  A brief introduction to topological phases is provided, considering several
two-band Hamiltonians in one- and two-dimensions. Relevant concepts of the
topological insulator theory, such as: Berry phase, Chern number, and the
quantum adiabatic theorem, are reviewed in a basic framework, which is meant to
be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ
models. The role of the electromagnetic interaction in the topological
insulator theory is addressed in the light of the Pseudo quantum
electrodynamics (PQED). The well known parity anomaly for massless Dirac
particle is reviewed in terms of the Chern number. Within the continuum limit,
a half-quantized Hall conductivity is obtained. Thereafter, by considering the
lattice regularization of the Dirac theory, we show how one may obtain the well
known quantum Hall conductivity for a single Dirac cone. The renormalization of
the electron energy spectrum, for both small and large coupling regime, is
derived. In particular, it is shown that massless Dirac particles may, only in
the strong correlated limit, break either chiral or parity symmetries. For
graphene, this implies the generation of Landau-like energy levels and the
quantum valley Hall effect.
