  In this article, we predict the occurrence of topological non-trivial phase
in hexagonal antiperovskite systems. By carefully investigating the evolution
of band structure, we have studied the pressure induced topological phase
transition in Ba$_{3}$BiP, Ba$_{3}$BiN, and Sr$_{3}$BiN compounds using the
hybrid functional calculations. The non-trivial topology has been verified by
computing Dirac like surface dispersion and topological invariant {\it Z$_2$}
index via parity analysis. The unconventional spin texture has been analyzed
which guaranteed the absence of impurity induced backscattering on boundary of
the sample while respecting the time reversal symmetry. Our simulation confirms
the chemical and mechanical stability of all the three compounds. The present
study introduces an important new class of hexagonal antiperovskite compounds
in the topological regime and are believed to capture ample of attention both
from theoretical as well as experimental front.
