ) B( τ )θ(τ τ ) B( τ ) A(τ )θ(τ ' τ ) Later times to the left G(τ,τ,τ, ) ~ T ( A(τ ) B(τ ) C(τ ) )

18 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

19 MATSUBARA GREEN S FUNCTION Definition of temperature GF For A(τ ) = Â( iτ ) = ehτ Ae Hτ G(τ ) = A(τ ) B τ > 0 B A(τ ) τ < 0 0 < τ β G(τ β) = G(τ ) we find the symmetry relation Definition of Matsubara GF for β G(ω ν ) = dτe iω ντ G(τ ) Symmetry implies ν = 0, ±1, ±2, 0 ω ν = π β β < τ β 2ν 2ν + 1 G(τ ) = 1 β ν e iω ντ G(ω ν ) = 1 for bosons ( ) = +1 for electrons

20 SPECTRAL REPRESENTATION Matsubara GF can be represented as SAME function as before A(x) = 1+ e βω Matsubara GF can be found from G as Z G(ω ν ) = dx A(x) nm iω ν x e βe n A nmb mn δ(x E m + E n ) G(ω ν ) = G(iω ν ) f (ω)

21 SPECTRAL REPRESENTATION Matsubara GF can be represented as SAME function as before A(x) = 1+ e βω Matsubara GF can be found from G as Z G(ω ν ) = dx A(x) nm iω ν x e βe n A nmb mn δ(x E m + E n ) G(ω ν ) = G(iω ν ) Inverse question: Can we determine G from MGF? G '(ω) = G(ω) + (1 + e βω ) f (ω) No, since has the same MGF G(ω ν ) = G(iω ν ) = G '(iω ν ) for an arbitrary analytical function f (ω) Unique definition through condition lim ω G(ω) ~ 1 / ω

22 THE SPECTRAL FUNCTION A(x) = 1+ e βω Z e βe n A nmb δ(x E + E ) G(ω) = dx A(x) mn n m ω x nm

23 THE SPECTRAL FUNCTION A(x) = 1+ e βω Z e βe n A nmb δ(x E + E ) G(ω) = dx A(x) mn n m ω x nm G(ω ν ) = G(iω ν ) = dx A(x) iω ν x

24 THE SPECTRAL FUNCTION A(x) = 1+ e βω Z e βe n A nmb δ(x E + E ) G(ω) = dx A(x) mn n m ω x nm G R (E) = G(E + iδ ) = dx A(x) E x + iδ G(ω ν ) = G(iω ν ) = dx A(x) iω ν x

25 THE SPECTRAL FUNCTION A(x) = 1+ e βω Z e βe n A nmb δ(x E + E ) G(ω) = dx A(x) mn n m ω x nm G R (E) = G(E + iδ ) = dx A(x) G A (E) = G(E iδ ) = dx E x + iδ A(x) E x iδ G(ω ν ) = G(iω ν ) = dx A(x) iω ν x

26 CONSEQUENCES OF ANALYTICAL PROPERTIES Kramers-Kronig relations (from analycity of G in upper half-plane) ReG R (E) = 1 ImG( P de ' E ) ImG R (E) = 1 ReG( P de ' E ) π E E π E E Real and imaginary parts of response functions are related

27 CONSEQUENCES OF ANALYTICAL PROPERTIES Kramers-Kronig relations (from analycity of G in upper half-plane) ReG R (E) = 1 ImG( P de ' E ) ImG R (E) = 1 ReG( P de ' E ) π E E π E E Real and imaginary parts of response functions are related Fluctuation-Dissipation relation (for Bose operators and A=B) ImG R (E) = π 2 e βe 1 ( ) dte iet B(t)B

28 CONSEQUENCES OF ANALYTICAL PROPERTIES Kramers-Kronig relations (from analycity of G in upper half-plane) ReG R (E) = 1 ImG( P de ' E ) ImG R (E) = 1 ReG( P de ' E ) π E E π E E Real and imaginary parts of response functions are related Fluctuation-Dissipation relation (for Bose operators and A=B) ImG R (E) = π 2 e βe 1 Dissipation (c.f. exercise) ( ) dte iet B(t)B

29 CONSEQUENCES OF ANALYTICAL PROPERTIES Kramers-Kronig relations (from analycity of G in upper half-plane) ReG R (E) = 1 ImG( P de ' E ) ImG R (E) = 1 ReG( P de ' E ) π E E π E E Real and imaginary parts of response functions are related Fluctuation-Dissipation relation (for Bose operators and A=B) ImG R (E) = π 2 e βe 1 Dissipation (c.f. exercise) ( ) dte iet B(t)B Fluctuations (spectral density)

