## Lecture course "Quantum Field Theory II", WS 2013/14

Quantum field theory is a universal tool which has applications ranging from condensed matter physics to elementary particle physics. A crucial role is played by symmetries which are at the heart of the universality shown by many physical systems. In this lecture the focus will be on aspects of symmetry breaking and collective phenomena. We will also expand on the ideas of renormalization, gauge theories and topological phases of matter.

**Outline of topics:**

- Brief review of perturbation theory
- Gaussian functional integration for bosons and fermions
- Saddle point and stationary phase approximation
- Perturbative expansions and Feynman diagrams
- RPA analysis of the interacting electron gas
- Broken symmetries and collective phenomena
- Hubbard-Stratonovich transformation
- Bose-Einstein condensation and superfluidity
- Superconductivity
- Non-linear sigma-models
- Spontaneous breaking of global symmetries and gauge symmetries, Goldstone's Theorem and the Anderson-Higgs mechanism
- Renormalization
- Gauge theory
- Topological phases of condensed matter theory
- Haldane's Conjecture
- ...

**Prerequisites:** Quantum Field Theory I.

**Time and location:** Wed 8-10h (Lecture Hall II) and Fri 14-16h (Seminar Room PH I). Exercises: Thu 10-12h (Seminar Room PH II).

Lecturer: PD Dr. Thomas Quella

Exercises: Dr. Dmitry Bagrets and Dr. Vladimir Osipov

**Note:**
Hand-written lecture notes are available on request. The lecture is following the book "Condensed Matter Field Theory" (2nd edition) by Altland and Simons. The books "Field Theories of Condensed Matter Physics" by Fradkin and "The Quantum Theory of Fields I & II" by Weinberg are suggested as an additional reference for advanced readers who wish to broaden their perspective.

**Schedule**

- Lecture 01: Motivation and overview, review of some basic principles of quantum field theory [Notes]
- Lecture 02: Principles of perturbation theory: Asymptotic series and Borel resummation, the beginning of Phi^4-theory [Notes]
- Lecture 03: Free Klein-Gordon propagators in various dimensions, perturbative expansion of the full correlator up to second order [Notes]
- Lecture 04: Feynman diagrams in momentum space, divergencies and their regularization, anomalies, real time propagators [Notes]
- Lecture 05: Random Phase Approximation for the weakly interacting electron gas at high densities [Notes]
- Lecture 06: Effective potential and screening in the weakly interacting electron gas, foundations of the auxiliary field method (Hubbard-Stratonovich transformation) [Notes]
- Lecture 07: Auxiliary field method applied to the weakly interacting electron gas at high densities [Notes]
- Lecture 08: Plasma oscillations, general aspects of Hubbard-Stratonovich transformations, Bose-Einstein condensation for non-interacting systems [Notes]
- Lecture 09: Bose-Einstein condensation for contact interactions from the Hamiltonian perspective, Bogoliubov transformations [Notes]
- Lecture 10: Bose-Einstein condensation for contact interactions from the path integral perspective, superfluidity [Notes]
- Lecture 11: Spontaneous symmetry breaking, Goldstone's Theorem Notes]
- Lecture 12: The origin of spontaneous symmetry breaking, Mermin-Wagner Theorem, Landau's Paradigm, the Anderson-Higgs mechanism for spontaneously broken gauge symmetries Notes]
- Lecture 13: Basic principles of superconductivity, Cooper pair formation, mean field description of the BCS Hamiltonian and determination of the gap (which is the T=0 order parameter) [Notes]
- Lecture 14: Path integral approach to the BCS superconductor, gap equation, existence of a critical temperature, microscopic derivation of the Ginzburg-Landau action describing the phase transition [Notes]
- Lecture 15: Properties of the electromagnetic field in a superconductor: Meissner effect and critical magnetic field, coherence length, types of superconductors [Notes]
- Lecture 16: Symmetries and symmetry classes of condensed matter systems [Notes]
- Lecture 17: Symmetries of mean field Hamiltonians (superconductors), the geometry of variational mean field groundstates [Notes]
- Lecture 18: The geometry of variational mean field groundstates, topological superconductors [Notes]
- Lecture 19: Kitaev's Majorana wire, linear response theory and correlation functions, Kubo's formula [Notes]
- Lecture 20: Advanced, retarded and time-ordered response functions, their relations and spectral decomposition (Lehmann representation) [Notes]
- Lecture 21: The resolvent and the density of states, properties of the self-energy, sum rules [Notes]
- Lecture 22: The Kramers-Kronig relations, application: a sum rule for the dielectric function, electromagnetic linear response [Notes]
- Lecture 23: The idea of renormalization, the example of the 2D Ising model on the triangular lattice [Notes]
- Lecture 24: RG fixed points and their stability in the 2D Ising model, renormalization of continuum systems [Notes]
- Lecture 25: Properties of scaling fields, perturbative renormalization of continuum systems, general form of the 1-loop beta-function [Notes]
- Lecture 26: The Gaussian fixed point and the RG equations of the scalar field in dimension 4-epsilon [Notes]
- Lecture 27: The Wilson-Fisher fixed point of the scalar field [Notes]
- Lecture 28: Scaling and critical exponents [Notes (to come)]

**Exercises**

- Exercise 01 (for 23.10.2013)
- Exercise 02 (for 30.10.2013)
- Exercise 03 (for 13.11.2013)
- Exercise 04 (for 20.11.2013)
- Exercise 05 (for 27.11.2013)
- Exercise 06 (for 4.12.2013)
- Exercise 07 (for 11.12.2013)
- Exercise 08 (for 18.12.2013)
- Exercise 09 (for 15.1.2014)
- Exercise 10 (for 22.1.2014)
- Exercise 11 (for 29.1.2014)
- Exercise 12 (for 6.2.2014, to appear)