Layer-number parity induced topological phase transition

Abstract

We demonstrate that stacking topologically trivial layers, under enforced symmetry restrictions, yields emergent topological phases with protected boundary states. Remarkably, the number of layers itself acts as a topological switch, enabling the system to host topological bound states in the continuum (BICs). We analytically show that the spectrum becomes gapless for an odd number of layers; combined with entanglement-spectrum calculations, this confirms that odd-layer systems indeed support topological BICs. We provide experimental confirmation of these phenomena in stacked acoustic lattices. Our findings establish a previously overlooked pathway to topology and demonstrate a readily applicable strategy for realizing exotic states in a wide range of artificial material systems.

Figures (4)

• Figure 1: (Color online) Trivial-to-topological phase transitions induced by layer stacking. Symmetry-constrained stacking in 2N-layer and (2N+1)-layer systems facilitates the evolution of topological states from isolated in-gap modes to BICs embedded within the extended state continuum.
• Figure 2: (Color online) Transition between topological in-gap states and BICs in multi-layer systems. (a--c) Bulk band structures weighted by the spectral function $\text{A(k, E)}$, energy spectra for finite systems under open boundary conditions, and spatial distribution of the magnitude of characteristic wavefunctions $|\psi|$ for 2-layer configurations. (d--f), (g--i), and (j--l) show corresponding results for 3, 4, and 5-layer configurations, respectively. The topological states (red dots) appear as in-gap edge states for even-layer systems ($N=2, 4$) and as BICs embedded within the continuum for odd-layer systems ($N=3, 5$). Parameters: $t = 1$, $t_1 = 1.34$.
• Figure 3: (Color online) Entanglement spectrum $\varepsilon_n$ for 2-, 3-, 4-, and 5-layer systems, color-coded by value. All parameters match those in Fig. 2. The integers indicate the count of mid-gap states at $\varepsilon_n = 0.5$ (highlighted by red points). The total system consists of $L=60$ unit cells, with a subsystem size $L_A = L/2$.
• Figure 4: Spectral and spatial characterization of multilayer acoustic lattices. (a) Photograph of the fabricated experimental samples. The inset provides a magnified view of the resonator geometry, highlighting the engineered intra- and inter-layer coupling mechanisms. Colored speaker icons denote the selective excitation positions for edge (orange) and bulk (blue) states. (b, e, h, k) Measured acoustic DOS for stack depths of $N = 2, 3, 4, \text{and } 5$ layers, respectively. The orange and blue shaded regions delineate the spectral domains dominated by edge-localized modes and bulk-distributed modes. (c, f, i, l) Measured spatial acoustic pressure-field profiles $|P|^2$ for representative topological states (marked by orange circles in the spectra), demonstrating robust localization at the lattice boundaries. (d, g, j, m) Spatial pressure-field profiles of the corresponding bulk states (marked by blue stars), exhibiting delocalized distributions across the primary lattice sites.