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Special Topics in Many-Body Theory, Spring 2016

The goal of this course, broadly speaking, is to understand how the seemingly simple laws of quantum mechanics and electromagnetism give rise to a rich variety of highly organized phases of matter. Some of these phases, such as superconductors and superfluids, are so different in their properties from our everyday experience that to their original discoverers they must have seemed nearly magical.

The prerequisites for the course are a solid understanding of quantum mechanics and one semester each of statistical mechanics and solid-state physics. Some examples of correlated states that we may discuss are superconductors and superfluids, the Fermi liquid description of metals, quantum ferromagnetism and antiferromagnetism, the integer and fractional quantum Hall effect, the Luttinger liquid theory of one-dimensional systems like carbon nanotubes, and the Kondo effect.
A large part of the course will be devoted to understanding both the various instabilities of the Fermi liquid (to attractive interactions, to magnetic order, in one dimension, etc.) and its exceptional stability to repulsive interactions in two and three dimensions. All of these emerge from what is sometimes known as the "theory of almost everything":
nonrelativistic kinetic terms for electrons and ions, plus the instantaneous Coulomb interaction.

The main theoretical techniques used will be second quantization as a way to write new types of many-body states, such as the BCS wavefunction, and many-body perturbation theory (Feynman diagrams) for Green's functions. Second quantization, which we will begin at the end of this lecture, is a compact way to write states with strong correlations or variable particle number; many-body perturbation theory is a clever way to compute corrections to physical quantities without having to deal with the entire wavefunction of $10^{26}$ particles.