Methods of QFT in Condensed Matter Physics
TMP-TA4, WiSe-2014/15
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 – instanton solution
2)
Derivation of the Feynman path integral
(PI) for the evolution operator.
3)
Quantum tunnelling: the Instanton solution – Part I (substitution by Dennis Schimmel).
4)
Quantum tunnelling: the Instanton solution – Part II (substitution by Dennis Schimmel).
Useful notes for Lect-s.
No.3 & 4 (by Prof. A. Nersesyan)
can be found here.
5)
ϕ4-theory for the
classical Ising model vs. quantum tunnelling. Linear
response theory, response functions (operator approach).
Correlation
functions, many particle systems, from PI to FI
6)
The PI for correlation functions, source
fields, generating functionals. QFT-Stat.Phys. correspondence.
7)
Coherent States (CS) for fermions and
bosons, Grassmann variables. How to express the
partition function as the FI over CSs.
Useful notes about CSs and
Gaussian integrals (kindly provided by Lode Pollet)
can be found here.
8)
FI for the generating functional,
boundary conditions for fermionic/bosonic fields; Gaussian integrals. Free
energy of a non-interacting gas.
Interacting
fermions: normal metals and superconductors
9)
Basic properties of the Fermi liquid,
life-time of single particle excitations; free energy of weakly interacting
fermions from FI.
10)
Selection
rule for the loop-diagrams, free energy in the RPA, polarization operator,
screening, plasmons.
11)
Hubbard-Stratonovich transformation, 3 interaction channels. The
density channel: the MF approximation.
12)
RPA
from fluctuations around the MF. Superconductivity: basic properties, the BCS
model.
if you do not remember basics of theory of superconductivity you may look at
the lectures
notes from Mesoscopics, SoSe-2013
13)
The
Cooper channel, ladder diagrams, the Cooper instability; the BCS GS, the BCS
theory as the MFA.
14)
FT
for superconductors: matrix GFs, the gap equation from the saddle-point. The Ginzburg-Landau theory from FI.
15)
Fluctuations
in the GL theory; spontaneous symmetry breaking and Goldstone modes, action for
the phase.
16)
Part
1 of L-16: Londons’ equations.
Luttinger
Liquid, Abelian bosonization
16)
Part
2 of L-16: Non-FL effects, peculiarity of
low-dimensional systems; 1D: Tomonaga-Luttinger model
(1d
massless fermions).
17)
Dzyloshinkii-Larkin diagrammatic solution: loop
cancellation, effective interactions; the Ward identity, non-FL GFs; Bosonization.
18)
Bosonization from FI:
alternative proof of loop cancellation, gauge transformation and Jacobian,
effective bosonic action.
19)
Luttinger Liquid: chiral & dual fields, correlations
functions, LDoS and ZBA, LL with a potential scatterer, RG method.
FT on the closed time-contour, NEGFs, kinetic
equation
20)
Real-time
formulation of FT for arbitrary systems, closed time contour. Example of a Bose
gas: partition function, discretized time and 4 GFs.
21)
Keldysh rotation,
independent GFs, continuous limit, causal structure, FDT, source fields, cumulants in discretized-/continuous representations.
22)
FT
for a Fermi gas on the closed contour: GFs, causality, source fields. Matrix
structure of the polarization operator, causality, FDT.
23)
Action
for interacting systems, the matrix Dyson equation; scale separation, the
Wigner transform, derivation of the semiclassical
kinetic equation.
24)
Screening,
the structure of the interaction propagator and of 
in the RPA; calculating 
. Collision integral
for interacting fermions, in-/out- terms.
Additional 4 lectures
Introduction
to the FT of disordered systems & Integer Quantum Hall effect (if time permits)
Last update: 27.01.2015