Quantum Field Theory in Condensed Matter PhysicsWhy is quantum field theory of condensed matter physics necessary? Condensed matter physics deals with a wide variety of topics, ranging from gas to liquids and solids, as well as plasma, where owing to the inter play between the motions of a tremendous number of electrons and nuclei, rich varieties of physical phenomena occur. Quantum field theory is the most appropriate "language", to describe systems with such a large number of de grees of freedom, and therefore its importance for condensed matter physics is obvious. Indeed, up to now, quantum field theory has been succesfully ap plied to many different topics in condensed matter physics. Recently, quan tum field theory has become more and more important in research on the electronic properties of condensed systems, which is the main topic of the present volume. Up to now, the motion of electrons in solids has been successfully de scribed by focusing on one electron and replacing the Coulomb interaction of all the other electrons by a mean field potential. This method is called mean field theory, which made important contributions to the explanantion of the electronic structure in solids, and led to the classification of insulators, semiconductors and metals in terms of the band theory. It might be said that also the present achievements in the field of semiconductor technology rely on these foundations. In the mean field approximation, effects that arise due to the correlation of the motions of many particles, cannot be described. |
Contents
1 Review of Quantum Mechanics and Basic Principles of Field Theory | 1 |
Second Quantization | 12 |
13 The Variation Principle and the Noether Theorem | 18 |
14 Quantization of the Electromagnetic Field | 23 |
2 Quantization with Path Integral Methods | 27 |
22 The Path Integral for Bosons | 37 |
23 The Path Integral for Fermions | 42 |
24 The Path Integral for the Gauge Field | 45 |
42 The Bogoliubov Theory of Superfluidity | 102 |
5 Problems Related to Superconductivity | 113 |
The Josephson Junction | 133 |
53 The SuperconductorInsulator Phase Transition in Two Dimensions and the Quantum Vortices | 146 |
6 Quantum Hall Liquid and the ChernSimons Gauge Field | 161 |
62 Effective Theory of a Quantum Hall Liquid | 167 |
63 The Derivation of the Laughlin Wave Function | 186 |
Appendix | 193 |
25 The Path Integral for the Spin System | 47 |
3 Symmetry Breaking and Phase Transition | 51 |
32 The Goldstone Mode | 60 |
33 Kosterlitz Thouless Transition | 68 |
34 Lattice Gauge Theory and the Confinement Problem | 78 |
4 Simple Examples for the Application of Field Theory | 91 |
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action amplitude approximation becomes bosonic system canonical conjugate Chern-Simons gauge field classical commutation relation components conclude condensed matter physics consider coordinate corresponds Coulomb gauge current density d²r d³r deduce defined degrees of freedom derivative described dimensions discussed effect eigenvalue electromagnetic field electron explicitly exponential expressed fermions Fourier transformation free energy gauge field gauge theory given Green function Hamiltonian imaginary instanton interaction introduced Josephson lattice points limit loop macroscopic magnetic field magnetic flux matrix obtain operator order parameter particle number partition function path integral performed phase transition problem quantization quantum field theory quantum Hall quantum mechanics right-hand side second term Sect Seff single-particle so-called space spin superconductor superfluidity symmetry breaking theorem two-dimensional vector potential vortex vortices wave function write XY model zero ΣΣ მხ