  This study demonstrates that the zeros of the diagonal components of Green
functions are key quantities that can detect topological nature of insulators.
We show that the crosses of the zeros of Green functions universally appear in
all six classes of conventional topological insulators. The present study
proves that the existence of the crosses of the zeros in the bulk Green
functions are guaranteed because of gapless edge states. The relevance of the
zeros for detecting the exotic topological insulators such as the higher-order
topological insulators is also discussed. The analysis demonstrates that the
zeros can be used to detect gapless surface/edge states and thus useful for
searching topological materials.
