  The integral and fractional quantum Hall effects are among the most important
discoveries in condensed matter physics in 1980s. The main results can be
summarized in the conductance matrix. When the filling factor is an integer or
some fractional value, the conductance is quantized. This quantization is
related to the first Chern number.
  The novel properties of the quantum Hall system raised an important question:
Can we have such important quantum properties in other systems? Especially, can
we find a quantum system without magnetic field but maintaining the same
properties in its conductance matrix? This is really the starting point for the
topological insulators, too.
  This issue was first discussed in our paper, published in Physical Review B,
1987. Here, we re-post this paper and want to emphasize that the physics
discussed in our paper is general and profound. The essence of quantum Hall
effect can be generalized to many quantum systems, either fermion system or
boson system. If the quantum system has discrete energy spectrum and depends on
two or more periodic parameters, the response coefficient will be the same as
the conductance of quantum Hall effect, dissipation free for the longitudinal
ones and quantized for off-diagonal ones.
