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We present a new perspective to the classification of topological phases infree fermion insulators by generalizing the Jackiw-Rebbi model to arbitrarydimensions. 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 completetopological classification of free fermion insulators. Gapless edge states area natural consequence of our construction and their topological nature can beunderstood from the fact that all gapless edge states at a given interfacetransform similarly under $P$ (all odd or all even). A naive non-topologicalmodel for states confined to the interface will allow both even and odd states.

Generalized Jackiw-Rebbi Model and Topological Classification of Free Fermion Insulators