  The study of topological property of band insulators is an interesting branch
of condensed matter physics. Two types of topologically nontrivial insulators
have been extensively studied. The first type is characterized by a nonzero
TKNN invariant or Chern number[1] which is directly related to the quantization
of Hall conductance in the integer quantum Hall effect. Haledane propose a
model with this type of band structure even in the absence of a macroscopic
magnetic field[2]. We refer to such materials "Chern insulator". The second
type called "Z2 topological insulators" is proposed recently[3, 4]. Quantum
spin Hall effect has been predicted and observed in such systems.[5, 6].
Despite the recent intensively study there are still some fundamental problems
that aren't quite clear about Z2 insulators even at the single-particle level.
For example, it's claimed that Z2 insulators will return to its origin state
after two cycles, thus coupling to the reservoirs is important for the Z2
insulators to continuously pump spin[7]. Theoretical and experimental results
show quantum spin Hall effect is an edge state transport property of the
materials and coupling to the reservoir seems not play an important role. So
the Z2 picture is not satisfactory in explaining these phenomena. We study the
relationship of the ground states of Z2 insulators and that of Chern insulator.
Combined with the results of recent researches on polarization of Chern
insulators[8] and topology of edge states[9] we propose a coherent physics
picture of topological insulators.
