Topology of Disconnected Elementary Band Representations

TL;DR

This paper demonstrates that certain elementary band representations can be gapped and trivial, providing counterexamples to previous claims and introducing new topological invariants like a movable Dirac cone and bent-Z2 index.

Contribution

It constructs explicit gapped elementary band representations in symmorphic and non-symmorphic groups and introduces new topological invariants to characterize their non-triviality.

Findings

Counterexamples to previous claims about non-symmorphic elementary band representations.

Introduction of a new topological invariant: a movable unremovable Dirac cone.

Identification of a bent-Z2 index as a topological invariant.

Abstract

Elementary band representations are the fundamental building blocks of atomic limit band structures. They have the defining property that at partial filling they cannot be both gapped and trivial. Here, we give two examples -- one each in a symmorphic and a non-symmorphic space group -- of elementary band representations realized with an energy gap. In doing so, we explicitly construct a counterexample to a claim by Michel and Zak that single-valued elementary band representations in non-symmorphic space groups with time-reversal symmetry are connected. For each example, we construct a topological invariant to explicitly demonstrate that the valence bands are non-trivial. We discover a new topological invariant: a movable but unremovable Dirac cone in the "Wilson Hamiltonian" and a bent-Z2 index.