Chapter 10 Dyson s equation, RPA and Ladder Approximations

Transcription

1 Chapter 1 Dyson s euation RPA and Ladder Approximations Dyson s euation low-high density fermion gases Summation tris to deal with divergent series. Conept of renormalization i.e. dressed partile is bare partile plus sreening loud. Solid State Many-Body Physis

2 Dyson s euation In previous setions we have seen how Hartree and Hartree-Fo ould be summed by only inluding ertain parts of the interation. Generalize this by summing over proper self-energy parts. Self-energy part : Any diagram without external lines whih an be inserted into a partile/hole line. Proper (irreduible self-energy part : A self-energy part whih annot be broen into two unonneted self-energy parts by removing one partile/hole line. Just lie we summed before we now get an expression for the propagator as 1 1

3 Dyson s euation ladder approximation G( ω ω ε 1 Σ( ω iδ Σ(ω is effetive potential seen by partile in state due to interations with other partiles. Low density system Short range repulsive interations between partiles F a<<1. Hole lines orrespond to momentum p and integral over p means lose to F. Contribution from hole lines small on sale of partile lines eep only lowest order in hole lines i.e. one.

4 Low density system ontinued These ladder diagrams (see 1.19 an be summed with help of K-matrix K( p' ε ' p ε; ω p p' i K( p 3 d p dε 4 (π p p' ε p ε; ω G ( p ε G ( p ω ε Can be summed to yield uasipartile effetive mass and lifetime whih orresponds with previously found results. Σ K K

5 High-density eletron gas (RPA approximation m n n m m n H ( ε High-density gas KE>>PE so latter ats as a small perturbation. Hamiltonian in a smeared out positive baground an be shown to be A typial proper self-energy part is Cirumvent this by ordering in series of ring type diagrams where r s ~1/ F is order parameter (note bubble terms are anelled by positive baground. Sum over repeated ring diagrams an now be fatored as normal RPA. Ω 4 (4 ( ( ( ( 1 ( e p ig p ig ig d d p π β ε β ε ω π β ε

6 RPA ontinued Σ RPA - ω-γ γ -i eff (RPA -iπ (ω RPA Double wiggle is effetive interation interpreted as sreened interation between two partiles. Diagrams with one interation part entering and one leaving are polarization diagrams in effet they form a virtual dipole due to interation between partiles. eff ( RPA ( ω Evaluating π (ω for << F and ω: 1 π ( ω eff ( RPA 4πe ; λ eff ( RPA e r e λr Although we added infinite diagrams result is finite due to the fat that effetive interations remain finite as.

7 General dressed interation Generalize RPA to inlude all possible polarization diagrams into partial sum for eff (RPA. Polarization part : Any diagram without external interation lines whih may be inserted into an interation line. Proper polarization part : A polarization part whih annot be broen into two unonneted polarization parts by removing one interation line. (Note bubble is not a polarization part. -iπ(ε π This is again just lie the Dyson series with funtional form eff ( RPA 1 π ( ω

8 Conlusion Dyson s euation is written so as to sum over all irreduible parts. All other diagrams an be built up from these. Proper (irreduible self-energy part : Any diagram without external lines whih an be inserted into a partile/hole line that annot be broen into two unonneted self-energy parts by removing one partile/hole line. Low density approximation ( F a<<1 hole ontribution muh smaller than partile Ladder approximation. High density approximation KE>>PE. Ions form positive baground whih is exatly anelled by bubble diagram in Dyson series. Sum over repeated ring type diagrams Random Phase Approximation. Although most of these diagrams are in themselves infinite fatorizing out the free propagator gives us an expression for the effetive interation whih we interpret as a sreened interation between two partiles -i eff (RPA 1 -

9 Conlusion Proper (irreduible polarization part : Any diagram without external interation lines whih may be inserted into an interation line. The proper polarization annot be broen into two unonneted polarization parts by removing one interation line. Effetive interation thus ensures that sum over all diagrams is finite. Interpretation that partiles are sreened by interation loud surrounding them all partile lines have beome uasipartile lines and all interations are dressed. Thus low- and high-density eletron systems an be analyzed. However for most physial systems lie somewhere in between these still problemati.

10 Chapter 11 Renormalization and Existene of Fermi Surfae Rewrite Σ suh that no propagator lines have self-energy parts and all free propagators have been replaed by lothed propagators. Selfonsistent renormalization Σ Exat form of above not nown. First solve above for bare propagator substitute into Dyson s euation to get first approximation for lothed propagator et. Self onsisteny is obtained when results stop hanging. This an also be applied to situations where system is subjeted to external fields.

11 Existene of Fermi Surfae Following a less than straightforward derivation in Mattu that for any systems where perturbation theory holds i.e. not superondutors ferromagnets et it is shown that uasi partiles exist. What does the Fermi surfae loo lie in an interating system. Naively assume that for system with strong interations momentum distribution would be smeared out near F due to ollision between partiles. Not seen experimentally. For systems where perturbation holds the Fermi surfae is physially well defined and it exists even in strongly interating systems.