Gap modes in Arnold tongues and their topological origins

TL;DR

This paper investigates the topological origins of stable gap modes in a modified Mathieu equation with a delta potential, revealing their connection to Arnold tongues and their potential physical realizations.

Contribution

It demonstrates that stable gap modes with topological origins exist in unstable regions of the Mathieu equation and generalizes this to a broad class of periodic Hamiltonians.

Findings

Stable gap modes are linked to topological features in Arnold tongues.

Localized potential perturbations can induce gap modes in unstable regimes.

The results apply to physical systems like electron wavefunctions and Alfvén eigenmodes.

Abstract

Gap modes in a modified Mathieu equation, perturbed by a Dirac delta potential, are investigated. It is proved that the modified Mathieu equation admits stable isolated gap modes with topological origins in the unstable regions of the Mathieu equation, which are known as Arnold tongues. The modes may be identified as localized electron wavefunctions in a 1D chain or as toroidal Alfv\'en eigenmodes. A generalization of this argument shows that gap modes can be induced in regimes of instability by localized potential perturbations for a large class of periodic Hamiltonians.