Quantum Many-Body Theory Reading Group

The reading group meets in the interaction room of the physics annex on Mondays at 3.30pm.


Session 1: Basics

Note that the dates are very speculative and subject to change but the content should stay (more or less) as it is.

Date Topic Reading Alternative Readings Homework
24-05-04 Second quantisation Fetter & Walecka p12-21   Fetter & Walecka Q1.1
31-05-04 The tight-binding model Mahan p21-33 Mihaly and Martin p25-30 (first quantisation only)

Lowe - chapter on the Huckel model (what chemists call the tight-binding model)

The Hamiltonian of the tight binding model is normally written as

H = -t Σ(ijσ)ciσ+c      

where: t is the hoping integral, c+ creates an electron on site i with spin σ and c annihilates an electron on site i with spin σ. (Here I've set the chemical potential to zero - this is half filling on hypercubic lattices.)

i) What is the dispersion relation (energy as a function of momentum, k) for the tight binding model (with nearest neighbour hoping only) in (a) 1D (b) 2D on a square lattice (c) 3D on a cubic lattice (d) arbitrary dimension on a hypercubic lattice (e) 2D on the anisotropic triangular lattice sketched below?

ii) What is the bandwidth (difference between the maximum and the minimum of the dispersion relation) in each of the cases (a-e) above - with (e) you need to carefully consider the effect of varying the ratio t1 on t2 between 0 and infinity?

iii) Calculate the Fermi surface for the 2D square lattice (a) at half filling (b) close to zero filling (c) close to complete filling.

iv) The anisotropic triangular lattice allows to extrapolate smoothly between three limits. What are the three limits? What are the corresponding values of t1 and t2?

v) Diagonalise the Huckel model for a benzene molecule.

07-06-04 This week was entirely taken up discussing last weeks homework      
14-06-04 No meeting: Computational Biology Workshop - - -
21-06-04 The -ve U Hubbard model, BCS theory and the BdG equations Ketterson & Song p257-264

You may also like to read p199-207 for a bit of background on BCS theory

de Gennes The Hamiltonian of the Hubbard model is normally written as

H = -t Σ(ijσ)c+c  + U Σ(i)ci up+cj upci down+cj down    

where: t is the hoping integral, c+ creates an electron on site i with spin σ and c annihilates an electron on site i with spin σ.

i) Fourier transform this Hamiltonian into k-space

ii) From the BdG eqns (K&S eqn 36.18) and the self consistency condition (37.11) derive the gap equation (26.29)

iii) From the Hubbard Hamiltonian derive the real space BdG equations

28-06-04 Discussion carried over from last week. n/a n/a n/a
05-07-04 The +ve U Hubbard model and the Stoner ferromagnetism Rickayzen p263-270   In the reading Rickayzen calculates the mean-field solution of his model. In the same way calculate the mean field solution of the Hubbard model assuming U>0 (repulsive). Hint - your solution should be very similar to Rickayzen's.
12-07-04 Green's functions Rickayzen Ch1 Fetter & Walecka Ch 3, AGD Ch 2&3, Mahan Ch 2 and 3 Your choice of questions from Rickayzen Ch1
19-07-04 Green's functions Rickayzen Ch2 Fetter & Walecka Ch 3, AGD Ch 2&3, Mahan Ch 2 and 3 Your choice of questions from Rickayzen Ch2
26-07-04 Green's functions and approximation schemes Rickayzen Ch3 Fetter & Walecka Ch 3 Look at one of the figures of Feynman diagrams in Ch 3 of Rickhayzen. Write down the corresponding Green's functions. Repeat this with other figures until you no longer have to think while you do it!

Your choice of questions from Rickayzen Ch3

02-08-04 Anomalous Green's functions and the Gorkov equations: a different approach to BCS theory Fetter & Walecka p 439-454 AGD Ch 7, Rickayzen Ch 8, Mahan Ch 9, Ketterson & Song Ch 55 Fetter & Walecka Q13.7
09-08-04 The mean field solution of the +ve U Hubbard model using Green's functions Rickayzen p271-282   Rickayzen p283/4 Q3


Authors Title Publisher City Year Comments
(AGD) Abrikosov, Gorkov and Dzyaloshinski Methods of Quantum Field Theory in Statistical Physics Dover New York 1963 The original and still, many believe, the best - very Russian in style, if you like Landau & Lifshitz you'll like this
Fetter and Walecka Quantum Theory of Many Particle Systems McGraw-Hill Boston 1971 "AGD made readable for western graduate students", Balazs Gyorffy. Covers a lot more ground than AGD though
Mahan Many Particle Physics Plenum Press New York 1990 Lots and lots and lots - covers a vast range of topics at a reasonable level of detail
Mihaly & Martin Solid State Physics: Problems and Solutions Wiley New York 1996 Does exactly what it says on the tin - covers mostly elementary topics, but has lots of questions and provides solutions to them all - very useful
Rickayzen Green's Functions and Condensed Matter Academic Press London 1980 A modern approach - very carefully written, covers most of the basics
Anderson Basic Notions in Condensed Matter Physics Benjamin/Cummings Menlo Park 1984 The title is a lie - this is not an introductory text. It is brilliant though. On one reading list I was given as a postgrad. the entire description of this book read "for inspiration" - that about sums it up.
de Gennes Superconductivity of Metals and Alloys Benjamin New York 1966 The first time the BdG approach appeared in the west. An absolute classic, very elegantly written but a little though unless read in conjunction with something else.
Ketterson and Song Superconductivity Cambridge University Press Cambridge 1999 Obviously only covers superconductivity - the authors claim that it is basically an updated version of de Gennes, but it is better than that. It gives a very modern treatment of superconductivity and is very strong on the BdG approach. Personally speaking I think that this is my favourite book on superconductivity - it's where I learnt the subject.