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

TL;DR

Quantum metamorphosis (QuMorph) introduces a scale-programmable framework based on hierarchically nested lattices (HNLs) in which a single knob, $α = J_2/J_1$, continuously morphs the spectrum and topology from IQH-like to AQH-like regimes via a cocoon state with dense mini-gaps. The approach enables hybrid edge–bulk states and scale-dependent topological invariants, including higher-order Chern characteristics and magic flat bands arising from multi-scale coupling. A concrete photonic implementation using ring-resonator circuits is proposed, with measurable spatial-spectral signatures and potential for multi-timescale nonlinear optics via the Lugiato–Lefever formalism, suggesting on-chip programmable multiscale photonics. Beyond photonics, the framework provides a universal design principle to engineer and explore multiscale emergent phenomena across moiré, cold-atom, and synthetic-dimension platforms, enabling controllable cross-scale transport and topology.

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.

Figures (17)

• Figure 1: The general notion of quantum metamorphosis (QuMorph) in hierarchically nested lattices (HNLs). (A) A $4 {\times} 4$ first-order nested square lattice with flux $\phi$ threading each plaquette, yielding an integer quantum Hall-like (IQH-like) lattice, hosting the renowned Hofstadter's butterfly as its energy-flux spectrum. The edge states appear as the sparse levels resembling a butterfly. (B) An anomalous quantum Hall-like (AQH-like) $4{\times} 4$ first-order nested square lattice, where the total flux through each unit cell is zero. The corresponding energy spectrum of this system resembles that of a mantis, with the edge states similarly appearing as sparse levels. (C) A second-order $4{\times} 4^{4{\times} 4}$ HNL architecture, in which individual copies of small IQH lattices (with alternating magnetic fields $\pm{B}$, marked by red/blue) serve as the sites of the larger AQH lattice, forming a $\text{IQH}^\text{AQH}$ HNL. Hopping rate between the sites within an IQH lattice (between copies of IQH lattices in the super AQH lattice) is set by $J_1$ ($J_2$). (D) Tuning the relative hopping $\alpha = J_2/J_1$ from $0$ to $\infty$ transforms the effective system from the IQH to the AQH phase, yielding the metamorphosis of flux-energy spectrum from the butterfly into the mantis. The crossover is a cocoon stage that lies in the middle. (E) Representative emergent phenomena at the cocoon stage: (left) programmable and (right) designed emergent phenomena.
• Figure 2: For an $\text{AQH}^\text{AQH}$: (A) Energy bands of a $4{\times} 4^{\infty{\times}\infty}$ HNL as a function of $\alpha$. The gray shades indicate the bulk modes of the entire HNL. White regions denote trivial gaps; blue regions are topological with $|C|=1$. The light (dark) gray shade represents the $\text{edge}^\text{bulk}$ ($\text{bulk}^\text{bulk}$) states. The cyan shade is an example of a gap with residue topology, within which an isolated (detached) edge band exists. The isolated edge band is created by two band crossings, resulting from higher orbital mixing. After the first crossing, the gap carries $|C|=1$; by the bulk-boundary correspondence, the separated bulk bands must be connected by an edge band. A second crossing at a different symmetry point restores $C=0$, lifting the connectivity requirement and leaving a detached edge band. (B,C) Magnified examples (circled) of multiple instances of band crossing and band flattening observed in (A), plotted along the two-dimensional energy-momentum dispersions. (B) shows the flattening of two adjacent bands; (C) shows a band crossing. Note that the flattenings are perfect, e.g., at $\alpha \approx 0.87$ a perfect flat band sits at $E \approx \pm 0.15 J_1$. The color encodes the point-wise Berry curvature $\Omega(\vec{k})$, normalized to $[-1,1]$ within each band. (D) A cross-section of (A) at $\alpha=1.73$, within the cocoon regime where orbital mixing yields emergent phenomena. (E) From top: isolated edge bands; magic perfect flat bands; isolated flat edge bands; and topologically embedded perfect flat bands. Note that in the latter, the flat band is surrounded by topological gaps and thus threaded by edge bands. Apart from these novel behaviors, the spectrum in (D) includes $\text{bulk/edge}^\text{bulk/edge}$ bands; Dirac cones; and connected (regular) edge bands, all in one instance of $\alpha$.
• Figure 3: (A-I) Bottom-up scalable hierarchical construction of HNLs in coupled photonic ring-resonator architectures, with spatial-spectral (field intensity profiles and drop-port transmission spectra) spectroscopic signatures for realizing QuMorph in HNL devices in an add-drop configuration (here only for the AQH-type HNLs). Different types of hybrid modes hosted in each HNL are labeled. In (I), resonant-frequency nesting scales clearly with increasing nesting order (only a single longitudinal mode of the single-ring resonator is considered). Frequency bands for each mode type are highlighted and labeled with gray and green, distinguishing the bulk and edge-type bands, respectively. (J) The coarse-grained local Chern marker at three different scales depicted for an $\text{AQH}^{\text{AQH}^\text{AQH}}$ HNL of size $7{\times} 7^{4{\times} 4^{6{\times} 6}}$. Color encodes the coarse-grained Chern marker averaged around each point. The bulk Chern value of the HNL flips as the coarse-graining scale increases from bottom to top.
• Figure 4: Calculated eigenvalues and drop-port transmission for $6{\times} 6$ photonic coupled-ring HNLs in an add–drop configuration: (A and C) IQH and (B and D) AQH. Insets show the indicated edge and bulk mode profiles. Drop-port transmission for $6{\times} 6^{6{\times} 6}$ HNLs: (E) $\text{IQH}^{\text{IQH}}$, (F) $\text{AQH}^{\text{AQH}}$, (G) $\text{IQH}^{\text{AQH}}$, and (H) $\text{AQH}^{\text{IQH}}$. Insets mark bulk (gray) and edge (green) sections of the nested frequency bands. (I-L) Representative hybrid spatial-mode profiles, illustrating $\text{edge}^{\text{edge}}$ and $\text{edge}^{\text{bulk}}$ states in each HNL type.
• Figure 5: (A) Field intensity profile of a typical $\text{edge}^{\text{edge}}$ mode in a $4{\times} 4^{4{\times} 4}$$\text{AQH}^{\text{AQH}}$ HNL in an add–drop configuration. (B) Linear-regime drop-port transmission schematic as a function of single-ring longitudinal mode number $\mu$. Three mode spacings, corresponding to the three timescales of the $\text{edge}^{\text{edge}}$ modes, are marked. Pumping one such mode can seed cascaded four-wave mixing (FWM) and comb generation (arrows). (C) Above–OPO-threshold pumping of the $\text{edge}^{\text{edge}}$ mode marked in (B), versus pump frequency. Red (blue) indicates pump power (integrated comb power, excluding the light in the $\mu$ = 0 mode) in the HNL. (D) Corresponding spatiotemporal dynamics. The white arrow marks a single super-nested soliton at pump frequency $\delta\omega_{p} = 0.2795\,J_1$. (E) Comb profile of the soliton state. (F) Comb spectrum reorganized by single-ring mode spacing and first-order nested frequency. Inset: three-timescale mode locking. (G) Temporal output showing pulse bursts with three periodicities. (H) Soliton intensity distribution and counter-clockwise (CCW) circulation around the array. The three phase-locked timescales are indicated. For clarity, only site rings are shown.