Lectures:
9-10am on Mon, Wed, Fri in Henn. 301.

Prerequisites: PHYS 502
Corequisites: PHYS 500

Grading scheme:

• 30%: assignments
• 30%: project
• 40%: final exam

Syllabus:

The goal is to introduce various techniques used extensively in condensed matter for computing one-electron (also many-electron) Green's functions. These include perturbational techniques (diagrammatics) as well as non-perturbational techniques (path integrals) for both zero and finite temperature. Well known approximations, such as Hartree-Fock and Random-Phase Approximation will be discussed. These techniques will be illustrated on a variety of models, such as jellium model, Hubbard model, Heisenberg Hamiltonian, electron-phonon coupling and the polaron problem, 1D Peierls instability and polyacetylene solitons, etc. If time permits, we will also discuss some effects of disorder (possibilities include dopants in semiconductors, Integer Quantum Hall Effect, etc).

• Techniques for independent particles:
  • The propagator; Dyson's equation; exact solution vs. diagrammatic expansion.
  • Example: Disorder-averaged Green's functions -- self-energy, self-consistent Born approximation.
  • Path integrals for spinless particles. Evaluation and applications:
  • double well potential and instantons;
  • particle coupled to a harmonic oscillator;
  • coherent state representation;
  • Path integral for spin; Schwinger bosons, coherent spin states.
  
  
• Techniques for many-body systems:
  • Brief review of 2nd quantization;
    • define field operators;
    • some representative models: Jellium model, the Hubbard Hamiltonian, Phonons and anharmonic terms, the Frohlich Hamiltonian
  • Variational self-consistent techniques; the Ritz Variational Principle.
    • Hartree - Fock and applications to Jellium model;
    • Hartree - Fock - Bogoliubov; Bogoliubov - de Gennes equations and brief mention of superconductivity
    • Boson Condensate Approximation
    • Boson Coherent State Approximation
    • Beyond the Hartree-Fock: the Random Phase Approximation; quasiparticle vs. collective excitations; applications to Heisenberg Hamiltonian (spin-waves or magnons); Jellium model (screening, particle-hole vs. plasmon excitations); brief mention of Hubbard Hamiltonian.
  • Many-body Green's functions: one-particle, many-particle, causal, advanced, retarded; physical meaning and uses.
  • Diagrammatic techniques; Dyson's equation; self-energy; effective interactions; meaning of HFA and RPA.
  • Finite temperature formalism
  • (if time permits) Path integral techniques; Grassmann algebras and fermion coherent states;
  • (if time permits) Non-equilibrium formulation: Keldysh formalism
  
  
• Applications: a subset of the following, depending on amount of time available and students' interests:
  • The polaron
  • Peierls instability; the polyacetylene solitons;
  • More about degenerate ground-states: the Majumdar-Gosh Hamiltonian
  • More about spin hamiltonians: the Bethe Ansatz, XY model and the Jordan-Wigner transformations, integer vs. half-integer spins, valence-bond states, ...
  • Impurities in semiconductors; phonon-assisted hopping, the Coulomb gap, ...
  • Quantum Hall Effect.

Recommended books:

I will provide notes for all the material we will discuss in this course. Some of the notes are typed, some are hand-written. I have used several books and sources for this course, listed below. I have most of them, and you can borrow them from me for short periods of time. The library should also have them.

The project:

This will be devoted to a topic of your own choosing. I will make some suggestions for possible ideas as we go along, but it is up to you to decide what is of interest to you. At the beginning of Oct. I will make sure that you all have a reasonably ambitious project. You will have to do some literature search and reading on your own and to write a 5 page review which will be given to all your colleagues before the last week of classes. In the last week, you will give a 20 or 30 minute presentation (depending on how many students we have). All students will grade all other presentations for precision and clarity, and I will take these grades into consideration for the "project" component of the final grade. You can see the projects from last year here.

Useful links:

• Online references
• One page guides to writing and speaking

The final exam:

This will be an open book exam. You are welcome to bring as much material with you as you wish. The exam will be 2.5h long. It will be "multiple choice", in that there will be more problems than you could probably fully solve. The challenge will be for you more to figure out how to solve the problems, rather than to write down the solutions in all gorry details.