“…[ 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 ]. Using the HOTIs and their lower-dimensional boundary states, various wave control methods are proposed, for example, multi-dimensional topological switching [ 5 ], valley-selective corner sates [ 32 , 33 , 34 , 35 ], topological sub-wavelength imaging [ 36 , 37 , 38 ], and even controllable emergence of corner states enabled by non-linearity [ 39 ], voltage [ 40 ], or temperature [ 41 ]. Despite these celebrated advances, the HOTI-enabled functionalities often highly depend on the lattice geometries due to the fact that nontrivial higher-order topology is usually protected by the crystalline symmetries, which vary from lattice to lattice.…”