Band Flattening and Overlap Fermion

TL;DR

This paper explores the mathematical structure of Dirac operators in various symmetry classes, deriving the overlap operator and related topological invariants, thereby advancing the theoretical understanding of topological phases in condensed matter physics.

Contribution

It provides a unified framework connecting symmetry classes, classifying spaces, and the overlap Dirac operator, including the Ginsparg--Wilson relation and mod-two index theorem.

Findings

Effective Dirac operators lie in classifying spaces per symmetry class.

Derived the overlap Dirac operator for each symmetry class.

Established the Ginsparg--Wilson relation and mod-two index theorem.

Abstract

We show that, for each symmetry class based on the tenfold way classification, the effective Dirac operator obtained by integrating out the additional bulk direction takes a value in the corresponding classifying space, from which we obtain the flat band Hamiltonian. We then obtain the overlap Dirac operator for each symmetry class and establish the Ginsparg--Wilson relation associated with $\mathcal{C}$ and $\mathcal{T}$ symmetries, and also the mod-two index theorem.