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The Nambu-Goldstone (NG) modes in a nonrelativistic system can be classifiedinto two types from their characteristic features: being of either an odd (typeI) or an even (type II) power energy-momentum dispersion. Conventionally, thetype-II NG modes may universally arise from spontaneous breaking ofnoncommutative symmetry pairs. Here, we predict a novel type of quadraticallydispersed NG modes that emerges in mixed $s$ and $p$ band Bose superfluids inan optical lattice and, unlike the conventional type-II NG modes, cannot besolely interpreted with the celebrated symmetry-based argument. Instead, weshow that the existence of such modes has a profound connection to thetopological transition on projective complex order-parameter space. Thedetection scheme is also proposed. Our Letter reveals a new universal mechanismfor emergence of type-II NG modes, which bridges intrinsically the Landausymmetry-breaking and topological theories.

Emergence of the Unconventional Type-II Nambu-Goldstone Modes with Topological Origin in Bose Superfluids