Lecture 18: Quasiparticles and Fermi Liquid Theory
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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.

One of the most influential developments that has affected our concepts of interacting Fermi systems is the Fermi Liquid Theory (FLT) of Landau. It postulates that there exist "quasiparticles" with the same quantum numbers as non-interacting particles, even though the particles may be strongly interacting as in a metal or 3He. Another key development is the "Random Phase Approximation" (RPA) which sums a selected set of diagrams to provide a theoretical basis for description of systems that appear to obey FLT. (Note that there is no rigorous proof that the sums converge - the arguments depend upon agreement with experiment and with the few exact calculations that have been done.)

Important References:
Pines "Elementary excitations in solids"; Pines and Nozieres "Quantum Liquids" Vol 1, Ch. 1.; Baym and Pethick "Landau Fermi Liquid Theory" Abrikosov, Gorkov, and Dzyaloshinski; Mahan; Jones and March, sec. 2.9

  1. Non-interacting systems
    1. Well-defined independent particle states and Fermi surface
    2. Many-body excitations are sum of independent particle excitations

      3.  
  2. Weakly interacting excitations
    1. FLT is a theory of excitations (not the ground state)
      • Assume independent particle excitations are weakly interacting (phenomenological)
      • Leads to excitations well-defined for energies near the Fermi energy
      • Width (inverse lifetime) proportional to (E - EF)2  - Simple phase space argument
      • Effective mass, Fermi liquid parameters
      • Renormalized susceptibilities
    2. Phase transition when interactions are sufficiently strong

      3.  
  3. Theory of Interacting systems
    1. Reminder of second quantization notation
    2. Hamiltonian for interacting electron gas
    3. Hartree
      • Ignore interactions (except for an average effective potential)
    4. Hartree-Fock
      • Interactions to first order, wavefunctions to zero order
      • Reasonable ground state energies
      • Disaster for excitations near Fermi Energy - due to long range Coulomb interaction
    1. Beyond Hartree-Fock - Correlation
      • Divergence of straightforward perturbation theory
      • Gell-Mann and Brueckner showed how diagrams can be summed - not covered here
       
  4. Random Phase Approximation (RPA)
    1. Interactions screened - physically motivated selected summation of diagrams
    2. Frequency and wavevector dependent screening - described by dielectric function
    3. Modifies Hartree-Fock in a way that qualitatively leads to Fermi liquid type behavior for excitations near the Fermi energy - in agreement with experiment
    4. Improved ground state energy compared to Hartree Fock

      5.  
  5. Conclusions