Arbitrary number of tunable localized states by synthetic band windings

Topological insulators host robust, symmetry-protected localized states at their boundaries, whose existence is dictated by bulk topological invariants. However, the number or spatial positions of these states are typically fixed by lattice geometry, limiting their tunability for practical applications. Here, we propose a synthetic band winding mechanism enabled by inserting additional lattice sites into a generalized Aubry-Andr\'e-Harper (AAH) model. This design allows independent control over the number and eigenfrequency of localized states. These newly introduced states hybridize with original edge bands, giving rise to multiple windings in the parameter-dependent band dispersion. Under time modulation, we uncover delayed and directional energy transfer processes enabled by these multiband windings, including novel Landau-Zener type pathways in finite systems. Extending this construction to two dimensions leads to a hierarchy of tunable corner and edge states. By permitting the additional sites to be inserted at arbitrary locations, the spatial positioning of localized modes also becomes fully programmable. Our framework offers a highly reconfigurable platform for engineered localization, with promising implications for topological photonic devices, multimode energy transport, and programmable waveguides.