Physics 598SCM - Fall 2004 - Instructor: Richard M. Martin
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Exams and Homework
Course MaterialsCourse Objectives:
Electronic Structure denotes the ground state and excitations of the electrons, which determine the properties of atoms, molecules, condensed matter, and (increasingly) man-made structures. The purpose of this course is to provide a unified description of current issues in understanding electronic structure of matter -- including important experimental information, theoretical approaches, computational methods with useful codes, and applications chosen to illustrate characteristic properties of matter.
The course is meant for students with a broad range of backgrounds: experimentalists and theorists in condensed matter physics, materials science, and engineering, as well as specialists in electronic structure theory. Electronic structure is a useful tool for many areas of science and this course will provide background to understand the basic ideas, approximations and limitations. For students engaged in research in electronic structure theory, the course will provide the opportunity to go more deeply into the theoretical methods.
The course will follow the text "Electronic Structure: Basic theory and practical methods," Cambridge University Press, 2004, written by the instructor Richard M. Martin. Additional material will be added in the latter part of the course on many-body aspects of electronic structure and methods to deal with electron interactions.
The topics of the course will be chosen with the point of view that the theory of electronic structure is one of the great challenges of theoretical physics: to develop theoretical approaches and computational methods that can accurately treat the interacting system of many electrons. We will emphasize density functional theory (DFT), Quantum Monte Carlo (QMC), and Green's function methods. DFT is by far the most widely-used method for quantitative calculations including simulations of complex systems, which we will discuss in some detail. However, present DFT approximations fail in strongly interacting cases, such as in transition metal oxides, where current understanding is based upon Hubbard and Anderson/Kondo type models. For explicit many-body calculations, QMC is a general method that provides the most exact results known for the ground state, whereas the spectra of excitations can be found using Green's function methods.
Background expected of students is knowledge of condensed matter physics at the level of Physics 460 or 560 (formerly 389 or 489), e.g., the theory of bands at the level of Aschroft and Mermin. It will also be helpful to have some knowledge of more advanced concepts at the level of Physics 561 (formerly 490), such as the ideas of quasiparticles and the random phase approximation. Knowledge of programming (Fortran, C, or C++) at only an elementary level is needed.
Out of class work will include homework problems and a term paper on a topic chosen by each student. The term papers will be collected, distributed to all students, and placed in the library (and on the Web) as the collected papers for the class.
The purpose of this course is to give the participants insight into the intellectual challenges in the theory of electrons in condensed matter; knowledge of the basic principles and practical measures which guide the theoretical developments; and familiarity with state-of-the-art computational methods. The course is organized the first part involving primarily independent electron approximations and the second part that addresses many-body issues and methods. An important part of the philosophy of the course is that the electronic structure problem is an interacting, correlated, many-body problem, and we will attempt to develop appreciation of the value and the limits of widely-used pictures based upon independent-electron and Fermi-liquid ideas.
We will emphasize by density functional theory, which in principle makes it possible to use independent electron methods to determine exact ground state properties of many-body electron systems. Density functional calculations are now the predominant method for systematic studies of such properties of condensed matter, which include binding energy, stable crystal structure, magnetic properties, vibrations of the nuclei, etc. However, there are serious issues in strongly correlated cases and in excited states. We will cover the general ideas and recent developments which make possible such calculations on both solids and liquids.
In the area of many-body problems we will emphasize key prototype problems and theoretical developments in solution of these problems. The two most useful methods discussed will be quantum Monte Carlo and Greens function approaches that build upon the independent electron states described above. We will give explicit examples of results that support the applicability of these methods to real materials, such as Mott insulators and high-temperature superconductors. We will also consider solutions for representative model problems, such as the Anderson and Hubbard models, and current controversies in the electronic states of highly correlated systems.
The background expected of students in this class is described below. It is not expected that a student has detailed knowledge of each topic - all needed concepts and details will be discussed in the course - but basic background knowledge of these topics is needed.Exams and Homework:
1) Basic knowledge of the description of periodic crystalline lattices, the reciprocal lattice, Brillouin Zones, the Bloch theorem and other general properties of bands in crystals, as described, for example, in Aschroft & Mermin Chs. 4,5 and 8.
2) Familiarity (not detailed knowledge) with the basic ideas of the two most physically appealing approaches to electronic bands in crystals, the nearly-free electron approximation and the tight-binding method, as described, for example, in Aschroft & Mermin Chs. 9 and 10.
3) Knowledge of the basic ideas of Hartree-Fock theory and second quantization. We will review briefly second quantization, which can found in Pines, Elementary Excitations in Solids, Appendix A; Ziman, Advanced Quantum Theory; Kittel, Quantum Theory of Solids; and many other places.
4) It will be useful to have some familiarity with the ideas of quasiparticles and the random phase approximation (RPA). These are important concepts that we will describe briefly using the concepts of self energies with appropriate references. The RPA is described in Pines, Elementary Excitations in Solids and in Nozieres and Pines, Quantum Liquids, Vol. I.
5) Interest in the challenging problems of condensed matter physics, such as metal-insulator, magnetic and superconducting transitions in strongly-correlated electron systems, and prediction of properties of matter in previously unknown phases.
There will be no exams. There will be homework sets which will be oriented toward basic knowledge of the key points in the course which are important for all students to master.Term Paper:
The most important work for each student to complete during the course will be a term paper which can be chosen from a list of topics or by mutual consent. The paper must describe in depth some problem related to electronic structure, with appropriate references to the literature. The paper can describe a general problem in the theory of electronic structure, an application to an experimental measurement or analysis, or a computational project carried out during the course. It does NOT have to be original research, but it must be original work on the part of the student. For example, it could be writing an original computer program and producing results on a problem that has already been solved and reported in the literature. Or it could be a non-computational project involving literature searches and in depth discussion of an electronic many-body problem or a materials problem. A paper related to experimental work could deal with analysis and/or the theoretical interpretation of the underlying mechanisms.Course Materials:
Suggested topic for the term paper will be distributed early in the course.
Each term project must be agreed upon in advance with Prof. Martin. The deadline for choosing the subject will be announced.
The paper must be typed and include references and figures like a research paper. It should be done using "REVTEX" for which software, sample papers, and instructions will be available. This is the format used for submission of papers to Phys. Rev.
The papers and computer programs will be due near the end of the course - but before the end, in time for some comments and revisions before distribution to the class. A collected set of works of the class will be published on our Web site.
Textbook: "Electronic Structure: Basic theory and practical methods," Cambridge University Press, 2004, which covers thoroughly the first parts of the course.
Additional material will be added in the latter part of the course on many-body aspects of electronic structure and methods to deal with electron interactions.
A list of useful books is given on the class web pages and the books are on reserve in the library.
Useful papers and other sources are given in the text and additional material will be posted and updated during the semester.
For the background level material, I consider the best text to be "Solid State Physics", by Ashcroft and Mermin, with Introduction to "Solid State Physics" by Kittel, a good alternative. I also consider as excellent "Principles in the Theory of Solids" by Ziman and "Introduction to Solid State Theory" by Madelung. Texts on many-body theory which I find useful are "Advanced Solid State Physics" by Phillips and "Many-Particle Physics", 2nd Ed. by Mahan.
Lecture notes will be posted on the Web.
Electronic Structure Resources on the web are provided at ElectronicStructure.org, includes extensive links to groups, information, software, additional material related to the text. A link is on the class home page.