Excitonic charge density waves in moiré 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 phasons, are collective, gapless phase fluctuations. We study a half-filled, four-band ladder model where a shift δ = p / q between the legs leads to a supercell of q composite cells. The moiré potential narrows minibands near the Fermi level, resulting in additional peaks in the density of states, whose separation is controlled by δ . 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 δ and the interleg 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 HfTe 3 is excitonic in nature.