Quantum Metamorphosis: Programmable Emergence and the Breakdown of Bulk-Edge Dichotomy in Multiscale Systems

Abstract: Multiscale synergy -- the interplay of a system's distinct characteristic length, time, and energy scales -- is becoming a unifying thread across many contemporary branches of science. Ranging from moiré and super-moiré materials and cold atoms to DNA-templated superlattices and 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 principle for emergent phenomena. Specifically, we introduce a scale-programmable framework for hierarchically nested lattices (HNLs) that can host quantum metamorphosis (QuMorph) -- a continuous evolution between system-dependent features governed by a dimensionless tunable parameter $α$ (the relative hopping). To exemplify, we show an HNL, in which as $α$ 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-dependent 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, and explore applications for multi-timescale nonlinear optics. Our work establishes a scalable and programmable paradigm for engineering multiscale emergent phenomena.