Ultraspectral sensitivity and nonlocal bound states in algebraic non-Hermitian skin effect

Non-Hermitian systems in two and higher dimensions universally exhibit algebraic skin effect, yet the exotic phenomena stemming from its quasi-long-range decay remain largely unexplored. In this paper, we uncover a new form of spectral sensitivity in a broad class of two-dimensional non-Hermitian systems with algebraic skin effect. This sensitivity can be induced by weak impurities, and a minimal setup is a pair of distant weak local potentials. Unlike conventional Green's function technique, we diagnose the ultrasensitivity using the forth-back propagator: a unique indicator whose divergence signals the occurrence of ultraspectral sensitivity. Asymptotics of the forth-back propagator reveals the existence of a critical geometric ratio, beyond which the ultrasensitivity is activated, while below the threshold the spectrum remains stable against weak impurities. This yields, in the thermodynamic limit, a phase diagram governed solely by geometric factors. Moreover, this type of sensitivity leads to the formation of dynamical nonlocal bound states. Our findings can be verified on experimental platforms such as acoustic and photonic crystals.