  We investigate the roles of symmetry and bulk-boundary correspondence in
characterizing topological edge states in generalized Jackiw-Rebbi (JR) models.
We show that time-reversal ($T$), charge-conjugation ($C$), parity ($P$), and
discrete internal field rotation ($Z_n$) symmetries protect and characterize
the various types of edge states such as chiral and nonchiral solitons via
bulk-boundary correspondence in the presence of the multiple vacua. As two
representative models, we consider the JR model composed of a single fermion
field having a complex mass and the generalized JR model with two massless but
interacting fermion fields. The JR model shows nonchiral solitons with the
$Z_2$ rotation symmetry, whereas it shows chiral solitons with the broken $Z_2$
rotation symmetry. In the generalized JR model, only nonchiral solitons can
emerge with only $Z_2$ rotation symmetry, whereas both chiral and nonchiral
solitons can exist with enhanced $Z_4$ rotation symmetry. Moreover, we find
that the nonchiral solitons have $C, P$ symmetries while the chiral solitons do
not, which can be explained by the symmetry-invariant lines connecting
degenerate vacua. Finally, we find the symmetry correspondence between
multiply-degenerate global vacua and solitons such that ${T}$, ${C}$, ${P}$
symmetries of a soliton inherit from global minima that are connected by the
soliton, which provides a novel tool for the characterization of topological
solitons.
