THEORY OF STRONGLY CORRELATED SYSTEMS

The lecturer can be contacted via e-mail: luca.salasnich@unipd.it

Syllabus The course consists of 48 hours: 40 hours of theory and 8 hours of exercises. The main goal is to learn quantum phenomena of condensed matter physics by means of the path integral approach for quantum particles and quantum fields at zero and finite temperature. Special emphasis is given to macroscopic quantum phenomena: Bose-Einstein condensation, laser physics, superfluidity, and superconductivity. The exam will be a colloquium at the blackboard with chalk, where the student will discuss two topics chosen on the spot by the examination board. The topics of the course are: Path integral for quantum particles (14 hours) Least action principle for particles and fields in classical and quantum mechanics. Lagrangian vs Hamiltonian formulation of classical and quantum physics. The quantum propagator and its properties. Feynman path integral construction of the quantum propagator for a particle in an external potential. Quantum propagator of the free particle. Saddle point and stationary phase approximation. Quantum propagator of a particle in a harmonic potential. Quantum statistical mechanics of many identical particles. Wick rotation and thermodynamics. Quantum tunneling. Path integral for bosons (18 hours) Bosonic coherent states and harmonic potential. Bosonic Matsubara frequencies. Gas of photons at thermal equilibrium. Partition function of interacting identical bosons in quantum field theory. Semiclassical approximation and imaginary time. Ideal Bose gas of massive particles: Bose-Einstein condensation. Interacting Bose gas: Bogoliubov spectrum. Dimensional regularization of Gaussian fluctuations. Condensate fraction of interacting bosons. Gross-Pitaevskii equation and superfluid hydrodynamics. Path integral for fermions (16 hours) Fermionic coherent states and Grassmann variables. Partition Function of interacting identical fermions in quantum field theory. Ideal Fermi gas and the fermionic Matsubara frequencies. Repulsive fermions: Hartree-Fock approximation. Basic properties of superconductivity. Attractive fermions: BCS approximation of pairing. BCS-BEC crossover. Josephson effect and Josephson junctions.

Main text [T] L. Salasnich, Path Integrals for Many-Body Quantum Systems. Ultracold Atoms, Superfluids, and Superconductors, Lecture Notes for UNIPD Students (2024). Suggested books [A1] R. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals. Emended Edition (Dover, 2010). [A2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, 2009). [A3] N. Nagaosa, Quantum Field Theory in Condensed Matter Physics (Springer, 1999). [A4] H.T.C. Stoof, K.B. Gubbels, D.B.M. Dickerscheid, Ultracold Quantum Fields (Springer, 2009). [A5] A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge Univ. Press, 2010). [A6] L. Salasnich, Quantum Physics of Light and Matter (Springer, 2017).