Hierarchical Topological States without Dimension Reduction

TL;DR

The paper addresses how to realize hierarchical topological states without breaking symmetry or reducing dimensionality. It introduces a hierarchical mechanism in SSH-like lattices, where repositioning domain walls creates higher-HL unit cells and gaps that host new topological states, characterized by generalized winding numbers $W_n$ and a $ ext{Z}_2$ index. The authors demonstrate 1D and 2D realizations, deriving explicit criteria (e.g., $(c_1/c_2)^{D-2d}$) and showing that edge and domain-wall states appear in a controlled, iterative fashion, confirmed by finite-lattice simulations. This framework extends topological classifications to arbitrarily high hierarchical levels, enabling scalable, symmetry-preserving engineering of complex networks of protected modes in mechanical and quantum-inspired systems with potential for advanced vibration control and information processing.

Abstract

Topological insulators exhibit boundary states protected by bulk band topology, a principle first established in quantum systems and later extended to classical waves, including phononics. Conventionally, an $n$-dimensional bulk with nontrivial topology hosts $(n-1)$-dimensional topologically protected boundary states, which may be further gapped out by breaking the symmetry that protects them, potentially leading to the emergence of $(n-2)$-dimensional, or even lower-dimensional topological states, as in higher-order topological insulators. In this work, we introduce an alternative mechanism for gapping out topological states and forming new topological modes within the resulting gap without further unit-cell symmetry breaking or dimension reduction. Using one- and two-dimensional Su-Schrieffer-Heeger (SSH) models, we show that controlled repositioning of topological domain walls enables the construction of hierarchical unit cells that gap out the original domain-wall states while preserving the underlying symmetry. This process produces higher-hierarchical-level topological states, characterized by a generalized winding number, and can be iterated to realize multiple - potentially infinite - hierarchical levels of topological states. Our approach expands the conventional topological classification and offers a versatile route for engineering complex networks of protected modes in higher dimensions.

Figures (15)

• Figure 1: One-dimensional hierarchical lattice and its topological properties. (a) Schematic of a second-hierarchical-level (2nd-HL) unit cell comprising 96 identical masses connected by alternating springs of stiffness $c_1$ (black) and $c_2$ (red). Two first-hierarchical-level (1st-HL) domain walls are marked by vertical green bars. Red and blue shading highlight distinct 1st-HL arrangements near domain walls and terminations. The 2nd-HL cell, bounded by gray dashed lines, spans two domain walls; $d$ and $D$ denote the number of mass pairs between the domain walls and within the full 2nd-HL cell, respectively. (b) Phonon dispersion for $c_1>c_2$ showing bulk bands (blue), 1st-HL topological domain-wall states (TDWSs, green), and 1st-HL unit cell dispersions (red) for various $\frac{d}{D}$ ratios. (c) Normalized 2nd-HL bandgap $\Delta \omega_E^2/\Delta \omega_B^2$ between TDWSs (E1-1/2) and bulk bands (B1-1/2) as a function of $\frac{d}{D}$. (d-s) Mode shapes at (d,f,h,j,l,n,p,r) $ka=0$ and (e,g,i,k,m,o,q,s) $\pi$ for bands B1-1/2 and E1-1/2, comparing configurations with (d-k) $\frac{d}{D}=0.07$ and (l-s) $\frac{d}{D}=0.93$.
• Figure 2: Evolution of the $\mathbb{Z}_2$ invariant with respect to $\frac{d}{D}$ when (a) $c_1>c_2$ and (b) $c_1<c_2$ at the boundaries of the second-order unit cell.
• Figure 3: Topological characterization of second-hierarchical-level (2nd-HL) unit cells. (a)2nd-HL winding number $W$ as a function of the spring constant ratio $\frac{c_1}{c_2}$ and structural ratio $\frac{d}{D}$, assuming $c_1$ precedes $c_2$ in the lattice sequence. (b) Schematic representations of the four 2nd-HL unit cell types labeled in (a), using the simplified notation of Fig. \ref{['fig:Interface_distance']}(a). Boundary colors indicate spring configurations: red for strong-spring-start ($c_1>c_2$) and blue for weak-spring-start ($c_1<c_2$). All cases have $D=37$ mass pairs, the number of 1st-HL unit cells between domain walls (green bars), $d$, being 34 for Type I, 2 for Type II, 33 for Type III, and 3 for Type IV.
• Figure 4: Spectral and modal characteristics of one-dimensional finite hierarchical lattices. (a) Schematics of four finite lattices, each containing 12 second-hierarchical-level (2nd-HL) unit cells from Fig. \ref{['fig:W3D']} (b). Black bars mark boundary masses with grounding springs to maintain an effective stiffness of $c_1+c_2$ for all masses. (b,g,j,n) Normalized eigenfrequencies ($\omega^2/\omega_0^2$) for lattices I-IV in (a), with zoomed-in views (right panels) highlighting 1st-HL bulk modes (B1, blue), 1st-HL bandgaps (green, hosting 2nd-HL bulk modes, B2), and 2nd-HL bandgaps (orange, potentially hosting 2nd-HL edge states, E2-1/2). (c-f,h-i,l-m,o-r) Representative mode shapes of B1, B2, and E2-1/2 (if present) modes for each lattice. Green squares denote 1st-HL domain wall masses.
• Figure 5: Spectral and modal characteristics of one-dimensional hierarchical domain wall states. (a) Schematics of three finite lattices featuring 2nd-HL domain walls (orange), each with six 2nd-HL per side. (b,g,l) Normalized eigenfrequencies ($\omega^2/\omega_0^2$) for Scenarios $\it{i}$-$\it{iii}$ in (a), with zoomed-in views (right panels) highlighting: 1st-HL bulk modes (B1, blue), 1st-HL bandgaps (green, hosting 2nd-HL bulk modes, B2), and 2nd-HL bandgaps (orange, potentially hosting 2nd-HL domain-wall states, DW). (c-e,h-j,m-o) Representative mode shapes of B1, B2-1/2, and DW (if present). Green squares, yellow stars, and purple triangles mark 1st-HL domain walls, 2nd-HL topological domain walls, and 2nd-HL trivial domain walls, respectively.