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

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

Definition of temperature GF ⎧   ⎪ − A(τ )B τ > 0 A(τ ) = Aˆ(−iτ ) = eHτ Ae− Hτ G(τ ) = ⎨ ⎪  BA(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. 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

G(1,2) G (1,2) U(1) Dictionary: 0 x Dyson equation: = + x G(1,2) = G (1,2) + d3G (1,3)U(3)G(3,2) 0 ∫ 0 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 C. 2 TWO-PARTICLE INTERACTION

V ( ) S−1( )VS ( ) H = H 0 + V I τ = 0 τ 0 τ ⎡ τ ⎤ ⎡ τ ⎤ S0 (τ ) = T exp − H 0 (τ ') SI (τ ) = T exp − VI (τ ') ⎣⎢ ∫0 ⎦⎥ ⎣⎢ ∫0 ⎦⎥ Expression for 1P-GF in interaction picture ⎡ ( )  (1)  (2)⎤ T ⎣SI β ψ I ψ I ⎦ G(1,2) = 0 T S (β) [ I ] 0 ready for expansion.... C. 2 TWO-PARTICLE INTERACTION

Dictionary: G(1,2) G0 (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 • ... = + + C. 3 TRANSLATIONAL INVARIANCE

G(k,ων ) G0 (k,ων )−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 • ... = + + C. 4 SELFENERGY AND CORRELATIONS

Summing all diagram which can be cut by cutting a single line = +

G = G0 + G0ΣG 1 Dyson equation G = −1 Physical implications G0 − Σ • real part of selfenergy implies energy shifts • imaginary part leads to ﬁnite lifetime Hartree-Fock approximations • leads to (diverging) real self energy • neglects correlations in 2P-GF C. 5 SCREENING AND RANDOM PHASE APPROXIMATION (RPA)

Dressing the interaction = +

Π(k,ων ) V0 (q) Polarization bubble V(q,ων ) = 1− Π(q,ων )V0 (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 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 CONTENT

1.Greens function in a random potential 2.Conductivity and polarization bubble 3.Vertex corrections and transport life time 4.Diffuson, Cooperon and weak localization 5.Localization (?) D.1 GREENS FUNCTION IN A RANDOM POTENTIAL

Electron motion in a random potential:

• randomly distributed impurities • semiclassical picture neglects wave nature • equivalent to Boltzmann-equation treatment (Drude conductivity), mean free path l • systematic expansion in wave nature: ~1/kFl D.1 GREENS FUNCTION IN A RANDOM POTENTIAL

Field theory method, Dyson equation: = + x G(1,2) = G (1,2) + d3G (1,3)U(3)G(3,2) 0 ∫ 0 • The impurities are randomly distributed and we have to average over the positions • This introduces correlations between scattering events, since scattering at different impurities averages out, but not at the same impurity! AVERAGING OVER RANDOM IMPURITY POSITIONS Averaging leads to correlations between scattering events at the same impurity G(1,2) = = + + +

+ + -> self energy + crossed lines SELF ENERGY FOR ELASTIC IMPURITY SCATTERING

Only diagrams, which cannot be “cut”, c.f. double counting Σ(1,2) = + + crossed lines Solving the Dyson equation 1 G(k,ω µ ) = −1 G0 (k,ω µ ) − Σ(k,ω µ )

′ k k ′ − − k k k,ω k′,ω k,ω Σ(k,ω µ ) = µ µ µ SELF ENERGY IN BORN APPROXIMATION

Self energy 1 Σ(k,ω µ ) = −i sgn(ω µ ) 2τ k Momentum life time (->mean free path) l = vFτ 1 2 = 2π Ni N(0) V(k − k 'F ) v′ τ k F

Impurity averaged 1 G R(A) (k,E) = Greens function i E − k + (−) 2τ k SOME CONSEQUENCES

Spectral density 1 1 A(k,E) = 2 2 2 πτ (E − k ) + (1 / 2τ ) Fourier trafo −π N(0) G R (r,E) = eikF r−r /2l kFr Density of states N(E) = N(0) SUMMARY D1: IMPURITY AVERAGED GREENS FUNCTION

