Hierarchically Nested Lattices (HNLs)

• Hierarchically Nested Lattices (HNLs) are recursive multi-scale designs where a base lattice is nested within outer lattices to produce emergent spectral, topological, and elastic properties.
• They enable programmable control by tuning intra- and inter-layer couplings, yielding phenomena like flat bands, edge hybridization, and scale-dependent invariants.
• HNLs are applied in quantum metamorphosis, robust lattice coding through Construction D', and architectured materials with customizable elastic and spectral responses.

Hierarchically Nested Lattices (HNLs) are a multiscale lattice paradigm in which structural, quantum, or coding behaviors are programmed by “nesting” distinct lattice architectures recursively within each other. The concept arises independently in quantum many-body physics, information theory, and mechanical metamaterial design, but shares an essential blueprint: the embedding of structure-within-structure at multiple scales such that the interplay between layers yields emergent properties—spectral, topological, elastic, or algorithmic—not achievable at any single scale alone. Recent developments formalize HNLs as a framework for programmable cross-scale synergy, exemplified by Quantum Metamorphosis (QuMorph), bioinspired architectured materials, and Construction D' nested lattice codes (Mehrabad et al., 17 Nov 2025, Boda et al., 19 May 2024, Zhou et al., 2021).

1. Hierarchical Construction Principles

In the quantum and condensed matter setting, an HNL is constructed by recursively embedding a base (“inner”) lattice model $H_i$ within the sites of an “outer” model $H_f$, defining higher nesting orders as needed. At second order, each site $r_f$ of $H_f$ becomes a copy of $H_i$, yielding sites $(r_f, r_i)$, with intra-copy and inter-copy couplings governed by $H_i$ and $H_f$, and hopping strengths $J_1, J_2$ respectively. The general $n$-th order nesting Hamiltonian is

$$H = \sum_{j=1}^n \alpha_j \bigl( \bigotimes_{k<j} I_k \bigr) \otimes H_j \otimes \bigl( \bigotimes_{k>j} I_k \bigr)$$

where $H_j$ is the $j$-th scale Hamiltonian and $\alpha_j$ is its relative energy/hopping scale (Mehrabad et al., 17 Nov 2025).

In coding theory, Construction D' realizes HNLs as lattices built from chains of nested binary codes $$C_0 \subseteq C_1 \subseteq \dots \subseteq C_{a-1} \subseteq \mathbb{F}_2^n$$. Lattice points $x \in \Lambda$ satisfy congruence relations with respect to the parity-check spaces of each $C_i$ modulo $2^{i+1}$, encoding multi-level structure (Zhou et al., 2021).

In architectured materials, HNLs are realized by nesting multiple scaled and oriented lattice units (e.g. cubes at various rotations and separations) to generate a self-similar framework of struts and pores (Boda et al., 19 May 2024). Each “nesting order” (NO) adds a concentric geometric motif, and “nesting orientation” (NOR) specifies its rotation, inspired by bioarchitectures such as osteons and golden spirals.

2. Mathematical Formalism and Spectral Evolution

The eigenspectrum of quantum HNLs interpolates between the spectra of the outer and inner models, controlled by the dimensionless parameter $\alpha = J_2/J_1$ for second-order nesting. In the simplest tight-binding form:

$$H = \frac{1}{1+\alpha} (I_f \otimes H_i) + \frac{\alpha}{1+\alpha}(H_f \otimes I_i)$$

In this framework, varying $\alpha$ yields distinct spectral regimes (Mehrabad et al., 17 Nov 2025):

• $\alpha \to 0$: Spectrum is degenerate copies of $H_i$ (e.g., Hofstadter butterfly).
• Intermediate $\alpha \sim 1$: Spectrum fragments into a dense mini-gap network ("cocoon" regime), with magic points of flat-band coalescence.
• $\alpha \to \infty$: System approaches $H_f \otimes I_i$ physics (e.g., "mantis" spectrum).

The transition points and density of states are tracked numerically by minimizing the sum of squared level spacings:

$$S(\alpha) = \sum_n [E_{n+1}(\alpha) - E_n(\alpha)]^2$$

Magic points are characterized by vanishing bandwidth and peak density of states, signaling topologically embedded flat bands.

3. Topological and Physical Invariants

Quantum HNLs exhibit scale-dependent topological invariants. For first-order (single-scale) models, bands are indexed by first Chern numbers

$$C^{(1)} = \frac{1}{2\pi}\int_{\rm BZ} d^2k\,\Omega(k)$$

In the cocoon regime ($\alpha \sim 1$), an emergent second Chern number arises from the combined 4D Brillouin zone, defined as

$$C^{(2)} = \frac{1}{32\pi^2} \int d^2k_i\,d^2k_f \,\epsilon^{\mu\nu\rho\sigma}\mathcal{F}_{\mu\nu}\mathcal{F}_{\rho\sigma}$$

Higher-order HNLs allow the construction of hierarchy-product invariants

$C^{(n)} = \prod_{i=1}^n C_i$

where $C_i$ are the scale-wise Chern numbers (Mehrabad et al., 17 Nov 2025).

In mechanical metamaterials, effective elastic isotropy is quantified by the Zener anisotropy ratio

$Z = \frac{C_{11} - C_{12}}{C_{44}}$

where $C_{ij}$ are the independent components of the cubic stiffness tensor $C^*_{ijkl}$. $Z=1$ indicates perfect isotropy, $Z<1$ tension/compression-dominated, $Z>1$ shear-dominated. Effective modulus is

$$\bar{E} = \frac{(C_{11}-C_{12})(C_{11}+2C_{12})}{C_{11}+C_{12} } \, / \, E_s$$

with $E_s$ the base material modulus (Boda et al., 19 May 2024).

