# Green's Function Method (PHYS*7090)

**Code and section:** PHYS*7090*01

**Term:** Winter 2020

**Instructor:** Elisabeth Nicol

## Details

## Course Information

### Instructor

Prof. Elisabeth J. Nicol (Guelph)

email: enicol@uoguelph.ca

### Time and Place

Tuesday/Thursday

1:00 - 2:20 pm

Mini Link Room - MACN 203

### Recommended Background

Students should have completed 701/7010 (Quantum Mechanics I) and 704/7040 (Statistical Physics I) and should have had a course in Solid State Physics (up to and including band theory and phonons) from the fourth year undergraduate level. Good knowledge of complex variable theory will be essential. I will assume that students have seen second quantization notation or will pick it up on their own (see note below). Students who are uncertain about their background preparation or do not have the required courses mentioned above, should consult the lecturer.

### Course Content:

The primary aim of this course is to provide the student with a working knowledge of Green’s function techniques as applied to many body physics with application to condensed matter. Much of the theoretical literature is based on such techniques. I will introduce the concept of the Green’s function and define its relationship to physical observables. The physical interpretation of the Green’s function as a propagator will be stressed. Equations of motion will be derived, boundary conditions introduced to define the ground state, and the equations solved by iteration. Feynman diagrams and rules will be developed and various aspects of the perturbation series analyzed systematically. The idea of the Dyson equation, screening in the random phase approximation, as well as other concepts in metals physics will be introduced. The mathematics will be generalized to include finite temperature techniques developed by Matsubara and others, which make use of an imaginary time axis formulation and discrete summations. Analytic continuation techniques will be introduced. Several applications to specific problems will be considered as time allows.

### Tentative outline

- [Second quantization and ideas of quantum field theory]*
- [Example of Hamiltonians in second quantization notation]*
- Green’s functions
- Zero temperature Green’s functions and Feynman diagrams

- Dyson eqn., self-energy, spectral function, quasiparticles, Hartree-Fock theory - Finite temperature Green’s functions and Matsubara sums

Applications (as time and interest allows):

- Electron spectroscopies: tunneling and photoemission
- Green’s functions for bosons:

- phonons, phonon self-energy, electron self-energy due to electron-phonon interaction - Green’s functions in BCS superconductivity

- anomalous Gorkov Green’s functions, BCS gap equation from self-energy - Screening in an electron gas, Kohn anomalies in phonon dispersion curves
- Impurity scattering
- Response functions: optical conductivity, etc.
- Green’s functions applied to the insulating magnet

* Second quantization is fairly intuitive to use although complicated to derive properly. As this material is commonly known by many of the students, I plan to skip this part and expect students who are missing this background to pick it up on their own.

### Evaluation

**The method of evaluation will be in the form of problems sets throughout the term worth 100% of the final grade.**

### Some recommended reference books on many body theory

Copies of the lecture notes will be made available, but in addition, one might wish to refer to some of the following texts:

- Methods of Quantum Field Theory in Statistical Physics, 2nd Ed., by A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Prentice Hall, 1963. Republished by Dover, New York, 1975. (A Russian classic that has been translated into English. It is now available in soft-cover through Dover which provides inexpensive editions of classic books. Often referenced in research papers and is nicknamed “AGD”.)
- Many-Particle Physics, 3rd Ed., by G.D. Mahan, Springer, 2007. (Very comprehensive and with modern topics. The one that is probably referenced the most in research papers.)
- Quantum Theory of Many-Particle Systems, by A.L. Fetter and J.D. Walecka, McGraw Hill, 1971. Republished by Dover, New York, 2003. (Has been referred to as “ADG made readable for western graduate students” but with more material on other topics. Includes applications in nuclear physics. Can be found in an inexpensive paperback edition from Dover.)
- Green’s Functions for Solid State Physicists, by S. Doniach and E.H. Sondheimer, Benjamin, 1974. In softcover, it was published by Imperial College Press, 1998. (A well-written book.)
- Green’s Functions and Condensed Matter, by G. Rickhayzen, Academic Press, London,
- 1980 (softcover)
- (A very good book. Rickhayzen was a good teacher. Out of print, used copies available.)
- A Guide to Feynman Diagrams in the Many Body Problem, 2nd Ed., by R.D. Mattuck,
- McGraw-Hill. Republished by Dover, New York, 1992.
- (Sometimes referred to “Feynman Diagrams for Dummies”. Available from Dover.)
- Introduction to Many-Body Physics, by P. Coleman, Springer, 2015.
- (Overall, it presents a more modern approach and is worth a look. However, it has problems
- with typos and notation issues. It also skips steps at times and is not deductive in its
- development of the topic. Hopefully a newer edition will resolve some of these issues.)

... and many more texts (check out the library or look online) .

## Academic Misconduct

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