1 G(k,ω ) = µ i iω µ − k + sgn(ω µ ) 2τ k = + = + Selfconsistent T-matrix Born approximation approximation THE FUTURE

Mo 13.6. (pentecost) no lecture, no exercise Th 16.6. exercise (Martin) Mo 20.6. 2x lecture Th 23.6. (corpus christi), no lecture Mo 27.6. lecture, exercise (Cecilia) Th 30.6. lecture Mo 4.7. lecture, exercise (Peter) Th 7.7. lecture Mo 11.7. lecture + exam preparation Th 14.7. no lecture D.2 CONDUCTIVITY AND TRANSPORT LIFE TIME

• Current as response to external electric ﬁeld • Linearization in electric ﬁeld - linear response • Use the Kubo formula to relate the conductivity to the two-particle Green’s function CURRENT AS LINEAR RESPONSE

 J E  ∂A   Conductivity α = σαβ β E = − ↔ E = iωA ∂t Current density operator  −ie −e j ⎡ † † ⎤ j (q) (2k q)c†c 1 = ⎣Ψ (∇Ψ) − (∇Ψ )Ψ⎦ ↔ 1 = ∑ + k k +q 2m 2m k e Electric current J(r) = j (r) − Ψ†(r)Ψ(r)A(r) 1 m e2 t J (r,t) = − n(r,t)A (r,t) − dt 'dr 'G jj (r,t;r ',t ')A (r ',t ') α α ∫ αβ β m −∞ e2 J (q,ω) = − nA (q,ω) − G jj (q,ω)A (q,ω) = −K (q,ω)A (q,ω) α m α αβ β αβ β CURRENT-CURRENT RESPONSE FUNCTION

2 jj e (2) − Gαβ (1,1') = (∇2′ − ∇1' )(∇2 − ∇1 )G (2,2';1,1') 2′→1′ 4m 2→1− Two-particle Greens function

G(2) (2,2';1,1') = − T Ψ (2)Ψ (2')Ψ (1')Ψ (1)

G(2) (2,2';1,1') = + ≈ EVALUATION OF CONDUCTIVITY

  Approximations (k ± q / 2) = (k) ± vFq / 2 Exchange of the order of frequency and momentum sum Current-current response function 2 jj 2e Gαβ (q,ων ) = − T ∑∑kα kβG(k + q / 2,ω µ )G(k − q / 2,ω µ + ων ) m µ k 2e2 = − T ∑ N(0)∫ d × m ν

k k 1  Fα Fβ   vFq i vFq i iω µ −  + + sgn(ω µ ) i ω µ + ων −  − + sgn(ω µ + ων ) ( )  2 2τ 2 2τ vF BOLTZMANN CONDUCTIVITY

2  3ne vFα vFβ  Jα (q,ω) = i   Eβ (q,ω) mv2 ω − v q + i / τ FF   ω  EF σ (q,ω ) Validity αβ q  k Result coincides with Boltzmann equation treatment F 1 For slowly varying ﬁelds: vFq  (ql  1) τ σ ne2τ Drude conductivity σ = 0 σ = 1− iωτ 0 m D.3 VERTEX CORRECTION AND TRANSPORT LIFE-TIME

So far neglegted: Captured in Vertex correction:

Integral equation for Vertex function: = + 2k + q Γ(k,q) INTEGRAL EQUATION

= +

2 q q (k,q) 2k q N V(k k ') G(k ' , )G(k ' , ) (k ',q) Γα = ( + )α + i ∑ − + ω µ + ων − ω µ Γα k ' 2 2

• Integral equation for the vertex function Γ(k,q) • Vertex function is a vector • Vertex function Γ ( k , q ) depends parametrically on frequencies • Equivalent to Boltzmann equation HOMOGENEOUS PART

depends weakly on k depend only on  q = 0 k′ 2 Γα (k) = 2kα + Ni ∑ V(k − k ') G(k ',ω µ + ων )G(k ',ω µ )Γα (k ') k ' We make the Ansatz Γ (k) = γ k γ =independent of k  α α Multiplying with k   2 k ⋅ k′ 2 N V(k k ') G(k ', )G(k ', )  γ = + γ i ∑ − ω µ + ων ω µ 2 k ' k Close to Fermienergy   2 k ⋅ k′ γ = 2 + γ N N(0) V(k − k ' )  F d G(k ',ω + ω )G(k ',ω ) i F 2 ∫ k′ µ ν µ k v′ F 1/2πτ′ SOLVING THE EQUATION

