A.A. di erogazione 2019/2020

Laurea Magistrale in FISICA
 (A.A. 2019/2020)


Anno di corso: 
Tipologia di insegnamento: 
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Primo Semestre
Ore di attivita' frontale: 
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Lezione (48 ore)

To develop the basic elements of many body theory as well as the expertise in using its analytical tools.
Upon completing the course the student will be able to
-operate with the Green functions and the functional formalism proper to condensed matter models
-explain at the theoretical level (i.e. not as a mere phenomenological description) the superfluidity, the spectrum of excitations in a Coulomb gas, the superconductivity, the integer and fractional Quantum Hall effect
he will then have acquired the basic skills allowing to
-find his way when reading research papers
-collaborate to a research work in condensed matter theory

Single particle quantum mechanics. Equilibrium statistical mechanics. Phase transitions.

1.Second quantization
Single particle Green functions and potential scattering. Identical particles and second quantization.
Zero temperature Green’s functions for free bosons and fermions.
Functional integral for bosonic systems.
Finite temperature Green’s functions, Matsubara frequencies. Linear response theory, Kubo formula and retarded Green’s function.
Functional integral for fermionic systems.
Grassmann variables and fermionic states: resolution of the identity and trace formula. Partition function for free fermions.
Bogolyubov theory of superfluidity. Spontaneous symmetry breaking. Bogolyubov transformations.
Effective action for a dilute Bose gas. Random Phase Approximation. Dispersion relation for superfluidity.
3.Instabilities and collective effects
D=1 Friedel oscillations. Spin chain and Jordan-Wigner transformation.
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.
Electron-phonon interaction, computation of the polaron’s self-energy. Peierls effect: phononic Green’s function and electronic singularity.
4.Coulomb gas
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.
Cooper instability. Bogolyubov Hamiltonian for superconductivity and gap equation.
Ginzburg-Landau free energy and equations of motion. Nambu-Gorkov spinors. Diagram technique for superconductivity. 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.
6.Quantum Hall effect
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.

The lectures are based on working through a sequence of problems, covering the main topics of the program.
The teacher first introduces the theoretical issues of the problem, then proceeds to its discussion and solution; in this process the students are fostered to play an active role.
This group activity, developed during the lecture time, is supplemented by the individual work: homework assignments, under teacher’s check, must be done on a biweekly basis. In particular, two of such assignments, corresponding to the midterm and to the end of the course, do contribute to the final grade.

The acquired skills are evaluated as follows: each of the two (midterm and end-of-the-course) homework papers contributes to 1/3 of the final grade.
The midterm homework covers points 1,2,3 of the program, and the end-of-the-course one refers to points 4,5,6.
In the final exam the student first gives an oral presentation of a program’s topic,
previously agreed with the teacher, and then sustains a discussion on related issues.

N. Nagaosa, Quantum Field Theory in Condensed Matter Physics, Springer 1999
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
Supplementary suggested textbooks (*):
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
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 201
S.Q. Shen, Topological insulators, Springer 2013
* No other teaching aids, such as printed or e-learning supported lecture notes, are available. Copies of the textbooks can be found at the Como Science Library.

Homework correction is made during office hours.
The course is based on the Nagaosa textbook, with the unavoidable supplement of Fetter-Walecka and Negele-Orland.

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A.A. 2020/2021

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