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Mathematical Methods of Condensed Matter Physics

Lecturer: 
Marcello Porta
Course Type: 
PhD Course
Academic Year: 
2020-2021
Period: 
October - June
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:

 

  1. Elements of spectral theory, with applications to lattice Schroedinger operators.
  2. Effect of disorder on quantum dynamics. The Anderson model, and its phase diagram.
  3. Relation between spectra and dynamics: the RAGE theorem.
  4. Anderson localization through path expansions.
  5. Quantum transport, heuristic linear response theory. Making it rigorous: the adiabatic theorem.
  6. Quantum Hall effect. Bulk-edge correspondence.
  7. Time-reversal invariant systems. Example: Kane-Mele model. Z2 classification and bulk-edge duality.
  8. Interacting lattice models. Grand canonical formulation, Fock space, perturbation theory.
  9. Cluster expansion, convergence of fermionic perturbation theory, analyticity of the Gibbs state.
  10. 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.
  11. Universality of transport in quantum Hall systems and semimetals.

 

  • References:

 

  1. M. Aizenman and S. Warzel. Random Operators. American Mathematical Society.
  2. G. M. Graf. Aspects of the integer quantum Hall effect. Proceedings of Symposia in Pure Mathematics (2007).
  3. G. M. Graf and M. Porta. Bulk-edge correspondence for two-dimensional topological insulators. Comm. Math. Phys. 324, 851-895, (2013).
  4. M. Porta. Mathematical Methods of Condensed Matter Physics. Lecture notes.
  5. A. Giuliani, V. Mastropietro and S. Rychkov. Gentle introduction to rigorous Renormalization Group: a worked fermionic example. arXiv:2008.04361
Research Group: 
Geometry and Mathematical Physics
Location: 
A-136 and Zoom, sign in to get the link
Next Lectures: 
Friday, January 22, 2021 - 09:00 to 11:00
Friday, January 29, 2021 - 09:00 to 11:00
Friday, February 5, 2021 - 09:00 to 11:00
Friday, February 12, 2021 - 09:00 to 11:00
Friday, February 19, 2021 - 09:00 to 11:00
Friday, February 26, 2021 - 09:00 to 11:00
Friday, March 5, 2021 - 09:00 to 11:00
Friday, March 12, 2021 - 09:00 to 11:00
Friday, March 19, 2021 - 09:00 to 11:00
Friday, March 26, 2021 - 09:00 to 11:00
Friday, April 2, 2021 - 09:00 to 11:00
Friday, April 9, 2021 - 09:00 to 11:00
Friday, April 16, 2021 - 09:00 to 11:00
Friday, April 23, 2021 - 09:00 to 11:00
Friday, April 30, 2021 - 09:00 to 11:00
Friday, May 7, 2021 - 09:00 to 11:00
Friday, May 14, 2021 - 09:00 to 11:00
Friday, May 21, 2021 - 09:00 to 11:00
Friday, May 28, 2021 - 09:00 to 11:00
Friday, June 4, 2021 - 09:00 to 11:00
Friday, June 11, 2021 - 09:00 to 11:00
Friday, June 18, 2021 - 09:00 to 11:00
Friday, June 25, 2021 - 09:00 to 11:00

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