Layer-Number Parity Induced Topological Phase Transition

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). Analytically, we reveal that an odd-layer configuration not only renders the spectrum gapless but also supports BICs protected by interlayer reflection symmetry. Combined with entanglement-spectrum calculations, this confirms that odd-layer systems indeed support topological BICs. We provide experimental confirmation of these topological states 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.