Geometry-dependent acoustic higher-order topological phases on a two-dimensional honeycomb lattice

Higher-order topological states, as emergent topological phases of matter, originating from condensed matter physics, have sparked a vibrant exploration of topological insulators. Their topologically protected multidimensional localized states are typically associated with nontrivial bulk band topology, and the significant impact of lattice geometry is unconsciously overlooked. Here, we construct coupled acoustic cavities on a two-dimensional honeycomb lattice to investigate the sensitivity of higher-order topological modes to the variations of edge contour. Fractional charge is utilized to accurately predict topological modes with distinct topological orders, in spite of the minimal bulk bandgaps inherent in the honeycomb lattice and bound states in the continuum. It is found that the presence and absence of the first-order and higher-order topological modes in the same topological phase are tightly linked to the sample boundaries, which can be demonstrated by both theoretical analysis and numerical calculation. Our study also discusses potential physical realization of geometry-dependent topological states across different platforms, providing inspiration for the prospective application of topological devices in acoustics.