Lecture 19: Green's Functions, Self-Energies, and Quasiparticles
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No notes on the web for lectures 18 and 19. Prof. Martin has notes and will provide copies to anyone who asks.

In the previous lecture we described Fermi Liquid Theory (FLT) and the Random Phase Approximation (RPA) that provide a basis for understanding normal metals in terms of quasiparticles near the Fermi energy. In this lecture we extend the arguments used in constructing the RPA to define the one-electron Green's functions and self-energies. These provide a general description of the spectral functions for adding and subtracting electrons to a many-body system, which are very important experimentally. We will describe the analytic properties of these functions and simple pictures in terms of Feynman diagrams at the level of the RPA omitting discussion of difficult vertex corrections etc.

Important References for Green's Functions and Self energies:
Mahan (the reference I use for the theory in the context of condensed matter); Fetter and Walecka (the reference I use for clear definitions); Abrikosov, Gorkov, and Dzyaloshinski; Doniach and Sondheimer

  1. Green's Function G(p,E)
    1. Definition of "one-electron Green's function" (applies in many-electron system)
    2. Example of one electron in vacuum or in a lattice with no other electrons
    3. Spectral function - weighted density of states = - (1/2 p) Im (G)
    4. One-electron Green's function in system of many non-interacting electrons

      5.  
  2. One-electron Green's function in system of many interacting electrons
    1. Poles at exact excitation energies
    2. Perturbation expansion in powers of interaction
    3. Time-ordered Green's functions and Feynman diagrams

      4.  
  3. Proper Self Energy S*
    1. Dyson's equation for G
      • Definition of S* in terms of diagrams that cannot be divided into two parts by cutting a single line (Fetter and Waleska, p 105-107)
      • Mathematical transformation of perturbation expansion in powers of S* where S* is itself a sum in powers of the interaction

        • G(p,E) = [ G0(p,E)-1 - S*(p,E)]-1
        where  G0(p,E) is for non-interacting electrons
      • Properties of G(p,E) directly given in terms of S*(p,E)
      • Imaginary part of S*(p,E) determines the width of quasiparticle peaks, etc.   proportional to (E - EF)2  -
    2. Phase transition when interactions are sufficiently strong; then one must change the G0(p,E) about which one expands

      3.  
  4. Properties of S*(p,E) near the Fermi energy
    1. Form like FLT
    2. Dependence of S*(p,E)  on p and E gives renormalized mass
    3. Renormalization factor due to energy dependence 
      Z(p) = (1- d/S*(p,E)dE)-1
    4. Discontinuity in n(p) at the Fermi surface reduced from non-interacting value by factor of Z; mass increased by that factor
    5. These are the key properties of a Fermi Liquid
      • Quasiparticles have one-to-one correspondence to non-interacting particles
      • Fermi surface well-defined
      • NOT derived but rather assumed that the expansion converges
      • Appears to agree with experiment in many cases

        •  
  5. Conclusions