Retarded interaction between opposite chiral edges in anomalous Hall crystals

An anomalous Hall crystal combines spontaneous electronic crystallization with a Chern insulating gap, supporting both chiral edge modes and low-energy electronic phonons. We show that this coexistence produces a distinct dynamical effect from ordinary Chern insulators: transverse bulk phonons can mediate a retarded interaction between counterpropagating chiral edge modes on opposite sides of the sample, realizing a Luttinger-liquid variant with delayed inter-edge coupling. Using microscopic time-dependent Hartree--Fock calculations for rhombohedral pentalayer graphene, we find that lowering the carrier density softens the long-wavelength transverse phonon mode. Near this instability regime, the resulting boundary-projected phonon continuum inevitably overlaps with the edge dispersion, thereby enabling their coupling. A smoking-gun probe is a nonlocal measurement: a drive applied to one edge can induce a response on the other edge, delayed by the transverse phonon time of flight across the sample.