University of Illinois at Urbana-Champaign · Department of Physics

Physics 561

Condensed Matter Physics II

Academic Year 2015/2016

Fall Semester 2015

Instructor: Professor Eduardo Fradkin

Office Address: Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 ESB, MC-704,
1110 W Green St, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-244-7704
Eduardo Fradkin's Homepage

Time: 11:00-12:20 am Tuesday-Thursday
Place: Rm. 158 Loomis
CRN: 30721
Credit: 1 unit.
Office Hours: Mondays 4:00-5:00 pm, Rm 2119 ESB (new time!)
TA: Mr. Xiao Chen
Address:Rm 3101 ESB
Office Hours:Thursdays 4:00- 5:00 pm, Rm 3101 ESB


Welcome to Physics 561, Condensed Matter Physics II! This course is the sequel to Physics 560, Condensed Matter Physics I. In that course you studied Condensed Matter Systems in regimes in which correlations and fluctuations do not play a significant role. Thus, much of the theory of electrons in metals that you studied in that course assumed that the electrons are free. Such one-electron theories are successful in may regimes but fail in many others. For example one-electron theories cannot explain screening in metals or the origin of superconductivity (regardless of whether Tc is high or low!). Similarly, non-interacting theories also fail in low dimensions or in systems in which external fields restrict the kinematics of the particles. In this course we will be interested in the understanding of strongly coupled and fluctuating systems and with the theoretical tools required to treat such problems. I will assume that you are fully familiar with the language and methods of second quantization, which are discussed in great detail in our second graduate course in Quantum Mechanics, Physics 581. I will also assume that you are familiar with the material of the Condensed Matter Physics I course, Physics 560. You should be aware that these two courses are a prerequisite for this course. I will not assume that the students are familiar with methods of quantum field theory at the level of Physics 582. However, students who are already with this material will profit more of this course.
Below you will find a detailed Course Plan (or Syllabus) for Physics 561. It is divided in items and there you will find links to my class notes. I will post them as they become available. You will also find links to the homework sets and to their solutions. There will be a total of five homework sets (more or less). The homeworks are very important. There you will find many applications to different problems in various areas of Condensed Matter Physics. You will not be able to master the subject unless you do (and discuss) the problem sets. There will not be a midterm exam. There will be a Final Exam in the form of a Term paper and an oral presentation.

Course Plan

Brief Review of Second Quantization, (pdf file).

Identical particles. Fermions and Bosons and their wave functions. Creation and anihilation operators. The free Fermi gas and the free Bose gas. Elementary methods of quantum field theory in interacting Fermi and Bose systems. User-friendly path integrals.
You may find it useful to read my review of this subject in my Quantum Field Theory Lecture Notes.

Green's Functions, Measurements and Correlation Functions

The theory of the propagator interacting systems. Feynman diagrams. (pdf file).
Observables and their measurement. Linear response theory. (pdf file).Spectral functions and their physical content. Analytic properties and their connection with the spectrum. Finite temperatures and finite frequency behavior.

The Weakly Interacting Electron Gas and Landau's Fermi Liquid Theory

The weakly interacting electron gas.(pdf file). Landau's Principle of Adiabatic Continuity and the Landau Theory of the Fermi liquid ( pdf file ). Quasiparticles and quantum numbers. Effective interactions and effective masses. Effective Hamiltonian and Landau parameters. The random phase approximation and screening in a weakly interacting electron gas. The Landau theory as a stable fixed point of a renormalization group.

Luttinger Liquids and Non-Fermi Liquid States in one dimension (pdf file)

Non-Fermi liquids: Luttinger liquids in one-dimensional and quasi-one-dimensional systems: quantum wires, carbon nano-tubes and quantum Hall edges.
Abelian bosonization. Non-perturbative theory of interacting fermi systems in one dimension. Luttinger liquids. Application to the Kondo problem and other quantum impurity systems.
Luttinger physics and its connecions with the quantum Hall effect, and quantum impurity problems. Tunneling into non-Fermi liquids.
Extensions to higher dimensions and the stability of The Landau theory.

Disorder, Diffusion and Localization, scanned notes
The role of disorder in weakly interacting Fermi systems. The Born approximation. Multiple scattering. The elastic mean free path. Quantum diffusion and Boltzmann conductivity. Ladder diagrams. The Ioffe-Regel criterion.
Anderson localization in strongly disordered non-interacting systems. Lifshitz tails. The localization length. The metal-insulator transitions. The Thouless length. (Almost) Absence of diffusion in two dimensions and scaling theories of weak localization. Non-linear sigma models and replicas.

