![]() However, these systems are limited to excitations without orbital angular momentum, which may produce unforeseen results. Light propagation in gradient media and curved spaces induce intriguing phenomena, such as focusing and self-imaging, thus delivering a wide range of applications. This study develops the method for low-dimensional constrained systems and exhibits the possibility of a new degree of control for waveguiding in nanostructures. Numerical analysis reveals that the inhomogeneity of the confinement significantly affects the transport properties through changing the geometric symmetry of the system. To demonstrate the impact of the inhomogeneity, we apply our method to investigate the coherent transport on a cylindrical surface where two helical ditches is imposed on the thickness. Tiny fluctuations in the thickness are envisioned to induce considerable magnitude of the effective potential. This effective potential is relevant to the ground-state energy perpendicular to the surface and the morphology of the confining potential. Here, by extending the thin-layer procedure, we consider the case of inhomogeneous confinement and derive the effective Hamiltonian where an extra effective potential appears. The motion of quantum particles homogeneously constrained to a curved surface is affected by a curvature-induced geometric potential. This study develops the method for low-dimensional constrained systems and exhibits the possibility of new degree of control for waveguiding in nanostructures. Numerical analysis reveals that the inhomogeneity of the confinement significantly changes the transport properties. To demonstrate the impact of the inhomogeneity, we apply our method to investigate the coherent transport on a cylindrical surface where a confining potential with two helical ditches is imposed. Tiny changes in the thickness are envisioned to induce considerable magnitude of the effective potential. This effective potential is relevant to the ground state energy perpendicular to the surface and the morphology of the confining potential. Here, we consider the case of inhomogeneous confinement and derive the effective Hamiltonian by extending thin-layer procedure, where an extra effective potential appears. The motion of quantum particles homogeneously constrained to a curved surface is affected by a curvature induced geometric potential. The interaction between the geometrical curvature and topology in the system provides a novel scheme for manipulating and trapping wave propagation along the boundary of curved TIs, thereby offering potential applications in flexible devices. To understand the underlying mechanism for the localization of the topological edge state, a tight-binding model considering the geometric potential effect is proposed. Moreover, we experimentally verify the localized edge state is still topologically protected by introducing defects near the localized position. We experimentally demonstrate that the topological edge state in the bulk gap is modulated by the curvature of space into a localized mode, and the corresponding pressure distributions are confined at the position with the maximal curvature. In this work, we design a 2D curved acoustic TI by perforation on a curved rigid plate to localize the edge state by means of the geometric potential effect, which provide a unique approach for manipulating waves. ![]() In general, two-dimensional (2D) TIs are designed on a flat surface with special boundary to manipulate the wave propagation. Topological insulators (TIs) with robust boundary states against perturbations and disorders have boosted intense research in classical systems.
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