  We present a new perspective to the classification of topological phases in
free fermion insulators by generalizing the Jackiw-Rebbi model to arbitrary
dimensions. We show that a generalized Jackiw-Rebbi model where the Dirac mass
($m$) satisfies $m(x)=-m(-x)$ is invariant under a parity transformation ($P$)
that relates the $x>0$ half to the $x<0$ half. Determining the form of $P$
gives rise to a Clifford algebra that has been shown to give a complete
topological classification of free fermion insulators. Gapless edge states are
a natural consequence of our construction and their topological nature can be
understood from the fact that all gapless edge states at a given interface
transform similarly under $P$ (all odd or all even). A naive non-topological
model for states confined to the interface will allow both even and odd states.
