MANY BODY PHYSICS
The knowledge of single particle quantum mechanics, as provided in the courses Quantum Physics I and II, is a strict requirement; perturbation theory applied to single-particle quantum mechanics is strongly recommended. Elementary notions of thermodynamics are also required, provided in the courses of Mechanics and Physics of Matter.
For the students who have obtained on average a grade of 18/30 or higher in the homework assignments, the final exam is a 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 exam. Students who were not able to reach a sufficient average grade in the 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 exam, which proceeds along the lines described above.
The basic issues of Many-Body Theory are introduced by examining and solving paradigmatic problems, mainly using Green’s functions techniques.The aim is to focus on the basic conceptual as well as technical hardships of the discipline, going beyond and extending topics covered in the courses of Quantum Physics I and II, Physics of Matter, Statistical Physics I. The course is preparatory for research in condensed matter, and may be of interest also for students inclined to general theory, as functional methods are applied to problems of general relevance.
As an outcome, the students are expected to
- acquire the knowledge of the basics of Many-Body Theory
- develop the capability to operate with the formalism of temperature Green’s functions.
1) Second quantization and Green’s functions
1.1) 4 hours: Perturbation theory in quantum mechanics; Phase transitions and order parameters
Single particle Green functions and potential scattering. Identical particles. Mean field hamiltonians. Spontaneous symmetry breaking. Bogolyubov theory of superfluidity. Bogolyubov transformations.
1.2) 8 hours: Many-body Green’s functions
Zero temperature Green’s functions for free bosons and fermions. Finite temperature Green’s functions, Matsubara frequencies. Linear response theory, Kubo formula and retarded Green’s function. Friedel oscillations. Spin chain and Jordan-Wigner transformation.
1.3) 6 hours: Correlations
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.
2)Functional integrals
2.1) 6 hours: Effective action for bosonic systems
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.
2.2) 10 hours: Fermions: incoherent vs collective excitations
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.
3)Superconductivity
3.1) 4 hours: Instabilities
Peierls instability: phononic Green’s function and electronic singularity. London phenomenological approach.Cooper instability.
3.2) 8 hours: Effective theories
BCS theory. Bogolyubov Hamiltonian for superconductivity and gap equation. Ginzburg-Landau free energy and equations of motion. Nambu-Gorkov spinors. Computation of the polarization tensor: rigidity of the macroscopic phase. Gauge invariance in the normal metal and Landau diamagnetism.
4) 4 hours (optional): Geometry of single-particle band spectra
Berry phase. Gas of electrons in D=2. Identical particles in D=2 and anyons. Magnetic translations. Integer quantum Hall effect.
The course consists of lectures (in remote if the emergency persists) 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 2 assignments during the course. As illustrated below, students obtaining at least an average grade of 18/30 in these homeworks are exempted from the written test in the final exam.
Office hours: in remote, scheduled on a private basis contacting the teacher at the following email: vincenzo.benza@uninsubria.it
Professors
Borrowers
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Degree course in: PHYSICS