Institute for Complex Systems - Sapienza - CNR

  • Full Screen
  • Wide Screen
  • Narrow Screen
  • Increase font size
  • Default font size
  • Decrease font size
ISC Sapienza Phase Transitions and Critical Phenomena

Phase Transitions and Critical Phenomena

E-mail Print PDF


Instructor: José Lorenzana

Place: Physics Department,  Sapienza, University of Rome.

Number of hours: 48

Level: Undergraduate (Laurea  magistrale)

Years: 2010/2011 2012/2013 2013/2014


The propose of the course is to learn the properties of some systems in the broken symmetry phase with emphasis on quantum systems.


1. Quantum statistical mechanics and density matrix formalism.

2. Second Quantization

Creation and destruction operators. Basis vectors for a system of N identical particles and Fock space. Hamiltonian density operator of a system of N particles in second quantization.

Reduced density matrices. Hartree-Fock approximation in the second quantization formalism.

3. Bosons and superfluidity

Phenomenology of superfluid helium (bosonic case). Landau criterion of superfluidity. Condensation of a Bose gas. Superfluid helium excitation spectrum. Condensation and off-diagonal long-range order (ODLRO) of the one particle reduced density matrix for a system of interacting bosons. Quantization of the circulation and critical velocity. Bogolubov model.

4. Fermions and superconductivity

Fermion gas. Interacting fermions and Landau theory of normal Fermi liquids.

Superfluid helium (fermionic case). Phenomenology of superconductors and London theory.

ODLRO in the order parameter and two particle reduced density matrix.

Meissner effect, magnetic flux quantization and Josephson effect in superconductors.

Bardeen, Cooper and Schrieffer theory as a generalized Hartee-Fock approximation.