# TEORIA DEI SISTEMI A MOLTI CORPI

- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info

The knowledge of single particle quantum mechanics is assumed to be acquired by students holding the three-year degree in physics, and is a strict propaedeutics constraint.

Elementary notions of equilibrium statistical mechanics are also required.

For the students who have obtained on average a grade of 18/30 or higher in the 3 homework assignments, the final exam is an oral presentation of an argument of the program agreed on with the teacher a week. earlier; the presentation is followed by the defense of the material exposed.The final grade is based on the evaluation of this oral exam.

Students who where not able to reach a sufficient average grade in the 3 home assignments are requested to undergo a written test. Upon passing this test with a sufficient grade (18/30 or higher) they have access to the final oral exam, which proceeds along the lines described above. Again the final grade is based on the evaluation of this oral.

The basic issues of Many-Body Theory are introduced by primarily examining and solving problems

of paradigmatic relevance for the material of the course. In most cases techniques of finite temperature

Green’s functions and of functional integration are used.

The various problems are first introduced and eventually summarized during the lectures.

The aim is to develop the institutional points of the subject by focusing on its conceptual and technical hardships.

As an outcome, the students are expected to acquire the knowledge of the basics of many-body theory as well as the capability to operate with the formalism of Green’s functions as used in this context. The course, mainly

based on acquiring working methods, is thus propaedeutical to theoretical research in this field.

1) 4 hours

Single particle Green functions and potential scattering. Identical particles and second quantization.

Bogolyubov theory of superfluidity. Spontaneous symmetry breaking. Bogolyubov transformations.

2) 8 hours

Zero temperature Green’s fuinctions for free bosons and fermions. D=1 Friedel oscillations.

Spin chain and Jordan-Wigner transformation.

Finite temperature Green’s functions, Matsubara frequencies. Linear response theory, Kubo formula and retarded Green’s function.

3) 6 hours

Computation of spin susceptibility: Pauli paramagnetism and Stoner instability.

Specific heat and self-energy for a gas of interacting electrons. Computation of the density-density

correlation function for a D=1 Fermi gas.

Free phonon propagator. Electron-phonon interaction, computation of the polaron’s self-energy.

4) 8 hours

Functional integral for bosonic systems and finite temperature Green’s functions. Effective action

for a dilute Bose gas. Random Phase Approximation. Dispersion relation for superfluidity.

Peierls effect: phononic Green’s function and electronic singularity.

5) 10 hours

Grassmann variables and fermionic states: resolution of the identity and trace formula. Partition function

for free fermions.

Interacting Coulomb gas: jellium model. Hubbard-Stratonovich transformation, ground state energy

in Random Phase Approximation. Lindhard function. Spectrum of incoherent excitations and screening;

plasma waves and Landau damping. Fermions in contact interaction and zero sound.

6) 8 hours

Cooper instability.

Superconductivity: BCS theory. Bogolyubov Hamiltonian for superconductivity and gap equation.

Ginzburg-landau free energy and equations of motion. Nambu-Gorkov spinors. Diagram technique forsuperconductivity. Computation of the polarization tensor: rigidity of the macroscopic phase.

Gauge invariance in the normal metal and Landau diamagnetism. Computation of the polarization tensor

in the superconducting phase; rigidity of the macroscopic phase.

7) 4 hours

Berry phase. Gas of electrons in D=2. Identical particles in D=2 and anyons. Magnetic translations.

Integer quantum Hall effect: quantization of conductivity as a topological property. Fractional Hall effect and Chern-Simons field.

A.L.Fetter,J.D.Walecka

“Quantum Theory of Many-Particle. Systems”

MacGraw-Hill 1971

J.W.Negele, H.Orland:

“Quantum Many-Particle Systems”

Addison-Wesley 1988

H.Bruus,K.Flensberg:

'Many-body quantum theory in condensed matter theory; An Introduction.'

Oxford U.Press 2007

A.Altland,B.Simons:

“Condensed Matter Field Theory”

Cambridge U.Press 2010

P.Coleman:

“Introduction to Many Body Physics”

Cambridge U.Press 2011

N.Nagaosa:

“Quantum Field Theory in Condensed Matter Physics”

Springer 1999

C.Mudry:

“Lecture notes on field theory in condensed matter theory”

World Scientific 2014

F.Han:

“Modern Course in the Quantum Theory of Solids”

World Scientific 2012

S.Q. Shen:

“Topological insulators”

Springer 2013

The course consists of front lectures organized as follows: after a short presentation

of the theoretical context, the focus goes on discussing and solving paradigmatic problems, with the active

involvement, when possible, of the students. Homework problems are individually assigned on a

monthly basis, for a total of 3 assignments during the course.

As illustrated below, students obtaining at least an average grade of 18/30

in these 3 homeworks are exempted from the written test in the final exam.

Office hours

Wednesday, 9-11 a.m.

Thursday, 3-5 p.m.