Methods of QFT in Condensed Matter Physics

TMP-TA4, WiSe-2013/14

 

Lecturer: Dr. Oleg Yevtushenko,    Tutor: Dennis Schimmel

 

Lecture topics:

1)         Introduction: From classical field theory to effective field theory of lattice vibrations.

 

                              PI in quantum mechanics, tunnelling – instant solution

 

2)         Derivation of the Feynman path integral (PI) for evolution operator.

3)         ϕ4-theory for the classical Ising model and quantum tunnelling: the Instanton solution.

4)         Properties of the instanton, zero-mode problem. An instanton-antiinstanton pair.

5)         Dilute gas of instantons, tunnelling/survival probability, validity of the instanton method.

 

                              Many particle systems, response functions, FI

 

6)         Linear response theory, response functions, the PI for the correlation function.  

7)         QFT-Stat.Phys. correspondence, source fields, generating functional for correlation functions and cumulants
     (substitution by Lode Pollet).

8)         Partition function: changing from Fock states to coherent states (CS), CS for fermions and bosons, Gaussian integrals
     (substitution by Lode Pollet).

 

                              Thermodynamics, Fermi gas and Fermi liquid

 

9)           Functional integrals (FI) for the partition function and observables; free energy of the Fermi gas from FI.

10)          Basic properties of the Fermi liquid, life-time of single particle excitations; free energy of weakly interacting fermions from FI.

11)          Selection rule for the loop-diagrams, free energy in the RPA, polarization operator, screening, plasmons.

 

                              HS transformation, MF, screening, superconductivity

 

12)          Hubbard-Stratonovich transformation, 3 interaction channels. The density channel: the MF approximation.

13)          RPA from fluctuations around the MF.

13)          Superconductivity: basic properties, the BCS model.

14)          The Cooper channel, ladder diagrams, the Cooper instability; the BCS GS, the BCS theory as the MFA.

15)          FT for superconductors: matrix GFs, the gap equation from the saddle-point. The Ginzburg-Landau theory from FI.

16)          Fluctuations in the GL theory; spontaneous symmetry breaking and Goldstone modes, Londons’ equations.

 

                              Luttinger liquid, bosonization

 

17)          Non-FL effects, peculiarity of low-dimensional systems; 1D: Tomonaga-Luttinger model (1d massless fermions), g-ology.

18)          Dzyloshinkii-Larkin diagrammatic solution: loop cancellation, effective interactions; the Ward identity, non-FL GFs; Bosonization.

19)          Bosonization from FI: alternative proof of loop cancellation, gauge transformation and Jacobian, effective bosonic action.

20)          Luttinger Liquid: chiral & dual fields, correlations functions, LDoS and ZBA, LL with a weak scatterer/link, RG method.

 

                              FT on the closed time-contour, NEGFs, kinetic equation

 

21)          Real-time formulation of FT for arbitrary systems, closed time contour. Example of a Bose gas: partition function, discretized time and 4 GFs.

22)          Keldysh rotation, 3 independent GFs and their symmetries, continuous limit, causal structure, fluctuation-dissipation theorem.

23)          Source fields, cumulants in discretized- and continuous representations. FT for a Fermi gas on the closed contour: GFs, causality, source fields.

24)          Matrix structure of the polarization operator, causality, FDT. Closed-contour action for interacting systems, the matrix Dyson equation.

25)          QM equation for the Keldysh component of the GF, scale separation, using the Wigner transform to derive the semiclassical kinetic equation.

26)          Screening, the matrix structure of the interaction propagator and of the polarization operation in the RPA. Calculation of the self-energy.

27)          Collision integral for interacting fermions, Pauli factors, in-/out- terms.

 

                              Disordered systems

 

27)          Disordered systems: models, disorder averaging, problem of denominator.

28)          3 FT tricks for disorder averaging; the Keldysh technique: the HS transformation, saddle point equation and its solution (SpS) with causal structure.

29)          Reduced symmetry and manifold of degenerate SpSs, massive and Goldstone modes, smooth fluctuations around SpSs, action of NLsM.

30)          Solving the nonlinear constraint, the Usadel equation, diffusion propagators, QM effects obtained from interaction of Goldstones, sources in the NLsM.

        Additional topic: disorder averaged density response -  NLsM derivation.

30)          Concluding remarks.

 

 

Last update: 13.02.2014