θ(−ω )θ(ω + ω ) d G(k ',ω + ω )G(k ',ω ) = 2πi µ µ ν c.f. D.2 ∫ k′ µ ν µ iων + i / τ

i θ(−ω )θ(ω + ω ) i θ(−ω µ )θ(ω µ + ων ) γ = 2 + γ µ µ ν γ − 2 = τ′ iω + i / τ τ′ ⎛ 1 1 ⎞ ν iων + i⎜ − ⎟ ⎝ τ τ ⎠ ′ 1 τtr   1   2 ⎛ k ⋅ k′ ⎞ Transport lifetime: 2 N N(0) V(k k ' ) 1  F = π i − F ⎜ − 2 ⎟ τ tr ⎝ k ⎠ ′ vF Weighted average of scattering processes (backscattering contributes stronger) CORRECTION TO DRUDE CONDUCTIVITY

2 3 2e kF γ / 2 1 σ(ω) = N(0) = σ tr m 3 ω + i / τ 1− iωτ tr • Vertex correction leads to a different scattering time in the conductivity than the momentum life-time in the single-particle Greens function

For s-wave scattering • V(k − k ') = V0 = const.   k ⋅ k′ ′ = 0 τ tr = τ vF Vertex correction important for anisotropic problems D.4 DIFFUSON, COOPERON AND WEAK LOCALIZATION

How does diffusive motion appear in the quantum theory? Diffusion ladder (Diffuson) How does quantum interference change the diffusion picture? Maximally crossed diagrams (Cooperon) What is the effect on the conductivity? Reduced conductivity (weak localization) SUMMARY: D. DISORDERED CONDUCTORS

Scattering at randomly distributed leads to correlations between scattering events G(1,2) = = + + +

+ + -> self energy + crossed lines SUMMARY: D. DISORDERED CONDUCTORS

1P-GF contains momentum lifetime deﬁned through t-matrix 1 G(k,ω ) = µ i iω −  + sgn(ω ) µ k 2τ µ = + k = + Selfconsistent T-matrix Born approximation approximation D. CONDUCTIVITY AND VERTEX CORRECTION

Conductivity: Quasiclassical (Drude) σ σ(ω) = tr 1− iωτ tr

Integral equation for vertex correction: Γα (k) = + Small for isotropic scattering D. QUANTUM COOPERON

Ladder diagrams for Diffuson and Cooperon =D

C= D. QUANTUM CORRECTIONS

Diffuson describes classical diffusion ⎛ ∂ ∂2 ⎞ − D D(x,t) = δ(x − x′)δ(t − t′) ⎝⎜ ∂t 0 ∂x2 ⎠⎟ Cooperon describes weak localization to conductivity

Quantum correction • 3D: small correction • 2D: universal ~e^2/h • 1D: strong localisation gives access to dephasing time CONTENT

† 1 †       Hamiltonian H ph = ω λqaλqaλq Qλq = aλq + aλq ∑  ( ) λq 2ω λq

   1   iqR Field operators (displacement) ˆ  j u j = ∑Qλqeλ (q)e NM  Green’s function qλ  D (qq ; , ) 2   T ⎡Q  ( )Q  ( ')⎤ λλ′ ′ τ τ′ = ω λqω λ ' q ' ⎣ λq τ λ ' q ' τ ⎦    Free Phonons Dλλ′ (qq′;τ,τ′) = δλλ′δqq′ Dλ (q,τ )

 β 2ω λq  iωντ  (0)  Dλ (q,ων ) = dτe Dλ (q,τ ) Dλ (q,ων ) = 2 2 ∫0  ων + ω λq E.1 ELECTRON-PHONON INTERACTION HAMILTONIAN

† †       H = ω qλaqλaqλ + k ck ck k, ∑ ∑ σ σ σ ω µ qλ kσ q,ων † †        + ck +qσ ck '−qσ 'V(q)ck 'σ 'ckσ ∑ kk′qσσ ′ † †       k + q,ω µ + ων + gqλck +qσ ckσ a−qλ + aqλ ∑ ( ) kqσλ      Interaction constant gqλ = iqeλ (q) ρ0Ve− ph (q)  2Mω λq Screened Coulomb Ze2 V(q) = 2 2 interaction (Thomas-Fermi) q + qTF E.3 PERTURBATION THEORY FOR ELECTRONS ∞ ∞ Selfenergy R 2 ⎡ 1 1 ⎤ Σ (E) = ∫ d∫ dω α F(ω)⎢ + ⎥ 0 0 ⎣ E +  + ω + iδ E +  + ω − iδ ⎦ Combined coupling and density of states of the phonons 2 2 dΩk dΩk ' α F(ω) = N(0) g δ(ω − ω   ) ∑∫ ∫ kˆkˆ ' λ λk − k′ λ 4π 4π Effective mass m* ∂ ∞ 1 = 1+ λ λ = Σ R (E) = 2∫ dω α 2F(ω) m ∂E E =0 0 ω Finite life-time E 1 2 R 2π 2 3 = − ℑ⎣⎡Σ (E)⎦⎤ = ∫ dω α F(ω) ~ E τ e− ph (E) 1+ λ 1+ λ 0 F. S U P E R C O N D U C T I V I T Y