In lattice codes, shaping gain—quantifying energy efficiency over a power-constrained channel—serves as the figure of merit, with values up to $1.25$ dB achieved for optimized convolutional-code shaped HNLs (Zhou et al., 2021).

4. Eigenstates, Hybridization, and Multistage Features

Quantum HNLs generically exhibit four classes of eigenstates—distinguished by their edge or bulk character at each scale:

• bulk$^{\rm bulk}$: Bulk-like at both scales.
• edge$^{\rm bulk}$: Edge at inner scale, bulk at outer.
• bulk$^{\rm edge}$: Bulk at inner scale, edge at outer.
• edge$^{\rm edge}$: Edge at both scales.

Sufficient tuning of $\alpha$ enables hybridization between states and produces isolated edge bands, flat bands, and topologically protected edge channels ("edge-of-edge" modes). These phenomena have no analog in monolithic lattices and reflect the multiscale transition of topology and localization (Mehrabad et al., 17 Nov 2025).

In architectured materials, this hierarchical design enables precise tuning of connectivity, pore geometry, and mechanical response. Mono-nest, bi-nest, and tri-nest architectures span the spectrum from strongly anisotropic to perfectly isotropic elastic behavior, with strut-diameter mismatches further refining the Zener ratio and frequency response (Boda et al., 19 May 2024).

Coding schemes based on Construction D' employ multistage decoding, where each level of the nested code chain is decoded in succession, and residuals are propagated downward, enabling complexity-efficient correction and shaping (Zhou et al., 2021).

5. Design Methodologies and Algorithmic Frameworks

Engineering HNLs in quantum, coding, or architectured-matter contexts involves selecting component (“inner” and “outer”) models with desirable spectra or symmetries, embedding them recursively, and tuning relative couplings or geometric parameters.

For quantum and photonic HNLs, the following procedure is implemented (Mehrabad et al., 17 Nov 2025):

• Select $H_i$, $H_f$ with desired topological features (e.g. IQH, AQH).
• Couple appropriate orbitals across scales (boundary, bulk, or selective).
• Traverse $\alpha$ space to access regimes of interest (mini-gap maximization, flat bands, edge hybridization).
• Compute local markers such as coarse-grained Chern numbers to diagnose scale-specific topology.
• For higher orders, extend the Hamiltonian using the tensor product formalism described above.

For Construction D' HNL codes, two encoding algorithms exist:

• Method A (ALT form): Uses an approximate lower-triangular parity-check matrix to enable back-substitution for encoding.
• Method B (level-by-level): Maps message bits to lattice points scale-wise, solving parity equations at each level.

Decoding proceeds by multistage reduction and re-encoding, with shaping performed by projection onto the Voronoi region of a coarser lattice (e.g., $E_8$, $BW_{16}$, Leech, or optimized convolutional-code lattices) (Zhou et al., 2021).

For mechanical HNLs, geometric design is parametrized by choice of nesting order (NO), nesting orientation (NOR), strut diameters ($d_i$), and symmetry operations (T4FAS). Finite element homogenization procedures yield macroscopic moduli and anisotropy (Boda et al., 19 May 2024).

6. Experimental Realizations and Applications

Quantum HNLs are physically implemented using scalable photonic circuits composed of CMOS-compatible ring resonators coupled via link rings, which imprint hopping phases and amplitudes. Second- and third-order nesting is feasible, with explicit layouts described for $4 \times 4$ AQH lattices recursively embedded (Mehrabad et al., 17 Nov 2025). Experimental observables include:

• Add–drop spectra and group delay mapping resolve band topology and mini-gap formation.
• Real-space imaging discriminates hybrid edge–bulk modes and spatial fractals.

Contemporary bioinspired HNLs for additive manufacturing exploit self-similar multi-nest arrangements to achieve programmable elastic isotropy/anisotropy ($Z$ tunable over $0.11$–$1.81$), density ($\bar{\rho}$ from $0.05$ to $0.55$), and mechanical modulus. Tuned for osseointegration, energy absorption, or heat management, these structures offer optimal property gradients analogous to bone, aerospace, and functional applications (Boda et al., 19 May 2024).

In information theory, Construction D' HNLs demonstrate shaping gains up to $1.25$ dB for power-constrained AWGN channels, with practical encoding/decoding complexity. These nested codes underpin robust, scalable communication under low-SNR conditions (Zhou et al., 2021).

In nonlinear optics, photonic HNLs enable multi-scale frequency comb generation and soliton dynamics, with three well-separated time scales realized in second-order devices for applications in dual-comb spectroscopy and mode locking (Mehrabad et al., 17 Nov 2025).

7. Summary and Outlook

Hierarchically Nested Lattices unify multiscale design across quantum, algorithmic, and mechanical disciplines, enabling programmable emergent behavior by recursive embedding and scale mixing. The tunability—via spectral, topological, or elastic markers—afforded by nesting order, orientation, and coupling allows unprecedented control over physical properties. Key advances include the realization of cross-scale topological invariants, hybridization of edge and bulk states, maximal shaping gain in lattice coding, and application-specific optimization in architected materials. Ongoing research addresses higher-order nesting, emergent phenomena in more complex unit-cell architectures, and further experimental implementations in photonics, coding systems, and bioinspired material platforms (Mehrabad et al., 17 Nov 2025, Zhou et al., 2021, Boda et al., 19 May 2024).