Richard M. Martin, Professor

301A Loomis, 333-4229, rmartin@roma.physics.uiuc.edu

Argyrios Tsolakidis, Grader

390R Loomis Interpass, 333-1976, tsolakid@students.uiuc.edu

Kate Shunk , secretary

327 Loomis, 244-0595, kate@physics.uiuc.edu

The purpose of this course is to provide a framework for graduate students to understand at an "advanced" level some of the important aspects of the physics of condensed matter. We will build upon the material typically included in the previous course, Physics 489, which describes many of the important properties of solids in terms of non-interacting independent particles. One goal of the present course will be to start from the basic strongly-interacting particles - nuclei and electrons - and develop a description of condensed systems in terms of "elementary excitations". Often these excitations can be viewed as weakly interacting so that they lead to many of the properties of non-interacting particles. We will review selected aspects of independent particles - such as the classifications of perfect crystals into metals, semiconductors and insulators - and address the extent to which we can understand such fundamental properties of solids starting from the full interacting many-body system of nuclei and electrons. In order to understand this renormalization to weakly interacting excitations and also to describe experimental measurements made on interacting systems, we will introduce some mathematical methods involving correlation functions and Green’s functions. In addition, cooperative effects among the interacting particles may lead to consequences that cannot be described by independent particles, namely cooperative transitions to new phases of matter, such as the superconducting state, magnetism, and metal-insulator transitions. We will study such cooperative effects not only because of their central importance in condensed matter but also because they are paradigms for understanding similar phenomena in many fields of science.

The first part of the course deals with the description of interacting systems in terms of "elementary excitations" or "quasiparticles". We will focus upon the most important qualitative conclusions, such as the nature of the Fermi surface, and the relation to experimentally measurable quantities. The second part is on cooperative phase transitions to states of "broken symmetry" in which there are qualitative changes in the nature of the elementary excitations and "order parameters" describing the new phases. In particular, we will consider superconductivity (the microscopic theory, order parameter and variations in the order parameter) and strongly interacting electrons in transition metal and rare earth compounds, which are of great current interest because of the interesting phenomena they exhibit, including metal-insulator transitions, magnetism, and high-temperature superconductivity.

The background expected of students in this class is a working knowledge of quantum theory (e.g., second quantization, which will be briefly reviewed), elementary complex variables, and knowledge of condensed matter physics at the level of Physics 489. The latter includes the mathematical description of periodic systems, Bragg scattering, common crystal structures, the nature of phonon dispersion curves and electron bands in the Brillouin Zone, one-electron description of the Fermi surface of metals and the bands in semiconductors and insulators, and other similar material covered, for example, in Aschroft and Mermin (A&M), Harrison, or other texts. The type of material I will assume as background is:

Office hours will be announced after the start of the semester and posted on the class WWW pages. Both Argyrios Tsolakidis and Richard Martin will be pleased to discuss with students at other available times. Please make an appointment for times other than the class period or office hours.

The material covered in the lectures will follow the set of Lecture Notes, which will be placed in the Physics Library as the semester progresses. The syllabus for the course and short summaries of the notes for each lecture, with the primary references, will be posted on the WWW pages, and will be passed out at the lectures. Some extensive notes will be posted on the WWW pages in postscript format, but some lecture notes will be handwritten and not be posted on the WWW pages.

There will be (approximately) six homework sets during the semester. Copies of homework problems and solutions will be placed in the Physics Library. Problems will posted on the class WWW; solutions will be posted on the WWW pages only if it is feasible for us to have them in electronic form. Solutions to the problem sets will be due on Thursdays of the week announced. They should be in the homework box by 5PM Tuesday. For the grader's benefit, late solutions will be penalized: 75% credit for solutions turned in between Thursday and the class period the following Tuesday; no credit after that time. You may miss one homework set with no penalty.

Each student is expected to complete a term paper on a topic that he/she has investigated in more depth than would be possible as a topic in the course. A list of suggested topics will be given out, but each person can choose his or her own topic. Be sure to discuss the topic of your paper with Prof. Martin before proceeding! The primary objective of the term paper should be to describe a physics issue in such a way that other students at this level can benefit by reading the paper. It may be a summary of what is known about a research topic from the literature; it may describe a problem and its solution or partial solution; or it may describe a computational method for solving a problem. Appropriate references to the literature should be included just as in a paper to be published. The term papers should be typed in the style of a paper using REVTEX, which is the accepted format for Physical Review papers and is widely available. The final version should include figures and be in a single postscript file. The staff will help you with aspects of REVTEX and making the postscript files. The term papers will be collected in a volume, reproduced, distributed to all students, and placed in the library. The postscript version will also be published electronically on the class WWW pages

The will be a mid-term and a final exam. The nature of the exams will be discussed well in advance of the exams.

