A Bound on Topological Gap from Newton's Laws

TL;DR

This paper presents a simplified derivation of a universal bound on the energy gap in topological matter, extending previous results to more general systems using properties of optical conductivity and fundamental physical principles.

Contribution

It introduces a more straightforward derivation of the topological gap bound, enabling generalizations to anisotropic, multi-charge, and zero Hall conductance systems.

Findings

Derived a universal bound on the energy gap in topological systems.

Extended the bound to anisotropic and multi-charge systems.

Connected the bound to optical conductivity and fundamental forces.

Abstract

A striking general bound on the energy gap in topological matter was recently discovered in Ref. [Onishi and Fu, Phys. Rev. X {\bf 14}, 011052 (2024)]. A non-trivial indirect derivation builds on the properties of optical conductivity at an arbitrary frequency. We propose a simpler derivation, allowing multiple generalizations, such as a universal bound on a gap in anisotropic systems, systems with multiple charge carrier types, and topological systems with zero Hall conductance. The derivation builds on the observation that the bound equals $\hbar$ times the ratio of the force by the external electric field on the charge carriers and their total kinematic momentum in the direction perpendicular to the force.