Abstract
Topological numbers are known to play a crucial role in charactering various phases in low dimensional systems. A typical example is the integer quantum Hall effect (IQHE) where quantized Hall conductances are given by Chern numbers [1]. One topic we will present below is unconventional IQHE recently observed experimentally in graphene, i.e., a single sheet of graphite. What is interesting here is that the honeycomb lattice structure yields a Dirac-like dispersion relation which gives an unconventional quantization rule to the Hall conductances [2, 3]. Namely, Dirac particles are indeed realized, which enables us to directly observe the Dirac Landau quantized levels in condensed matter physics. Another topic concerns a new class of topological insulating phases. The IQHE state at each plateau, where the Fermi energy lies in a gap between Landau levels, can be considered as a topological insulator characterized by the Chern number. Here, broken time-reversal symmetry due to external magnetic fields plays a role in topological properties of IQHE. Actually, without magnetic fields, the system recovers time-reversal symmetry and then, Hall conductance vanishes, telling that such systems are topologically trivial. However, it is incorrect [4, 5]: Even without magnetic fields, insulators can be topologically nontrivial due to spin-orbit couplings. In such systems with time reversal symmetry, quantum spin Hall effect (QSHE), a spin-related version of IQHE, could occur and the groundstate is classified by Z_2 invariants.