The grades will be determined approximately as follows: problem sets (30%), term paper (25%), mid-term exam (15%), and final exam (30%).

There is no one textbook that I consider covering the material for this course. I have listed the primary text as Mahan, Many Particle Physics because it is a good text for many-body methods and certain important problems; I will use Mahan as my primary reference for Green’s function methods. However, it does not give the motivation and background for the issues in Solid State or Condensed Matter Physics. I regard two other texts as very good; I will use them, but they are "old-fashioned" and not appropriae as the main texts: Elementary Excitations in Solids by Pines, which is a good introduction to the physics of quasiparticles, and Superconductivity of Metals and Alloys by de Gennes, which is an excellent text for the phenomena of superconductivity. Also Abrikosov, Gorkov and Dzaloshinki is a real classic now available in low-cost Dover paperback. Finally, a text like Solid State Physics, by Ashcroft and Mermin, is needed. I recommend this one because I consider it the best background text for 489 level material with some of the 490 level material. (If you have another text, e.g., Kittel, Introduction to Solid State Physics, or Ziman, Principles in the Theory of Solids, then use that instead.) Below I also list the books I have found most useful in preparing the detailed contents of my lectures and other useful books. All of these books are on reserve in the library.

A recent review article (Rev. Mod. Phys. 70, 1039 – 1263 (1998)) on "Metal-Insulator Transitions and Correlated Metals in d-Electron Systems" by M. Imada, A. Fujimori, and Y. Tokura, will be used in the class for the portion on strongly-correlated electrons in the d and f systems. This review covers both theory and experiment, and it will certainly not be possible to cover it in detail, which would require an entire course. We will attempt to cover this in enough detail that students can choose topics for further study for a term paper. Copies of the article will be available in the library. This review paper is used with permission of the authors, Reviews of Modern Physics, and the AIP.

Texts and Reference Books

* Material I will use in preparing course

Richard M. Martin

Numbered note sets will approximately coincide with the individual lectures. Copies of the note sets will be in library. Outlines will be on the WWW pages with links to other material.

Introduction: Outline of course; background material that leads us to the need for many-body pictures and methods to understand the physics of condensed matter - both theory and experiment. (Pines Ch. 1; Class Notes)

1. Adiabatic approximation: Division into slow nuclear excitations vs fast electronic excitations
2. Ground state of the electrons: Total energies; phase stability; Force ( Hellmann-Feynman) theorem; Lattice dynamics for nuclear motion with electrons in instantaneous ground state of electrons
3. Nuclear elementary excitations: Phonons (bosons) ; dispersion curves
4. Electronic elementary excitations: Independent electrons (fermions); bands
5. Including electron-electron interactions: Hartree approx., Hartree-Fock approx., exchange,

1. Second quantization - natural language for interacting particles
2. Hartree approx., Hartree-Fock approx., exchange, second quantization, pair correlation function, structure factor, difficulties of treating correlation
3. Hohenberg-Kohn Theorem: Self-consistent Kohn-Sham Hartree-like theory for exact ground state energy and density of electrons; potential for alternative path to include many-electron effects

1. Linear Response Theory: Classical damped oscillator, causality, analyticity, and Kramers-Kronig relations, sum rules, fluctuation-dissipation theorem, relations to Green's functions, spectral representation, inelastic scattering
2. Dielectric response function: Dynamic structure factor, scattering of charged particles, sum rules, ground state energy
3. Green’s functions in many-body perturbation theory: Interaction Representation, time ordering, Wick’s theorem, Dyson’s Equation
4. Quasiparticles and Self energies: Spectral functions, broadening, Luttinger theorem
5. Random Phase Approximation, Quantum Liquids and Fermi Liquid Theory

Strongly-interacting electrons and metal-insulator transitions, Hubbard-type models, Anderson Model and Kondo effect, Phase transitions and Broken Symmetries, Quantum Hall Effects (brief)

Electron-Phonon interaction in metals, BCS theory, Landau,Ginzburg,Gorkov theory of order parameter, Type vs type II superconductors, High-Tc systems (brief)