Topological gap modes in a singularly perturbed periodic lattice and induced localization*

We show that when a local, singular (delta-function) perturbation is applied to a periodic lattice modeled by the Mathieu equation, localized gap modes emerge. The topological characteristics of these gap modes are demonstrated. The resulting localization effect is deterministic and attributed to one lattice defect or one impurity ion, in contrast to Anderson

localization, which in general is associated with system-wide disorder and randomness. Connections to high-toroidal-mode-number toroidal Alfvén eigenmodes in tokamaks are elucidated using the ballooning representation.