Table of Contents

1 Green’s function and Observables

               T=0, spectral representation, sum rules
               Matsubara Green functions
               Connection to observables
               Feynman diagrams

2 Meanfield theory of superconductivity, BCS Theory

               Hartree-Fock
               Gorkov equations
               Singlet order parameters
               Pairing interaction
               Self-consistency
               Critical temperature
               Current carrying state
               Abrikosov vortex lattice
               Higher angular momentum pairing

3 Functional methods

               Green’s function from the path integral
               Grassmann numbers
               Coherent state path integral
               Gaussian integration
               Coherent state path integral
               Superconductivity from the path integral
               Hubbard Stratonovich transformation
               Saddle point equation

4 Collective modes

               Higgs and Leggett mode from the path integral
               Anderson-Higgs mechanism
               Symmetry action in Landau Ginzburg theories
               Anderson’s theorem: impurity scattering

5 Higgs spectroscopy: experimental and theoretical methods

               General response formalism from effective action
               Linear response
               Linear coupling to Leggett mode
               Fluctuation dissipation theorem
               Gauge invariance
               Nonlinear response
               Kernel method of quasi-equilibrium
               Sum-Frequency Generation (SFG), Difference-Frequency-Generation (DFG)
               Excitation schemes for collective modes: Quench, drive (THG)
               Polarization dependence
               Raman spectroscopy

A manuscript of the lecture will be delivered.