Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance

We study the phases of Bloch insulators with time-reversal symmetry on the basis of the homotopy of the ground-state wave functions in momentum space and find that there are two topological classes characterized by a Z(2) invariant. The results are in agreement with a recent study based on counting the zeroes of a certain Pfaffian function related to the ground-state wave function. It is shown that there is a link between the formulation of the topological invariant presented here and the number of robust edge states. A formula is also provided which greatly simplifies the computation of the invariant in a large number of cases. The present study provides guidance for the search of systems which belong to the nontrivial topological class and also establishes a link between the quantum spin Hall effect and the integer quantum Hall effect.