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

Abstract

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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 equation and
the Dirac-like Hamiltonian is reviewd 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. Note 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 Chern-Simons QFT with fractional
charge/statistics. Outline: possible applications to graphene, applications of
AdS_4/CFT_3 duality to FQHE, topological insulators and superconductors.