Thouless pumping moves particles by exactly one lattice spacing per cycle — quantized transport, topologically protected. It usually applies to extended states: Bloch waves that fill the lattice. Solitons, by contrast, are localized. They don’t fill anything. Can you pump a soliton?

Yes — using helicoidal spin-orbit coupling in a two-component BEC. The spin-orbit coupling acts as a moving potential that drags the soliton along the lattice, one unit cell per cycle. The transport is quantized and robust: disorder in the lattice doesn’t break it. The soliton doesn’t need to know the global topology. It just rides the local field.

But there’s a limit. In the semi-infinite gap, solitons with too many atoms stop being transported. The nonlinearity that makes the soliton a soliton — its self-interaction — eventually competes with the topological pumping. Below a critical atom number, topology wins and the soliton moves. Above it, the soliton is too heavy for the pump.

The on/off switch is the Zeeman splitting. Remove the longitudinal component of the Zeeman field and quantized transport vanishes entirely. The splitting breaks a symmetry that the pumping requires. Without it, the two spin components see the same potential and the pump has no grip.

A topological machine with a weight limit and a kill switch. The mechanism is quantum, but the engineering constraints are classical.