Quadratic Band Touching and Nontrivial Winding Reveal Generalized Angular Momentum Conservation

Angular momentum conservation stands as one of the most fundamental and robust laws of physics. In discrete lattices, however, its realization can deviate markedly from the continuous case, especially in the presence of nontrivial momentum-space band touchings. Here, we investigate angular momentum conservation associated with quadratic band-touching points (QBTPs) in two-dimensional lattices. We show that, unlike in graphene lattices hosting linear band-touching points, the conventional angular momentum is no longer conserved near QBTPs. Instead, we identify a generalized total angular momentum that remains conserved for both linear band-touching points and QBTPs, inherently determined by the topological winding number at the band-touching point. Using a photonic Kagome lattice, we experimentally demonstrate the conservation of generalized total angular momentum through pseudospin-orbital angular momentum conversion. Furthermore, we show that this conservation principle extends to a broad class of discrete lattices with arbitrary pseudospin textures and higher-order winding numbers. These results reveal a fundamental link between pseudospin, angular momentum, and topology, establishing a unified framework for angular momentum dynamics in discrete systems.