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