30 B.3 SINGLE PARTICLE GREEN S FUNCTION Definition: G( r,τ; r, τ ) = T ( Ψ( r,τ ) Ψ( r ) ',τ ') Ψ( r,τ ) = e Hτ Ψ( r )e Hτ Ψ( r,τ ) = e Hτ Ψ ( r )e Hτ Spectral representation β G( r, r ;ω ) = dτe iωντ G( r, r ;τ ) ν = dx A( r, r, x) iω x ν 0 A( r, r ; x) = 1 + e βω Z nm e βe m Ψ nm ( r )Ψ ( r ')δ(x E + E ) mn m n

31 MOMENTUM REPRESENTATION G( k,τ ) = T ( c k (τ ) c k ) β G( k,ω ) = dτe iωντ G( k,τ ) ν = dx A( k, x) iω x ν 0 Free particles G( 1 k,ω ) = ν iω ν k A( k, 1+ e βx x) = Z Density of states nm e βe m c knm 2 δ(x Em + E n ) N(E, k) = 1 π ImG R ( k, E) = A( k, E) N(E, k) = δ( k E) N(E) = 1 π Im G R ( k,e) = A( k,e) k k

32 WICK S THEOREM Recursion relation for GF of noninteracting particles G 0 (n) (1,2,,n;1',2',,n') = n i=1 ( ) i 1 G (1,i ')G (n 1) (1,2,,n;1',, i',,n') 0 0 Example: G 0 (2) (1,2;1',2') = G 0 (1,1')G 0 (2,2') G 0 (1,2')G 0 (1',2)

33 WICK S THEOREM II For Fermions G 0 (n) (1,2,,n;1',2',,n') = G (1,1') 0 G (1,2') 0... G (1,n') 0 G (2,1') 0 G (2,2') 0 G (n,1') G (n,n') 0

34 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

35 C. DIAGRAMMATIC METHODS Systematic pertubation theory in external potential, two-particle interaction or/and electronphonon interaction Symbolic language in terms of Feynman diagrams enables efficient algebraic manipulations

36 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)

37 C. 1 SINGLE-PARTICLE POTENTIAL Dictionary: G(1,2) G 0 (1,2) U(1) x Dyson equation: = + x G(1,2) = G 0 (1,2) + d 3G 0(1, 3)U(3)G(3,2)

38 C. 1 SINGLE-PARTICLE POTENTIAL Dictionary: G(1,2) G 0 (1,2) U(1) x Dyson equation: = + x G(1,2) = G 0 (1,2) + d 3G 0(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

39 C. 2 TWO-PARTICLE INTERACTION H = H 0 + V S (τ ) = T exp 0 τ 0 H (τ ') 0 V I (τ ) = S 0 1 (τ )VS 0 (τ ) S (τ ) = T exp I τ 0 V (τ ') I ready for expansion...

40 C. 2 TWO-PARTICLE INTERACTION H = H 0 + V V I (τ ) = S 0 1 (τ )VS 0 (τ ) S (τ ) = T exp 0 τ 0 H (τ ') 0 Expression for 1P-GF in interaction picture S (τ ) = T exp I τ 0 V (τ ') I G(1,2) = T S (β) ψ (1) ψ (2) I I I 0 [ ] 0 T S I (β) ready for expansion...

41 C. 2 TWO-PARTICLE INTERACTION Dictionary: G(1,2) G 0 (1,2) V(1,2) = + +

42 C. 2 TWO-PARTICLE INTERACTION Dictionary: G(1,2) G 0 (1,2) V(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... = + +

43 C. 3 TRANSLATIONAL INVARIANCE G(k,ω ν ) G 0 (k,ω ν ) V(q) = + +

44 C. 3 TRANSLATIONAL INVARIANCE G(k,ω ν ) G 0 (k,ω ν ) V(q) 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... = + +

45 C. 4 SELFENERGY AND CORRELATIONS Summing all diagram which can be cut by cutting a single line = + G = G 0 + G 0 ΣG Dyson equation G = Physical implications real part of selfenergy implies energy shifts imaginary part leads to finite lifetime 1 G 0 1 Σ Hartree-Fock approximations leads to (diverging) real self energy neglects correlations in 2P-GF

46 C. 5 SCREENING AND RANDOM PHASE APPROXIMATION (RPA) Dressing the interaction = + Π(k,ω ν ) Polarization bubble V(q,ω ν ) = V 0 (q) 1 Π(q,ω ν )V 0 (q) Physical implications screening of the interaction by creation of electron-hole pairs equivalent diagram occurs in charge response to external potential -> relation to dielectric function Consequences: Finite life time & Thomas-Fermi screening

47 C. DIAGRAMMATIC METHODS Systematic pertubation theory for Greens functions in external potential, two-particle interaction or/and electron-phonon interaction Symbolic language in terms of Feynman diagrams enables efficient algebraic manipulations Selfenergy leads to an improved approximation by summing infinite series of certain diagram classes Hartree-Fock diagrams provide a first rough approximation Screening of the interaction and finite lifetime in the RPA