Advanced Solid State I

Green's functions in condensed matter physics

**Lectures:** 10:45 am - 12:05 pm, Monday and Wednesday, JFB 210.

**Prof. Oleg Starykh**, office: 304 JFB, email: starykh 'at' physics.utah.edu

This is a course about Green's functions, with focus on their numerous applications in condensed matter and statistical physics. Its outline follows G. Rickayzen's book "Green's functions and Condensed Matter" (Dover Publications; Reprint edition (May 15, 2013); ASIN: B00CWR4Z8W; here is the link to its Amazon page). In addition, detailed lecture notes will be provided by the lecturer.

Course work consists in lecture-taking (hand-written lecture notes are to be posted online) and in-class discussions, bi-weekly problem sets, and end-of-the-semester presentation (topics for these will be suggested later in the semester).

**Historical introduction:** an interesting and informative article in Physics Today **56**(12), 41 (2003),
about George Green - the inventor of Green's functions.
Read it here online, or look here for a
pdf file.

Week 1: Green's functions in quantum mechanics Lectures 1 and 2

Homework 1, due Monday, February 2

Week 2: Linear response theory Lecture 3

Energy absorption rate and Im part of the retarted susceptibility Lecture 4

Electron spin resonance (ESR) in spin-rotation invariant system Lecture 5

Single-particle time-ordered (causal) GF for free Fermi gas Lecture 6

Formal properties of various GFs, and their spectral representation Lecture 7

Homework 2, due Monday, February 16

Practice: Friedel oscillations in 1d Fermi gas at finite T Lecture 8

Charge susceptibility of Fermi gas and 2-particle GF Lecture 9

Homework 3, due Monday, March 9

Using Matsubara technique to calculate density-density GF Lecture 10

Joule heating, conductivity and charge susceptibility Lecture 11

** Required reading for the week of March 2 - 6: Chap.3 of the textbook, pp. 54-96**,
diagrammatic series, self-energy, random-phase approximation, etc.

Electrons in random potential - short-ranged (Lect. 12) and long-ranged and correlated (Lect.13) Lecture 12-13

Homework 4, due Monday, March 30

** Chap.5 of
Doniach-Sondheimer's book, required reading for homework 4 **

Dielectric function, random phase approximation Lecture 14

** Chap.3 of
Mahan's book, required reading for Lect. 14**

Homework 5, due Monday, April 13

Screening, Friedel oscillations Lecture 15

RPA for charge and spin fluctuations (I) Lecture 16

Diagrammatic derivation of RPA for charge and (transverse) spin fluctuations (II) Lecture 17

Response of interacting system: particle-hole continuum and collective excitations. Zero sound. Ferromagnetic metal: Stoner continuum and spin waves (I) Lecture 18

** Additional make-up lecture, Friday, April 10, 10 am, same room **

Ferromagnetic metal: Stoner continuum and spin waves (II) Lecture 19

Pairing instability: linear response, ladder series for BCS reduced Hamiltonian. Electron-phonon mechanism. Lecture 20

Cooper problem for the pair with finite angular momentum. Lecture 21. Here is the classic solution of a 2+N electron problem by Leon Cooper (1956).

Leggett's 1975 review of He3, a must-read about ** p-wave ** superconductivity;
and a brief summary about phases of Helium-3 from Chalmers Univ.

Superconductivity out of repulsion: Kohn-Luttinger mechanism for magnetized electron gas with contact interaction. Lecture 22

This lecture is based on a nice paper by S. Raghu and S. A. Kivelson. I find this detailed review of this topic from the author of several very important papers, Andrey Chubukov, to be very useful. Here are few of those papers, going back to 1988 (1, 2, 3).

Exam = collection day for your **study projects**: Tuesday, May 5, 10:30 am - 12:30 pm, JFB 210.