  It is known that, under short-range interactions many topological
superconductors (TSC) and topological insulators (TI) are trivialized, which
means the boundary state of the system can be trivially gapped out by
interaction without leading to symmetry breaking or topological ground state
degeneracy. This phenomenon is also referred to as "symmetric mass generation"
(SMG), and has attracted broad attentions from both the condensed matter and
high energy physics communities. However, after the trivialization caused by
interaction, some trace of the nontrivial topology of the system still
persists. Previous studies have indicated that interacting topological TSC and
TI could be related to the "zero" of Green's function, namely the fermion
Green's function $G(\mathrm{i} \omega \rightarrow 0) = 0$. In this work,
through the general "decorated defect" construction of symmetry protected
topological (SPT) states, we demonstrate the existence of Green's function zero
after SMG, by mapping the evaluation of the Green's function to the problem of
a single particle path integral. This method can be extended to the cases
without spatial translation symmetry, where the momentum space which hosts many
quantized topological numbers is no longer meaningful. Using the same method
one can demonstrate the existence of the Green's function zero at the "avoided
topological transition" in the bulk of the system.
