Math Courses

Equilibrium Quantum Lattice Systems

by Daniel Ueltschi (University of Warwick)


The course will include 6 classes, of 2 hours each, see this page for the current schedule. 

Here are the topics:

1. Spin systems.
    Hilbert spaces, spin operators, symmetries, hamiltonians. 
    Finite-volume Gibbs states. Correlation functions. 
2. Free energy. 2D systems.
    o  Infinite volume limit of the free energy.
    o  No long-range order in two dimensions, in systems with continuous symmetry.
3. Spontaneous magnetisation.
    The proof of the existence of spontaneous magnetisation using the method of reflection positivity and infrared bounds.
4. Fermionic and bosonic systems.
    Fock space, creation and annihilation operators. Hamiltonians rewritten using creation and annihilation operators.
5. Bose-Einstein condensation.
    o  Symmetry breaking and the concept of off-diagonal long-range order.
    o  Bose-Einstein condensation for the ideal gas.
6. Probabilistic representations of quantum lattice systems
    Loop representation for the Heisenberg model. Poisson-Dirichlet conjecture. Calculations of the "spin-Laplace transform".
All lectures will be held at the Main Lecture Hall in Viale Crispi 7 building.