1.Cooper instability 2.Bardeen-Cooper-Schrieffer (BCS) theory 3.Quasiclassical approximation 4.Impurity scattering and dirty limit 5.Applications Phenomenology (discovery by H. Kamerlingh Onnes, 1911) • perfect conductor σ(ω = 0) = ∞ persistent current with decay-time >100000y • perfectly diamagnetic (for small external ﬁelds)

M S = −H → B = µ0 (H + M ) = 0 → χ = −1 • occurs below a critical temperature in some metals and ceramics Al (1.2K), Hg (4K), Nb (9K), MgB2 (40K), YBCO (92K),HBCCO (164K) F.1 COOPER INSTABILITY

Attractive interaction leads to instability of the Fermi sea

Response to amplitude of a pair b = c c q ∑ k k +q↓ k↑

k,ω µ k′,ω µ q,ω = 0 q,ω = 0

−k + q,−ω µ −k′ + q,−ω µ F.1 COOPER INSTABILITY

Simpliﬁed interaction: ⎧ 2 ⎪ V for ω , ω′ < ω g D ≈ ⎨ µ µ D ⎩⎪ 0 else Divergent building block for a certain critical temperature 1 1 = VT G(k,ω )G(−k,−ω ) = VT c ∑ µ µ c ∑ ω 2 + 2 k, ωµ <ω D k, ωµ <ω D µ k 1 − Solution: N (0)V Tc = 1.13ω De • The Fermi sea becomes unstable at the critical temperature! • Finite Tc for arbitrarily small interaction! F. 2 B C S T H E O RY

Mean ﬁeld approximation for the interaction term † † c↑c↓c↓c↑ = c†c† c c + c†c† − c†c† c c + c†c† c c − c c + c†c† − c†c† c c − c c ↑ ↓ ↓ ↑ (↑↓↑ ↓) ↓ ↑ ↑ ↓ (↓↑↓ ↑) (↑↓↑↓ )(↓ ↑ ↓ ↑) δ δ ~δ 2 neglected † † † † † † ≈ c↑c↓ c↓c↑ + c↑c↓ c↓c↑ − c↑c↓ c↓c↑

Mean ﬁeld hamiltonian c c = b δ −k +q↓ k↑ k q,0 Δ = V∑bk k H  c† c c† c† c c * * b MF = ∑ k kσ kσ − ∑( k↑ −k↓Δ + −k↓ k↑ Δ ) + Δ ∑ k kσ k k

Quadratic hamiltonian, can be diagonalized by the method of “equation of motion” F. 2 B C S T H E O RY

⎛ c ⎞ † † k↑ Nambu formalism: ψ = c c ψ k = ⎜ ⎟ k ( k↑ −k↓ ) ⎜ c† ⎟ ⎝ −k↓ ⎠ Gorkov-Greens function: ⎛ ⎞ ck↑ (τ ) ⎛ G(k,τ ) F(k,τ ) ⎞ Gˆ (k,τ ) = − T ⎜ ⎟ ⊗ c (0) c (0) = ⎜ ⎟ ⎜ c (τ ) ⎟ ( k↑ −k↓ ) ⎜ −F*(k,τ ) G(k,τ ) ⎟ ⎝ −k↓ ⎠ ⎝ ⎠ ⎛ ⎞ k −Δ ⎛ 0 −Δ ⎞ Hamiltonian: Hˆ = ⎜ ⎟ Δˆ = ⎜ * ⎟ ⎜ * ⎟ ⎝ Δ −k ⎠ ⎝ Δ 0 ⎠ −iω −  τˆ + Δˆ ˆ ˆ −1 µ k 3 G(k,ω µ ) = (iω µ − H ) = 2 2 2 ω µ + k + Δ F. 2 B C S T H E O RY

iω µ + k Δ G(k,ω µ ) = − 2 2 2 ; F(k,ω µ ) = 2 2 2 ω µ + k + Δ ω µ + k + Δ Consequences: • Pair amplitude/anomalous GF F(k,ω) is ﬁnite for ﬁnite Δ • Normal GF G is different from the free fermion form • Δ has to be determined selfconsistently, i.e. it depends on F Unusual spectral properties (viz. quasiparticles): Excitation energies determined from poles of retarded GF