Quantum Impurities in metals, the Anderson and Kondo Problems

theory of a single magnetic impurity in a Fermi liquid. The Kondo effect. Reduction to a one-dimensional problem. Perturbation theory and Poor Man's Scaling.

The BCS Theory of Superconductivity

Electrons and Phonons. The BCS Instability. Cooper pairs. The BCS wavefunction. Superconductors and Superfluids. Landau-Ginzburg theory. Electromagnetic Response and the Meissner effect. The Josephson effect and macroscopic quantum coherence. Anisotropic superconductivity: He3 and High temperature Superconductors. Inapplicability of BCS theory to high Tc superconductors and to their ``normal" state.

Correlated Fermi Systems: The Hubbard Model

Hartree Fock theory. Stoner ferromagnetism. Spin and Charge Density Waves.
The Hubbard Model at half-filling. Mott insulators. The strong coupling limit of the Hubbard model and the quantum Heisenberg model. The Néel state. Spin-wave approximation and antiferromagnetism.
Phase transitions in magnets at finite temperature. Mean field theory. Fluctuations. Landau-Ginzburg theory. Critical phenomena. Scaling and the Renormalization Group.

One-dimensional Quantum Antiferromagnets.

The spin 1/2 quantum antiferromagntic chain. Critical behavior. Bosonization. The role of spin. Quantum disordered states and Haldane gaps.
The Sigma Model picture of antiferromagnets in one and two dimensions. Quantum critical behavior. Brief review of the renormalization group. Solitons and their quantum numbers. Quantum phase transitions.

The Quantum Hall Effects

Hilbert space on the lowest Landau level. The integer quantum Hall effect. Quantization of the Hall conductance. Chern numbers and adiabatic topological invariants. Disorder and correlations.
The Fractional quantum Hall effects. The Laughlin wave function(s). Quasiparticles and quasiholes. Fractional charge and fractional statistics. Effective hydrodynamic theories of the quantum Hall effects. The role of topology. Flux attachment. Composite bosons and composite fermions. Paired FQH states.
Inhomogeneous and compressible quantum Hall states.

Doped Mott Insulators

Quantum and classical phase separation: correlations and fluctuations.
Stripe phases and electronic liquid crystal states. Non-Fermi liquid behavior.
Connections with mechanisms of high temperature superconductivity.

Spin liquids, and other exotic beasts.

Quantum disordered states in strongly correlated systems. Frustrated quantum antiferromagnets. Phases of frustrated quantum antiferromagnets. Spin liquids and liquid crystal phases.
Gauge theory approaches to Strongly Correlated Systems. When are these states physical?
Quantum Dimer Models. Confinement and deconfinement of excitations. Spin-Charge separation: when, where and why.


Homework set No. 1 pdf file
posted Monday August 31, 2015 , Due Monday September 14, 2015

Solutions to Homework Set No.1

Homework set No. 2 pdf file
posted Tuesday September 22, 2015 (edited on October 8), Due Monday October 12, 2015

Solutions to Homework Set No.2

Homework set No. 3 pdf file
posted Saturday October 24, 2015 , Due Sunday November 8, 2015

Solutions to Homework Set No.3

Homework set No. 4 pdf file
posted Sunday November 15, 2015, Due Monday December 2, 2015

Solutions to Homework Set No.4

Term Paper

List of Suggested Term Papers: Please click here to see a list of suggested Term Papers.

The Term paper will be due at 9:00 am on Wednesday December 16, 2015 , which is not the official day of the Final Exam for Physics 561 (12/11/2015), in room 3110 ESB.

The Final Exam will consist on an oral presentation of your paper for the entire class. Each one of you will have 20 minutes to present your work. The presentations will take place on Wednesday December 16, beginning at 9:00 am, in Rm. 3110 ESB. Please check the Final Exam Program to see at what time you are scheduled to give your talk. Please note that all students will be expected to attend all the talks.

You will assume that you are giving a talk for a reasonably educated audience of people that took this class. The talks must be reasonably self-contained, and the assumptions and your results must be clearly stated. You will also turn in your written paper on the day of the presentation. At the end of the Semester I will put all the Term Papers together in a printed volume which will be distributed among the students registered in the class. The talks will be computer presentations. You may prepare your slides using LaTeX, PowerPoint or Keynote. I will have my MacBook at hand and all the presentations will be stored on my computer so we can save time. I will need to have your presentation sent to me before 12:00 noon that day by email so I can set it up on my computer. I will have a projector installed in the room.

