Quantum Metamorphosis: Topology in Multiscale systems

Multiscale synergy---the interplay of a system's distinct characteristic length, time, and energy scales---is becoming a common thread across many contemporary branches of science. Ranging from moir\'{e} and super-moir\'{e} materials and cold atoms to nested photonic networks, multiscale synergy produces behaviors not obtainable at any single scale alone. Yet a general framework that programs cross-scale interplay to steer spectra, transport, and topology has been missing. Here, we elevate multiscale synergy from a byproduct to a general design rule for emergent phenomena. Specifically, we introduce a scale-programmable framework for hierarchically nested lattices that can host quantum metamorphosis (\emph{\textbf{Q}u\textbf{M}orph)}---a continuous evolution between system-dependent features governed by a dimensionless tunable parameter $\alpha$. To exemplify, we show a nested lattice in which as \(\alpha\) changes, the spectrum metamorphoses from integer-quantum-Hall–like to anomalous-quantum-Hall–like, passing through a cocoon regime with proliferating mini-gaps. This multiscale mixing yields multiple novel phenomena, including hybrid edge–bulk states, scale-aware topology, topologically embedded flat bands, and isolated edge bands. We propose a feasible photonic implementation using commercially available coupled-resonator arrays, outline spatial–spectral signatures to map QuMorph, as well as exploring applications for multi-timescale nonlinear optics. Our work establishes a programmable avenue for engineering multi-scale emergent phenomena.