Prof. dr Dietmar Ebert, Uniwersytet Humboldta w Berlinie

Integer/Fractional Quantum Hall Effect (IQHE/FQHE) and Graphene

This is an introductory lecture for the IQHE based on Landau quantization of free electrons in a magnetic field and for the FQHE resulting from electron correlations due to dominant Coulomb interactions. First, the Landau quantization of the nonrelativistic Schroedinger Hamiltonian and the effective Dirac-like Hamiltonian is reviewed and the resulting energy spectra of electrons are presented. Next, the quantization of the Hall conductivity and its relation to topological invariants and the first Chern number are discussed. In particular, the analogy to the Gauss-⁠Bonnet theorem in geometry is demonstrated by using the concepts of the Berry phase and the Berry connection/⁠curvature. Notice also that the FQHE inspired the introduction of new composite objects-⁠-⁠"Composite Fermions (CF)"-⁠-⁠which are electrons with attached flux quanta (vortices). It will be shown that the CF of the FQHE can be described by an effective Outline: applications of AdS_4/⁠CFT_3 duality to FQHE, topological insulators and superconductors.