  This study demonstrates that the zeros of the diagonal components of Green
functions are key quantities that can detect non-interacting topological
insulators. We show that zeros of the Green functions traverse the band gap in
the topological phases. The traverses induce the crosses of zeros, and the
zeros' surface in the band gap, analogous to the Fermi surface of metals. By
calculating the zeros of the microscopic models, we show the traverses of the
zeros universally appear in all six classes of conventional non-interacting
topological insulators. By utilizing the eigenvector-eigenvalue identity, which
is a recently rediscovered relation in linear algebra, we prove that the
traverses of the zeros in the bulk Green functions are guaranteed by the band
inversions, which occur in the topological phases. The relevance of the zeros
to detecting the exotic topological insulators such as the higher-order
topological insulators is also discussed. For the Hamiltonians with the
nearest-neighbor hoppings, we also show that the gapless edge state guarantees
the zeros' surfaces in the band gap. The analysis demonstrates that the zeros
can be used to detect a wide range of topological insulators and thus useful
for searching new topological materials.
