Excitonic Charge Density Waves in Moire Ladders

An incommensurate charge density wave (CDW) is a periodic modulation of charge that breaks translational symmetry incongruently with the underlying lattice. Its low-energy excitations, the phason, are collective, gapless phase fluctuations. We study a half-filled, four-band ladder model where a shift $\delta = p/q$ between the legs leads to a supercell of $q$ composite cells. The moir\'e potential narrows minibands near the Fermi level, resulting in additional peaks in the density of states, whose separation is controlled by $\delta$. The inclusion of short-range Coulomb interactions leads to an excitonic incommensurate CDW state. We identify the oscillations in its amplitude with a gapped Higgs collective mode and a lowest-energy Goldstone mode, realized by long-lived neutral phasons whose propagation velocity is governed by the shift $\delta$ and the inter-leg tunneling amplitude. Our results show that even the slightest interlayer mismatches can strongly modify both charge-ordering patterns and low-energy bosonic excitations in layered materials, and suggest that the enigmatic CDW phase in the quasi-one-dimensional compound $\rm HfTe_3$ is excitonic in nature.