Chiral topological order describes topologically ordered phases that break time-reversal symmetry — most notably fractional quantum Hall states — in contrast to non-chiral topological orders such as many gapped spin liquids. The defining signature is a nonzero chiral central charge: the phase’s edge supports gapless modes that all propagate in the same direction around the boundary, rather than balanced left- and right-moving modes that could gap out.
This one-directional edge transport is topologically protected and shows up in measurable quantities such as quantized thermal Hall conductance. Because chirality is an edge property with a bulk origin, diagnosing it from bulk data alone — as the modular commutator does — is a direct test of whether a candidate ground state realizes a chiral phase.