Abstract
We present a theoretical study and experimental realization of a system that is simultaneously a four-dimensional (4D) Chern insulator and a higher-order topological insulator (HOTI). The system sustains the coexistence of (4-1)-dimensional chiral topological hypersurface modes (THMs) and (4-2)-dimensional chiral topological surface modes (TSMs). Our study reveals that the THMs are protected by second Chern numbers, and the TSMs are protected by a topological invariant composed of two first Chern numbers, each belonging a Chern insulator existing in sub-dimensions. With the synthetic coordinates fixed, the THMs and TSMs respectively manifest as topological edge modes (TEMs) and topological corner modes (TCMs) in the real space, which are experimentally observed in a 2D acoustic lattice. These TCMs are not related to quantized polarizations, making them fundamentally distinctive from existing examples. We further show that our 4D topological system offers an effective way for the manipulation of the frequency, location, and the number of the TCMs, which is highly desirable for applications.
Highlights
The topological phase is an important development and unexplored freedom of traditional band theories [1,2]
The studies of higher-order topological insulators” (HOTIs) have led to several significant developments, these secondorder topological corner modes (TCMs) generally do not coexist with first-order topologically protected gapless edge modes [11]
topological hypersurface modes (THMs) are protected by the second Chern numbers of the 4D bulk bands, and topological surface modes (TSMs) are protected by nonzero combinations of first Chern numbers, each belonging to a Chern insulator existing on orthogonal subdimensions
Summary
The topological phase is an important development and unexplored freedom of traditional band theories [1,2]. THMs are protected by the second Chern numbers of the 4D bulk bands, and TSMs are protected by nonzero combinations of first Chern numbers, each belonging to a Chern insulator existing on orthogonal subdimensions When both synthetic coordinates are fixed, the 4D system is observable as 2D real-space systems, wherein the THMs become 1D topological edge modes (TEMs) and the TSMs manifest as 0D topological corner modes (TCMs). We identify that the THMs and TSMs can be mathematically traced to the topological boundary modes of the 2D Chern insulators [25,35,36] This new perspective leads to a striking capability for realizing TCMs and for manipulating their frequencies, locations, and number. Such a capability is experimentally demonstrated by the realization of two distinctive types of TCMs; one is a “separable bound state in a continuum (BIC)” [38], and the other is the realization of multiple TCMs in one corner
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