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

TL;DR

This paper introduces a generalized Jackiw-Rebbi model across dimensions, linking parity symmetry to Clifford algebra, providing a comprehensive topological classification of free fermion insulators and explaining the origin of gapless edge states.

Contribution

It extends the Jackiw-Rebbi model to arbitrary dimensions, establishing a topological classification framework based on Clifford algebra and parity symmetry.

Findings

Clifford algebra determines topological phases.

Gapless edge states are topologically protected.

Parity symmetry relates different regions and classifies insulators.

Abstract

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.