Methods of QFT in Condensed Matter Physics
TMP-TA4, WiSe-2012/13
Lecturer: Dr. Oleg Yevtushenko, Tutor: Dennis Schimmel
Lecture topics:
1)
Introduction: From classical field theory
to effective field theory of lattice vibrations.
2)
Feynman path integral for evolution
operator. Response functions and Kubo formula.
3)
Example of linear response: Conductivity.
Path integrals for time ordered correlation functions.
4)
Path integrals on Keldysh
contour. 6 Green’s functions, their analytical structure. Analytical
continuation.
5)
ϕ4-theory for the
classical Ising model and quantum tunnelling: Instanton solution.
6)
Properties of the instanton,
zero-mode problem. A pair instanton-antiinstanton.
7)
Dilute gas of instantons,
tunnelling/survival probability. A particle in a dissipative environment.
8)
“QFT-Stat.Mech”
correspondence, quantum oscillator vs. classical string. Source fields.
Coherent states.
9)
Functional integrals for partition
functions and observables. Matsubara vs. Keldysh
techniques. Free energy of a gas.
10)
Fermi
liquid theory, quasi-particles and their life-time, perturbation theory in the
interaction constant.
11)
Free
energy of interacting fermions, leading diagrams. Polarization operator, RPA,
screening, plasmons.
12)
Hubbard-Stratonovich transformation, 3 interaction channels. The
density channel: MFA.
13)
RPA
from density fluctuations. Beyond the FL theory: peculiarity of low-dimensional
systems.
14)
Tomonaga-Luttinger model (1d
massless fermions): failure of the perturbation theory.
15)
Dzyaloshinskii-Larkin
diagrammatic solution. FT approach: loop cancellation and changing variables.
16)
Bosonization: chiral and
dual fields, hydrodynamic action and Luttinger
liquid.
17)
Correlation
functions in clean LL, LDoS and ZBA, application of
conformal methods to LL, LL with an impurity.
18)
Models
of disordered systems, disorder averaging. “The problem of denominator” and its
FT solutions.
19)
The
SuSy trick. Basics of super-algebra. Disorder
averaging of the retarder/advanced Green’s functions.
20)
SuSy Hubbard-Stratonovich transformation. Disorder averaged DoS (“semi-circle”) from a saddle point integration.
21)
SuSy Mean-field
solution for continuous disordered systems. SuSy FT
for 2-point correlation functions.
22)
Averaging
2-point correlation functions. Supermatrix 0d
nonlinear σ-model and its applications.
23)
Diffusive
nonlinear σ-model. The universal limit. Diffusons
and Cooperons as Goldstone modes, perturbative
expansion.
24)
FT
on a closed time contour, Green’s functions from the matrix- and continuous representations.
Keldysh rotation, FDT.
25)
Causal
structure of action. Source fields, generating functional and cumulants. Action of arbitrary Fermi-/Bose- gases.
26)
Keldysh technique:
response functions, polarization matrix, interactions, Dyson equation.
27)
Derivation
of the kinetic equation: Wigner transform, kinetic term, collision integral.
Concluding remarks.
Crash course in theory of disordered systems:
1)
Models
of disorder
o
Gaussian
disorder, Edwards model, lattice models.
2)
Electrons
in a random potential
o
Disorder
averaged Green’s Functions, Dysons equation,
self-energy, elastic collision time.
3)
Quantum
diffusion
o
Drude-Boltzmann
approximation,
o
diffusion
approximation, ladder diagrams (Diffusons),
o
Coherent
propagation (Cooperons),
o
Duffuson-Cooperon perturbation
theory.
4)
Coductivity of disordered
metals
o
the
Kubo in terms of Green’s functions,
o
average
classical dc conductivity, the Einstein relation,
o
Thouless energy, mean
level spacing and conductance.
5)
Weak
localization correction and dephasing.
Last update: 4.02.2013