  We argue that various kinds of topological insulators (TIs) can be
insightfully characterized by an inspection of the charge centers of the hybrid
Wannier functions, defined as the orbitals obtained by carrying out a Wannier
transform on the Bloch functions in one dimension while leaving them Bloch-like
in the other two. From this procedure, one can obtain the Wannier charge
centers (WCCs) and plot them in the two-dimensional projected Brillouin zone.
We show that these WCC sheets contain the same kind of topological information
as is carried in the surface energy bands, with the crucial advantage that the
topological properties of the bulk can be deduced from bulk calculations alone.
The distinct topological behaviors of these WCC sheets in trivial, Chern, weak,
strong, and crystalline TIs are first illustrated by calculating them for
simple tight-binding models. We then present the results of first-principles
calculations of the WCC sheets in the trivial insulator Sb$_2$Se$_3$, the weak
TI KHgSb, and the strong TI Bi$_2$Se$_3$, confirming the ability of this
approach to distinguish between different topological behaviors in an
advantageous way.
