Fig. 4: Idealised schematics of flow-induced director profiles.

a Typical columns of vertical cross section of each state, showing the characteristic director profile. \({B}^{\pm }\) states are mirror images of each other. b Polar plot, representing the directions traced by the four states of flow when \(z\) coordinate traverses the height of the channel. \(B\) state only deviates back and forth in the direction of flow, chiral \({B}^{\pm }\) states deviate sideways, tracing a clockwise or anticlockwise path when viewed from the top, and the \(D\) state completes a full \(\pi\) turn. The mid-cell deviation \({\theta }_{0}\) measures the degree of symmetry breaking in the chiral \({B}^{* }\) phase (denoted by black vertical arrows). The sketch depicts a few examples of how the profile behaves at different amplitudes and handedness. See Fig. 5 for quantitative simulation results. c Schematic depiction of transitions and solitons between the observed states. Left: the achiral \(B\) state transitions into the chiral \({B}^{\pm }\) states via a second-order phase transition. Transitions between \({B}^{\pm }\) states are carried by nonsingular solitons (dashed line). Right: transitions from all \(B\)-type states into the \(D\) state are first-order and are separated by a disclination line. The dowser state is a polar quasi-planar state, which contains its own dynamics of point defects (red dots).