Main learning goal: 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/or 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 function, 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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Course description: 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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