Physics

LARA BENFATTO
LARA BENFATTO

GENERAL OBJECTIVES:
The course aims to introduce the foundations of Superconductivity and Superfluidity. A preliminary part will be devoted to the phenomenological London and Ginzburg-Landau theories. The latter will be used to introduce the more general topic of spontaneous symmetry breaking in second-order phase transition, and the Anderson-Higgs mechanism for superconductivity. After discussion of the second-quantization for many-body fermionic and bosonic systems the focus will be on the microscopic models for superconductors (BCS Bardeen_Cooper e Schrieffer theory) and superfluids.
The final part will consist in a brief overview of current research topics on unconventional superconductors.

SPECIFIC OBJECTIVES:
A - Knowledge and understanding
OF 1) To know the basis of the superconducting phenomenon, its phenomenological and microscopic description and its experimental applications
OF 2) To understand key concepts as spontaneous symmetry breaking and order parametr for a phase transition, with particular emphasis on continous symmetries.
OF 3) To know basic applications of second quantization to fermionic and bosonic many-particle systems
B - Application skills
OF 4) To be able to describe the superfluid phenomen both for fermions and bosons, and its theoretical and experimental implications
C - Autonomy of judgment
OF 5) To be able to integrate the knowledge acquired in order to apply in the more general context of unconventional superconduvtivity and interacting fermionic systems
E - Ability to learn
OF 6) Have the ability to read scientific papers in order to further explore some of the topics introduced during the course.

a) A fundamental prerequisite is that students must have the knowledge requested by the first level University degree in Physics or in Astronomy and Astrophysics. Specific competences are requested in electromagnetism, analytical mechanics, statistical mechanics and quantum mechanics

b) A fundamental prerequisite is that students must have the knowledge requested by the mandatory courses of the first semester of the Master’s degree in Physics, in particular Condensed Matter (band theory and semiclassical model of transport in metals)
f) It is useful to have a good knowledge of the lagrangian formalism and second quantization, as introduced in the Relativistic Quantum Mechanics course.

Part I: Review of fundamental concepts
Transport properties in metals. Gauge invariance in first and second quantization. Maxwell equations in matter.

Part II: Phenomenology of superconductors
Zero resistance state, Meissnerr effect, gap opening.
London theory of superconductivity. Paramagnetic and diamagnetic current, rigidity and gauge-invariance breaking.
Pippard's generalization of London model.
Magnetic phase diagram of Type-I superconductors. Condensation energy, latent heat, specific-heat jump, intermediate state.
Magnetic phase diagram of Type-II SC. Structure of the vortex, energy of a vortex line, estimate of Hc1.

Part III: Ginzburg-Landau theory
Landau theory of phase transition. Example of a scalar order parameter. Spontaneous symmetry breaking and order parameter.
Susceptibility, thermal fluctuations and fluctuation-dissipation theorem. Levanyuk-Ginzburg criterium. Mean-field critical exponents.
Ginzburg-Landau model for the complex order parameter. Derivation of the GL equations. Superconducting phase and gauge invariance. Flux quantization.
Collective excitations: Higgs and Goldstone mode, Anderson-Higgs mechanism for the gauge field.
Josephson model for S-I-S junction and SQUID.

Part IV: the microscopic BCS theory
Second quantization for many-body systems. Hamiltonian in second quantization and thermodynamic potentials for independent particles.
Dielectric function in the presence of phonons and effective attractive potential. Cooper problem.
BCS Hamiltonian and mean-field solution. Estimate of the gap and Tc. BCS wave function. Coherent states.
Dynamical properties within the Anderson pseudospin precession.

Parte V: superfluidity
Phenomenology and phase diagram of He4. Landau criterium. Bose condensation and interacting bosons.
Bogoliubov spectrum. Depletion of the condensate and superfluid density.
Phonon and roton spectrum. Vortices in neutral superfluids

Introductory topics related to courses of the first level University degree and of the first semester of the master degree are available on standard textbooks of condensed matter and (relativistic) quantum mechanics
Ashcroft and Mermin, Solid State Physics
J. J. Sakurai Modern Quantum Mechanics
K. Huang, Statistical Mechanics
F. Mandl and G. Shaw, Quantum Field Theory

Superconductivity and superfluidity is discussed in the following books
P.G. De Gennes, Superconductivity of Metals and Alloys
J. F. Annet, Superconductivity, Superfluids and Condensates
M. Tinkham, Introduction to superconductivity
P.Coleman Introduction to Many-Body Physics

Precise indication on the relevant references and additional extra material will be indicated in correspondence with the topics covered in the course on the elearning page: https://elearning.uniroma1.it/course/view.php?id=12850

The lecture format is through blackboard. Selected topics will be assigned as homeworks. Their correction will be done during the lectures by the students, with the support of the teacher.

The lecture format is through blackboard. Selected topics will be assigned as homeworks. Their correction will be done during the lectures by the students, with the support of the teacher.

The final grading will be based on an oral exam (about 45 minutes) aimed at assessing the competences acquired by the student during the course.
The oral exam consists of an interview on the topics illustrated in the course. To pass the exam the student must be able to present a topic or repeat a calculation discussed during the course and to apply the methods learned to examples and situations similar to those already discussed.
The evaluation will take into account:
- correctness of the concepts exposed;
- clarity and rigor of presentation;
- ability to analytically develop the theory.

London theory
Phase diagram of superconductors
Ginzburg-Landau theory
Cooper problem
Gap self-consistence equation at T=0
Gap self-consistence equation at T>0