Andrea Marini. Introduction to the Many-Body problem (I): the diagrammatic approach

Transcription

1 Introduction to the Many-Body problem (I): the diagrammatic approach Andrea Marini Material Science Institute National Research Council (Monterotondo Stazione, Italy) Zero-Point Motion

2 Many bodies and many trajectories... From complexity to... diagrams! Partial summations: self-energies & bubbles Zero temperature vs finite temperature The equation of motion approch Outline

3 Why so many bodies? 1 1 H = i h( x i, pi ) i j x i x j 2

4 Why so many trajectories? 1 1 H = i h( x i, pi ) i j x i x j 2 = Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't. Freeman Dyson

5 Second quantization and fields x, t )= ϕi ( x) ci (t ) ψ( i Anihiliation Field operator Complete basis operator 1 1 H = i h(x i ) i j x i x j 2 x ) h( x ) ψ ( x ) 1 d x y ψ ( x ) ψ ( y) v ( x y) ψ( y ) ψ( x) H = d x ψ( 2

6 Time-Dependent representations Schördinger Heisenberg Ψ H (t) =ei H t Ψ s (t ) OH (t)=ei H t O S e i H t Ψ s (t ) =e i H (t t ) Ψ s (t 0 ) 0 H =h ' H (t, 0) OH (t )=U (0, t ) O i (t ) U i h t Ψ I (t ) =e Ψ s (t ) i h t i ht O I (t )=e O s e Interaction Taylor expansion T-product Ψ I (t ) =U (t,t 0 ) Ψ I (t 0 ) (t, t 0 )=1 U n ( 1)n t ' I (t 1 )... H ' I (t n )] dt 1... dt n T [ H t n! t 1 >t 2 >... t n 0

7 Adiabatic Switching-on: the Gell-Mann & Low theorem η t ' H η= he H H η 0 Theorem: If the limit then lim η 0 U η(0, ) Φ0 Ψ 0 Φ0 U η (0, ) Φ0 Φ0 Ψ 0 it is an eigenstate of the Hamiltonian... Time... exists

8 Green's Functions Why this definition? To get the meaning think to the GL theorem and to expression for U(t)!... and... Are Green's functions usefull? i lim O t ' t n( x t ) ( i) lim t ' t G (x t, x t ') d x [O( x)g ( x t, x t ' )] E= i d x limt ' t lim x ' x [i t T ( x)]g( x t, x ' t ') 2

9 Lehmann representation n Ψn ( N 1) Ψ n (N 1) =1 Ψn Ψ 0 ψ(x) ξ 0n (x) Ψ 0 Ψ 0 ϵn ( N 1)= En ( N 1) E 0 ( N 1) μ=e 0 (N 1) E0 ( N ) f (ω) dt e i ω t η t f (t ) Eigenvalues of h IP limit

10 Wick's Theorem and Feynmann Diagrams (I) (,t ) U (t, t 0 )=1 U n (t, t ' ) U (t ', ) U (t, 0) OH (t )=U (0, t ) O i (t ) U n U (t, t 0)= 1 n ( 1) n! t t 0 ' I (t 1 )... H ' I (t n )] dt 1... dt n T [ H ( 1)n t ' I (t 1 )... H ' I (t n )] dt 1... dt n T [ H t n! = 0 H ' (t )= 1 d x1 x2 ψ I (x1 t ) ψ I ( x2 t )v ( x1 x2) ψ I (x 2 t ) ψ I (x1 t ) 2 The n-th order in the perturbative expansion of the Green's Function contains 4n2 operators integrated on all internal time and space variables T [ ψ I ( x 1 t 1 ) ψi ( x ' 1 t 1 ) ψ I (x ' 1 t 1 ) ψ I ( x 1 t 1 )... ψi ( x n t n ) ψi (x ' n t n ) ψ I ( x ' n t n ) ψ I ( x n t n ) ψ I ( x t) ψi ( x ' t ' )]

