A Non-topological Extension of Bending-immune Valley Topological Edge States

Breaking parity (P) symmetry in C$_6$ symmetric crystals is a common routine to implement a valley-topological phase. At an interface between two crystals of opposite valley phases, the so-called valley topological edge states emerge, and they have been proven useful for wave transport with robustness against 120$^\circ$ bending and a certain level of disorder. However, whether these attractive transport features are bound with the valley topology or due to topological-irrelevant mechanisms remains unclear. In this letter, we discuss this question by examining transport properties of photonic edge states with varied degrees of the P-breaking that tune the valley topology, and reveal that the edge states preserve their transport robustness insensitive to the topology even when the P-symmetry is recovered. Instead, a unique modal character of the edge states -- with localized momentum hotspots around high-symmetric $K$ ($K'$) points -- is recognized to play the key role, which only concerns the existence of the valleys in the bulk band structures, and has no special requirement on the topology. The "non-topological" notion of valley edge states is introduced to conceptualize this modal character, leading to a coherent understanding of bending immunity in a range of edge modes implemented in C$_3$ symmetric crystals -- such as valley topological edge states, topological edge states of 2D Zak phase, topological-trivial edge states and so on, and to new designs in general rhombic lattices -- with exemplified bending angle as large as 150$^\circ$.