Monday, April 13. Today: equation of motion method (EOM) Dyson equation and Self-energy. Many-Body Green s Functions

Transcription

1 Monday, April 13 Today: equation of motion method (EOM) Dyson equation and Self-energy

2 Unperturbed propagator Take a system of non interacting fermions The unperturbed propagator is: ( ) or

3 Unperturbed propagator The completeness for states with N±1 particles includes: states with more p-h excitations are not connected to by single / operators Thus, for example: 1 for α in = 0 for α not in

4 Unperturbed propagator Thus, the unperturbed propagator for a set of non interacting fermions is written as, And in Lehmann representation:

5 Unperturbed propagator If one chooses a different basis {α }, then where: In a general basis the propagator maintain its poles (excitation energies) but it is no longer diagonal!

6 Unperturbed propagator g (0) αβ(t-t ) has an inverse operator: Thus:

7 Unperturbed g 4-pt propagator The 4-points unperturbed propagator is: By Wick theorem, one has:

8 Equation of motion for g αβ Take the Hamiltonian, Equation of motion for the operator: derivative creates an additional ph excitation weighted by V

9 Equation of motion for g αβ Take the derivative w.r.t. time t :

10 Equation of motion for g αβ Apply g (0) αβ(t-t ) :

11 Equation of motion for g αβ Feynman diagram conventions: g αβ (t-t ) = β α u αβ, t αβ = α β g (0) αβ(t-t ) = v αβ,γδ = β α g (0) αβγδ (t 1,t 2,t 3,t 4 ) = α β g 4-pt γ α γ β δ δ

12 Equation of motion for g αβ The EOM for g is: Equivalent diagram: α α α α Expansion is in terms of g = - + g 4-pt EOM breaks a leg into three thus a GF with 2 more points β β hierarchy of equartions! β β

13 4-points vertex The 4-pt Green s function, can be expanded as: non-interacting but fully correlated 1- body propagators two-particle interactions

14 4-points vertex The 4-pt Green s function, can be expanded as: corresponding diagram: CONVENTION: repeated indices are summed and times are integrated

15 Dyson equation The EOM for g(t-t ) is: where: is a particular 2-times ordering of the 4-point GF. Substitute the expansion of g 4-pt in terms of non interacting propagators and Г 4-pt

16 Dyson Equation where: g 2p1h-1p (t-t ) [g g g g] + g g Г g g this extends the Hartree-Fock potential to a fully correlated density

17 Dyson Equation Dyson equation: Irreducible self-energy: diagrammatically:

18 Dyson Equation Dyson equation: Diagrammatically: Σ = + Σ = Σ Σ Σ Σ Σ + Σ Σ Σ Σ +

19 Dyson Equation The reducible self-energy sums to all orders, Then:

20 Conservation laws M あ croscopic quantities can be calculated from the single particle propagator: particle number tot. momentum angular momentum

21 Conservation laws There exist two-different forms of the Dyson equation: One usually chooses and approx. for Г to build an approximate!!!!

22 Conservation laws There exist two-different forms of the Dyson equation: One usually chooses an approximation for Г and then builds an approximation of!!!!

23 Conservation laws Theorem (Baym Kadanoff 1961): Assume that the propagator g αβ (t-t ) solves both forms of the Dyson equation (that means = ) and Г αβ,γδ =Г βα,δγ. Then <N>, <P>, <L> and <E> calculated with g αβ (t-t ) are all conserved:

24 Dyson equation - II The EOM for g(t-t ) is: where: g 2p1h-1p Want it symmetric try deriving w.r.t. to t as well

25 Dyson equation - II Focus on he g 2p1h-1p (t-t ) propagator: Then:

26 Dyson equation - II From the derivative on one finds, where: still a two-time functions but propagates three-excitations at the same time

27 Dyson equation - II By making use of the following relations Dyson eq. equation of motion for One arrives at the result for g 2p1h-1p : Substitute back into the EOM for g(t-t )

28 Dyson equation - II g 2p1h-1p One arrives at the result for g 2p1h-1p : Substitute back into the EOM for g(t-t )

29 Dyson equation - II Finally, one finds again the Dyson equation, with selfenergy given (in a symmetric form) by where: Irreducible 2p1h/2h1p propagator

30 Dyson equation - II Graphic representation of the 2p1h/2h1p irreducible propagator R(ω): μ ν λ μ ν λ μ ν λ R (2p1h/2h1p) = g 2p1h - g 2p1h-1p α β γ Propagation of 3 excitations α β γ g 1p-2p1h Subtract the α β γ contribution in which a particlehole anihilate each other The Dyson equation will account for it!

31 Dyson equation - II Different forms for the self-energy if Г 4-pt is approximated in such a way that these two are equivalent, then conservation laws are fulfilled. The exact Г 4-pt depends of 4 times variables R 2p1h is specialized to two-times only!

32 Dyson equation - II After Fourier transformation, the Dyson equation and the self-energy are expressed as and these are all two-times (one-frequency) functions. Hence, there appear no integral over energies.

33 Dyson equation - II The full Lehmann representation of the single particle propagator is In real systems these is always a continuum for large particle and hole energies The one body equation for the residues is the same in both discrete and continuum spectrum residues can be extracted as usual, with a limit:

34 Dyson equation - II Note: both g (0) αβ(ω) and have poles BUT these are different from those of the propagator g αβ (ω) Thus, we can extract the residues of the one body propagator from inside the Dyson equations: which yields, NB: this is valid for particle states as well. one-body equations overlap wave function to the final state k-, expressed in the basis {α} Schrödinger-like equation for quasiparticle amplitudes!!

35 Dyson equation - II Example of the Dyson equation written in coordinate space: Same eq. as above: wave fnct. in r space spin, isospin, use, inverse prop. g (0)-1 in frequency space Then: it cancels the U(r) contained in!!