2 2 E + k Ek = ± k + Δ Energy gap G(k,E) = 2 2 2 k 2 k =0 E −  − Δ  = − E Ek = Δ > 0 k k 2m F F. 2 B C S T H E O RY

Finite pairing amplitude (determined self-consistently) Δ −1/λ T Δ(T = 0) ~ ω De Δ = λ ∑ 2 2 −1/λ µ ω µ + Δ Tc ~ ω De

Universal BCS ratio Δ = 3.53kBTc Excitation gap Density of states ⎧ E 2 2 E > Δ ⎪ 2 2 Ek = ± k + Δ N(E) = ⎨ E − Δ ⎪ ⎩ 0 E < Δ F. 3 Q UA S I C L A S S I C A L APPROXIMATION • Integrating out fast oscillation • Equation for the envelope function

Wigner representation (strongly peaked at Fermi momentum)       Gˆ (r, p,ω ) = d 3ρGˆ (r + ρ / 2,r − ρ / 2,ω ) µ ∫ µ Quasiclassical Greens function   i   gˆ(r,v ,ω ) = d τˆ Gˆ (r, p,ω ) F µ π ∫ p 3 µ EILENBERGER EQUATION

   −ivF∇gˆ(r,vF ,ω µ ) =    ⎡i ˆ i ˆ ˆ '(r, )ˆ ,gˆ(r,v , )⎤ ⎣ ω µτ 3 − Δ + Σ ω µ τ 3 F ω µ ⎦ • transport like equation • homogenous (additional condition necessary • impurities, phonon scattering are in self-energy • occurence of coherence lengths ξ0 or ξT obvious

vF vF ξ = ξ = limp = vFτ imp 0 2Δ T 2πT EILENBERGER EQUATION

Homogenous Differential equation 2   ˆ supplemented by normalization gˆ (r,vF ,ω µ ) = 1     Vector potential: gˆ(r ) ⎡ ieAˆ ,gˆ(r )⎤ ⎡ ˆ ,gˆ(r )⎤ ∇ → ⎣∇ − τ 3 ⎦ = ⎣∇ ⎦

Selfconsistency: ˆ  ˆ   Δ(r ) = λT ∑ go.d.(r,vF ,ω µ )  vF µ Current density:    ˆ ˆ   j(r ) = −ieN(0)πT ∑ vF Trτ 3g(r,vF ,ω µ )  vF µ IMPURITY SCATTERING

Impurity selfenergy in selfconsistent Born approximation ˆ  2 ˆ ˆ   ˆ Σimp (r,ω µ ) = Ni V τ 3G(r, p,ω µ )τ 3 ∑ p i   ˆ = gˆ(r,vF ,ω µ )  τ 3 vF 2τ imp Homogenous solution ⎡ 1   ⎤ 0 = ω τˆ + Δτˆ + gˆ(v ,ω )  ,gˆ(v ,ω ) ⎢ µ 3 1 F µ v F µ ⎥ gˆ(ω )  = gˆ(ω ) 2 F µ µ ⎣⎢ τ imp ⎦⎥ vF 1 Anderson theorem: ˆ ˆ ˆ g(ω µ ) = (ω µτ 3 + Δτ1 ) Thermodynamics not affected Ωµ (Tc, Delta, density of states) THE DIRTY LIMIT

    ˆ  Expansion in anisotropy gˆ(r,vF ,ω µ ) = gˆ0 (r,ω µ ) + vF g1(r,ω µ ) +… Using Eilenberger and normalization ˆ    g1(r,ω µ ) = −τ impgˆ0 (r,ω µ )∇gˆ0 (r,ω µ )    gˆ (r, ) gˆ (r, ) ⎡ ˆ ˆ ,gˆ (r, )⎤ −∇( 0 ω µ ∇ 0 ω µ ) = ⎣ω µτ 3 + Δ 0 ω µ ⎦ Usadel equation

2 1 2 Current density: σ = 2e N(0)D D = vFτ imp  πσ 3  ˆ ˆ  ˆ  j(r ) = i T ∑Trτ 3g0 (r,ω µ )∇g0 (r,ω µ ) 2e µ