The paper must be formatted in LaTeX, which is the standard program for the production of science papers. Other lower quality formats, such as Word, will not be accepted.

The paper must be at least ten (10) pages long, double spaced pages, not including the title page, in 10pt. font. The title page must include the title, your name and an abstract. The paper must include a section with introductory material in which you give the background information and the main motivation. There should also be a main section in which you discuss the principal content, including the details of the model, the approximations that you use and the techniques that are needed to understand the results. Here you will present the main results and you will discuss whatever calculations you had to do. You may put the details of these calculations in an Appendix if these calculations are too involved and disrupt the natural logical flow of the paper. You should have section with your Conclusions and another one with your References.

You can either use the "article" documentclass (which is standard in LaTex 2e) or you can use the APS package (RevTeX 4), which also runs on LaTeX 2e; in this case please declare the document as a "preprint".
Figures: If you wish to use figures in your paper you are welcome to do so but they must be in eps ("encapsulated postscript") format. They must also be included in the text.
LaTeX Resources:. There are lots of resources for the use of TeX and LateX. The best books are The TeX Book by Donald Knuth (Addison Wesley) and Guide to LaTeX, by Helmut Kopka and Patrick W. Daly (Addison Wesley). A good summary can be found in this document on LaTeX2e.
You can also find examples of documents in TeX in the website of the Journals of the American Physical Society. Otherwise you may want to use the following example of a paper in LaTeX: latex file, pdf file


Eduardo Fradkin. ``Field Theories of Condensed Matter Systems, Second Edition", Cambridge University Press, 2013.

Sebastian Doniach and E. H. Sondheimer, ``Green's's Functions for Solid State Physicists", Imperial College Press/ World Scientific.

Philip Phillips, ``Advanced Solid State Physics", Westview Press.

A. Abrikosov, L. Gorkov and I. Dzyaloshinsky. ``Methods of Quantum Field Theory in Statistical Physics", Dover.

A. Fetter and J. D. Walecka. ``Quantum Theory of Many Particle Systems", McGraw-Hill.

R. P. Feynman. ``Statistical Mechanics", Addison-Wesley.

Gordon Baym, ``Lectures in Quantum echanics", Benjamin/Addison Wesley.

L. P. Kadanoff and G. Baym, ``Quantum Statistical Mechanics", Addison Wesley.

Gordon Baym and Christopher Pethick, ``Landau Fermi Liquid Theory", J. Wiley and Sons.

J. Robert Schrieffer, ``Theory of Superconductivity", Addison Wesley.

D.C. Mattis, ``The Theory of Magnetism", Harper & Row, and Springer-Verlag.

Pierre Gilles de Gennes, ``Superconductivity of Metals and Alloys", Addison Wesley.

Paul Chaikin and Tom Lubensky, ``Principles of Condensed Matter Physics", Cambridge University Press.

N. Ashcroft and D. Mermin, ``Solid State Physics", Holt, Rinehart and Winston.

P.W.Anderson, ``Basic Notions in Condensed Matter Physics", Addison Wesley.

L. D. Landau and E. M. Lifshitz, ``Statistical Physics", Volumes I and II, Pergamon Press.

Subir Sachdev, ``Quantum Phase Transitions", Cambridge University Press.

J. Cardy, ``Scaling and Renormalization in Statistical Physics", Cambridge University Press.

D. Pines and P. Nozieres, ``The Theory of Quantum Liquids", Volume I and II, Addison Wesley-Perseus.

J. Negele and H. Orland, ``Quantum Many Particle Systems", Addison Wesley.

N. Goldenfeld, ``Lectures on Phase Transitions ad the Renormalization Group", Addison Wesley.

Dieter Forster, ``Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions", Addison Wesley.

Paul Martin, ``Measurements and Correlation Functions", Gordon & Breach.

A. Auerbach, ``Interacting Electrons and Quantum Magnetism".

Charles Kittel, ``Quantum Theory of Solids", J. Wiley and Sons.

Richard Prange and Steven Girvin, ``The Quantum Hall Effect", Springer-Verlag.

Michael Stone, ``Quantum Hall Effect", World Scientific.

Alexei Tsvelik, ``Quantum Field Theory in Condensed Matter Systems", Cambridge University Press.

Alexander Gogolin, Alexander Nersesyan and Alexei Tsvelik, ``Bosonization and Strongly Correlated Systems", Cambridge University Press.

D. C. Mattis, ``The many Body Problem", World Scientific.

Last updated 12/11/2015