Quantized Thouless Pumping of Dark Solitons

Nonlinearity enables the emergence of localized waves such as solitons that maintain their shapes during propagation. Solitons are broadly classified into bright and dark solitons. While a bright soliton exhibits a density peak, a dark soliton presents as a defect on a continuous wave background. A distinctive feature of dark solitons is the abrupt phase change in their wave function, which can host Majorana zero modes in topological fermionic superfluids. Recent studies have shown that bright solitons can undergo quantized transport through Thouless pumping, where the bright soliton functions as a Wannier function. However, it remains unclear whether Thouless pumping can also occur for dark solitons, which fundamentally differ from bright solitons. Here, we theoretically demonstrate the occurrence of both integer and fractional Thouless pumping for dark solitons within both a continuous model under optical lattices and a tight-binding model. Specifically, we find that a dark soliton is transported by one or half a unit cell, following the center-of-mass position of a Wannier function, as a system parameter is slowly varied over one cycle. Our work opens new avenues for exploring Thouless pumping for defects with phase changes, such as dark solitons, vortex solitons, ring dark solitons, and vortices.