Observation of multiple off-site corner states induced by next-nearest-neighbor coupling in a sonic crystal

The unexpected connections between classical wave phenomena and higher-order topological phases have attracted the interest of researchers across various fields. Although attaining lower-dimensional localized states, such as corner states, has been a common achievement in the realm of sonic crystals, a notable limitation arises where wave energy primarily localizes at the outermost site with the degenerate eigenfrequencies, thereby restricting further applications in acoustics. In this paper, we address this limitation by introducing previously overlooked extra next-nearest-neighbor interactions into the system. Remarkably, the inclusion of these extra nonlocal interactions generates an array of unconventional corner-localized states, each with distinct off-site localization positions. Experimental validation of these theoretical predictions is meticulously conducted using fabricated sonic crystals. Notably, in contrast to natural crystals, the advantageous characteristics of artificial crystals provide a promising platform for probing the effect of long-range interactions in topological phases and materials, which is facilitated by distance-independent coupling strengths. Our findings not only expand the boundaries of sound confinement but also provide momentum for exploring long-range interactions in the field of artificial crystals.