Lecturer:
Course Type:
PhD Course
Academic Year:
2020-2021
Period:
October - April
Duration:
60 h
Description:
- The course will discuss rigorous results in quantum mechanics and in statistical mechanics, relevant for condensed matter physics. Topics to be covered include:
- Elements of spectral theory, with applications to lattice Schroedinger operators.
- Effect of disorder on quantum dynamics. The Anderson model, and its phase diagram.
- Relation between spectra and dynamics: the RAGE theorem.
- Anderson localization through path expansions.
- Quantum transport, heuristic linear response theory. Making it rigorous: the adiabatic theorem.
- Quantum Hall effect. Bulk-edge correspondence.
- Time-reversal invariant systems. Example: Kane-Mele model. Z2 classification and bulk-edge duality.
- Interacting lattice models. Grand canonical formulation, Fock space, perturbation theory.
- Cluster expansion, convergence of fermionic perturbation theory, analyticity of the Gibbs state.
- Approach to criticality: the rigorous renormalization group. Applications: interacting graphene, nonintegrable perturbations of the 2d Ising model. Construction of a nontrivial RG fixed point: lattice models with long range hoppings.
- Universality of transport in quantum Hall systems and semimetals.
- References:
- M. Aizenman and S. Warzel. Random Operators. American Mathematical Society.
- G. M. Graf. Aspects of the integer quantum Hall effect. Proceedings of Symposia in Pure Mathematics (2007).
- G. M. Graf and M. Porta. Bulk-edge correspondence for two-dimensional topological insulators. Comm. Math. Phys. 324, 851-895, (2013).
- M. Porta. Mathematical Methods of Condensed Matter Physics. Lecture notes.
- A. Giuliani, V. Mastropietro and S. Rychkov. Gentle introduction to rigorous Renormalization Group: a worked fermionic example. arXiv:2008.04361
Research Group:
Location:
A-136 and Zoom, sign in to get the link