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Main learning goals:
After completing the course you are familiar with quantum-field-theoretic techniques as used to determine equilibrium properties of many-body systems. You can apply these techniques to quantum condensed-matter systems, such as electrons in solids, and ultracold atoms.
- After completing this course, you are familiar with the formalism of second quantization. You can apply it to interacting systems of conserved particles to construct many-body wave functions and mean-field theories, such as Hartree-Fock theory, and mean-field theories for phase transitions (e.g. BCS transition and/or magnetic phase transitions).
- After completing this course, you are familiar with the expression for the partition function and correlation functions in terms of coherent-state path integrals. You can derive and compute diagrammatic perturbation expansions based on this formulation.
- After completing this course, you are familiar with Matsubara frequencies and Matsubara summations. You can evaluate these to compute response functions containing ladder or bubble diagrams.
- After completing this course, you are familiar with real-time and imaginary-time linear response theory and their mutual relation via a Wick rotation. You can apply this theory to derive expressions for linear-response coefficients in terms of correlation functions, i.e., a so-called Kubo formula.
- After completing this course, you are familiar with some advanced techniques and concepts in statistical field theory, such as, for example, spontaneous symmetry breaking, Landau theory of phase transitions, and Hubbard-Stratonovich transformations.
After following this course, you are familiar with some advanced topics in quantum matter such as, for example, Bose-Einstein condensation, BCS theory, RPA theory for screening in an electron gas, disordered electrons and/or renormalization-group methods.
Required pre-knowlegde:
- You are familiar with, and are able to work with, the following concepts and model systems of statistical physics: canonical and grand-canonical partition functions, chemical potential, ideal quantum gases, Bose-Einstein and Fermi-Dirac distribution function.
- You are familiar with, and are able to work with, the following concepts and model systems of quantum mechanics: single-particle Schrodinger equation, Hilbert spaces, spin, (time-dependent) perturbation theory, wave function and dispersion of free particles, wave functions and energy levels of the harmonic oscillator, description of the harmonic oscillator in terms of raising and lowering operators.
- You are familiar with, and are able to work with, the following concepts from classical mechanics and classical field theory: Lagrange and Hamilton formalism for particles and fields, Maxwell’s equations.
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Understanding the collective behavior of quantum and classical many-body systems from their microscopic constituents is a central theme in physics. A powerful theoretical tool that facilitates this understanding is quantum field theory. This course gives an introduction to the use of quantum-field-theoretic methods to determine the equilibrium properties of many-body systems. The effects of both classical and quantum fluctuations are treated by methods involving second quantization, many-body wave functions, mean-field theories, coherent-state path integrals, and diagrammatic perturbation expansions. Particular topics and applications discussed include such examples as Debye screening and plasma oscillations, Landau theory of phase transitions, superfluidity, superconductivity, and ferromagnetism.
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