Exceptionally deficient topological square-root insulators

Abstract: One of the most surprising features of effectively non-Hermitian physical systems is their potential to exhibit a striking nonlinear response and fragility to small perturbations. This feature arises from spectral singularities known as exceptional points, whose realization in the spectrum typically requires fine-tuning of parameters. The design of such systems receives significant impetus from the recent conception of \emph{exceptional deficiency}, in which the entire energy spectrum is composed of exceptional points. Here, we present a concrete and transparent mechanism that enforces exceptional deficiency through lattice sum rules in non-Hermitian topological square-root insulators. We identify the resulting dynamical signatures in static broadband amplification and non-Abelian adiabatic state amplification, differentiate between bulk and boundary effects, and outline routes to implementation in physical platforms