11 Wick's Theorem and Feynmann Diagrams (II) P Φ0 T [ A B C D... Z ] Φ0 = all conbinations ( 1) [ Φ0 T [ A B ] Φ0 Φ0 T [C D] Φ0...] (Gian Carlo) Wick's Theorem H ψ ( x 1, t 1 ) ψh ( x 1, t 1 ) ψi (x, t ) ψh ( x 2, t 2 ) ψh ( x 1, t 1 ) Φ0 T [ ψ I ( x1 t 1) ψ I ( x ' t ' )] Φ0 ψi ( x ',t ' )

12 Feynmann Diagrams: Hartree-Fock 1 ( ) 2 Fock Hartree

13 Exact Partial Summations (I): The self-energy Irreducible (skeleton) Partial summation = X X Reducible Self-consisteny G(0) (1) (2) G G G (3) Σ Hartree G(0) G(1)G(2)

14 Exact Partial Summations (II): The self-energy Σ Hartree = X X We extend both the left and right side of this identity and we get the Dyson equation = Σ Σ =

15 Feynmann diagrams rules (a) Draw all topologically distinct connected diagrams using (2n1) propagators (using as a bulding block the elemental scattering event) ρ( x ',t ) (b) Label each vertex with a space and time variable (c) Integrate all internal variables (d) Assign a F ψ H ( x,t ) ( 1) where F is the number of closed fermion loops ψh ( x,t )

16 Approximate Partial Summations (I) Feynmann rules Use Physical arguments to choose specific classes of diagrams!!! High density Short-range regime? interactions? Low density regime? Conserving approximations

17 Approximate Partial Summations (II): bubble diagrams 0 High density limit Sum the most divergent (dominant) diagrams

18 Approximate Partial Summations (III): The screened interaction 1 ( x t) 1 W (1,1 ') ϵ (1,2) v (2,3) W (1,1 ') v (1,1' )v (1,2) χ(2,3) v (3,1' ) 1 ϵ (1,1') δ (1,1' ) v (1,2)χ (2,1 ')

19 Approximate Partial Summations (IV): The GW approximation = High density limit HEG = sity n e d TMA ( short-range interactions) Real Real materials materials i G ( r 1, r 2 ; t )W ( r 1, r 2 ; t) GW

20 The Finite Temperature Regime K H μ N H (Grand-canonical Hamiltonian) OH (t)=ei H O S e i H OH (τ)=e K τ O S e K τ β K Z G Tr [e ] β=1 / k B T (Modified Heisenberg) 1 β K G ρg Z e Wick's theorem can be proven in the finite temperature case Finite temperature Feynmann Diagrams

21 The electron-phonon self-energy: the Fan approximation 1 y) ψ( x) d x y ψ ( x ) ψ ( y) v ( x y ) ψ( 2 electron-electron x )[ x. x V ion ( x )] d x ψ ( x ) ψ( electron-phonon D total The nucleus is screened 1 W (1,1 ') ϵ (1,2) v (2,3) Σ( r 1, r 2 ; t ) GW self-energy by the electronic dielectric function Fan self-energy

22 Diagrammatic vs Hedin-Lundqvist approach (I) LH and SL, Solid. State Phys. 23, 1 (1969) H H ϕ( x)ρ( x, t ) ρ( x ',t ) ψ H ( x, t ) ψ H, ϕ (x,t ) ψ H ( x,t ) ψ H ( x,t ) δ ψ H ( x, t) ρ( x ', t) ψ H ( x, t) δϕ

23 Diagrammatic vs Hedin-Lundqvist approach (II) δ ψh (1) δϕ Σ(1,2)=i d3 v (1,3)G(1,3 ; 2,3 ) Σ(1,2)=i d3 W (1,3 )G(1,4) Γ(4,2; 3) ϕ V (1)=ϕ(1)V Hartree (1) 3) 1 GW= Γ(1,2;

24 Plan of the Many-Body lectures D. Varsano & F. Giustino C. Attaccalite E. Cannuccia A. Marini & C. Attaccalite A. Marini

25 References