Active-Hydraulic Flows Solve the Six-Vertex Model (and Vice Versa).

By confining colloidal active fluids in microchannel networks, we demonstrate that their degenerate flows correspond to the configurations of the six-vertex model. We use this quantitative correspondence to control and explain the active flows that emerge in square-grid networks. In particular, we show that the Lagrangian trajectories of active particles realize the Baxter-Kelland-Wu mapping and form completely packed loops, whose geometry can be exactly predicted and explained. We then go beyond the square-grid geometry and introduce a general framework for predicting the geometry of active-hydraulic flows in arbitrary networks.