Gap modes in Arnold tongues and their topological origins

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 Alfvén 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.

Figures (6)

• Figure 1: A gap mode in the Arnold tongue around $\delta=0.25$, $\lambda = 0.7$.
• Figure 2: A comparison of a decaying gap mode to a quasiperiodic mode on the edge of the continuum gap.
• Figure 3: Gap modes in the Arnold tongues around $\delta=0.25\text{, }1.0\text{, and }2.25$, $\lambda = 1.0$.
• Figure 4: Points are numerically calculated values of $\delta_1$. The solid curve is the analytical form from Eq. \ref{['d1-anal']}.
• Figure 5: Spectra for Gaussian (blue) and Lorentzian (orange) perturbations with the same width $0.25$, $\lambda = 1.0$.