Are normal band insulators topologically distinguishable?

Over the past 15 years topological classification of solids has occupied the center stage of condensed matter physics, identifying a variety of topologically distinct quantum phases of matter in real materials. Although the sector of topological crystals is sufficiently rich and diverse, populated by strong, weak, crystalline and higher-order topological insulators, for example, all the normal or trivial insulators are considered to be equally mundane. Here we ask the following question: Can normal insulators be topologically distinct? and provide an affirmative answer to it. We consider a paradigmatic toy lattice model, featuring topological quantum anomalous Hall insulators and trivial or normal insulators. Within the framework of this model, we show that the parent normal insulators, realized via band gap closing around the gamma and M points, accommodate distinct topological superconductors, which can be probed by dislocation lattice defects. Namely, only the superconductor resulting from the latter parent normal insulator responds to bulk dislocation defects by binding topological Majorana modes near its core. Therefore, normal insulators can be distinguishable once superconductivity develops in the system. We anchor this prediction by computing their response to dislocations as well as from quantized thermal Hall conductivity.