Fakultät für Physik




TMP-TA6 many body physics (SoSe 2011) – Lecture

Date Notes Subject

Introduction to Second Quantization: Chapter 2.1, A. Altland and B. Simon, "Condensed Matter Field Theory" (2006).

pdf Appendix C. Second Quantization: M. P. Marder, "Condensed Matter Physics" (2000).
03.05.11 pdf Green's Functions GF1-16: Motivation, thermal averaging, Schrödinger, Heisenberg Interaction pictures, time-ordering operators, imaginary time
06.05.11 pdf GF17-23': Thermal GF, Periodicity in imaginary time direction, Matsubara transformation
10.05.11 pdf Kubo formula (O. Yevtushenko)
13.05.11 pdf GF24-36: Kubo Formula, definitions of G<, G>, retarded/advanced/causal GF, when is analytic continuation allowed?,
periodicity in complex time direction, relation between thermal and causal GF by analytic continuation
17.05.11 pdf GF37-42, (GF43-47 nonexistant), GF48-53: Spectral representations, relation between Matsubara and retarded/advanced via analytic continuation
in frequency domain, interpretation of spectral function, spectral sum rules
20.05.11 pdf GF54-68: complex conjugation, obtaining retarded/advanced from Matsubara by analytic continuation in time domain, expressing G>, G< in terms
of spectral function, fluctuation-dissipation theorem, conditions for analytic continuation from G(i omega_n) to G^{R/A}(w +/- i0), Matsubara sums
24.05.11 pdf Perturbation Theory PT1-8: Single-particle potential, equations of motion, Dyson equation, diagrammatic rules, definition of interaction term
27.05.11 pdf PT9-15: Expectation value of H in terms of spectral function, interaction picture in imaginary-time domain, definition of n-point correlators,
their periodicity properties, translational invariance in time implies frequency conservation
31.05.11 pdf PT16-29: Wick's theorem for thermal averages proved by cyclic permutations under trace, Wick's theorem for thermal GF, proved from previous
result, and proved using equations of motion
03.06.11 pdf PT30-37: Diagrammatic perturbation theory: partition function, 1-point function, Feynman rules
07.06.11 pdf PT38-51: 2nd order diagrams, connected diagrams, combinatorical factors, two-point functions, transforming to momentum and frequency representation, Hartree and Fock diagrams, translational invariance in time and space, Feynman rules
07.06.11 pdf PT52-57: Dyson equation, self-energy, quasiparticle weight and lifetime, Hartree-Fock approximation, two-particle connected diagram
17.06.11 pdf PT58-64: Hartree-Fock wavefunctions, density-response to external potential
21.06.11 pdf PT65-74: Screening of external potential, Random-Phase Approximation, polarization bubble, Lindhardt formula, plasma resonance,
Thomas-Fermi screening length
21.06.11 pdf Dis1-13: Disorder potential, self-averaging, impurity averages, Feynman rule for disorder averaging, 1-particle GF, elastic scattering time
24.06.11 pdf Dis14-25: Disordered systems: Higher order diagrams
28.06.11 pdf Dis26-32: Disordered systems: Conductivity before disorder averaging
01.07.11 pdf Dis33-41: Disordered systems: Conductivity after disorder averaging
05.07.11 Cancellation of diamagnetic term (using identity derived by gauge transformation)
08.07.11 pdf K0-7: Kondo Model, poor man scaling
12.07.11 pdf AM1-13Anderson model, multi-level dots, Schrieffer-Wolff transformation
15.07.11 pdf Numerics: NRG, DMRG and Matrix Product States
19.07.11 pdf Superconductivity: p. 1-6: Basic properties, electron-phonon interaction, phonon-mediated attraction, Cooper instability
22.07.11 pdf Superconductivity: p. 7-14: Properties of the vertex function, critical temperature, statistical approach, anomalous Green's functions
25.07.11 pdf Superconductivity: p. 15-19: Gorkov's equations, reduced BCS model, spectrum of excitations, wave-function of condensate
29.07.11 pdf Superconductivity: Matrix Green's functions of Nambu, DoS of excitations, gap equation, T-dependence of gap, Anderson theorem
25.07.11 Remarks about the exam (Tuesday, August 9, from 10:00 - 13:00, Room 348/349)
You can bring along any material (books, lecture notes) you want.
The exam will attempt to test both your understanding of the material covered in lectures & excercises and your fluency with the basic calculational tools developed during the course. However, due to the limited time available, no very long calculations will be required. For example, if you are asked to evaluate a Matsubara sum, this will be doable in a few lines (if you know what to do!)
Typical questions could involve, for example:
- doing the bosonic version of a problem for which the fermionic case was done in lecture or tutorial, or vice versa;
- discussing a specific example of a problem or theorem that was discussed in full generality in lecture or tutorial;
- writing down and evaluating the algebraic expression associated with a given Feynman diagram (including explaining the combinatiorial factor, if any);
- discussing the physical interpretation of a given diagram;
- extracting physical quantities (life-times, dispersion relation of excitations) from a given diagram and/or the corresponding correlation function;
- explaining the justification for (and/or limitations of) certain "standard" approximation schemes;
- ...
A more explicit set of hints will be published on this website on Monday morning, August 8, by 9:00 am, to help you to fine-tune your preparations on the last day before the exam.
08.08.11 More detailed hints: the following topics will feature in the exam:
- formal properties of quantum fields, thermal, retarded, advanced Green's functions
- know how to do Matsubara sums in detail, including ones involving Green's functions with higher order poles
- understand calculation of charge susceptibility in great detail; exam will contain an analogous calculation of a different physical quantity
- Hartree-Fock theory
- Feynman rules for disordered systems, which diagrams matter for calculation of electron lifetime, which don't, why not?
- Kondo problem, poor man scaling, derivation of RG equation
- Superconductivity: Gorkov equations, density of states, gap equation, calculation of Delta(T=0), Tc