Beyond the Non-Hermitian Skin Effect: Scaling-Controlled Topology from Exceptional-Bound Bands.

We establish a novel mechanism for topological transitions in non-Hermitian systems that are controlled by the system size. Based on a new paradigm known as exceptional-bound (EB) band engineering, its mechanism hinges on the unique critical scaling behavior near an exceptional point, totally unrelated to the well-known non-Hermitian skin effect. Through a series of ansatz models, we analytically derive and numerically demonstrate how topological transitions depend on the system size with increasingly sophisticated topological phase boundaries. Our approach can be generically applied to design scaling-dependent bands in multi-dimensional lattices, gapped or gapless, challenging established critical and entanglement behavior. It can be experimentally demonstrated in any non-Hermitian platform with versatile couplings or multi-orbital unit cells, such as photonic crystals, as well as classical and quantum circuits. The identification of this new EB band mechanism provides new design principles for engineering band structures through scaling-dependent phenomena unique to non-Hermitian systems.