Energy-phase relation of a boundary mode in a charge density wave state

Individually, charge order and topology have been widely studied in various condensed matter platforms. Nevertheless, the manifestation of the topology in charge ordered systems remains largely unexplored. Here, using scanning tunneling microscopy, we unveil the bulk and boundary phenomenology of the charge density wave (CDW) in a topological material. This material, considered a candidate for a Weyl semimetal, exhibits a phase transition to CDW. Below the CDW transition temperature we find an in-gap boundary mode (edge state) at a monolayer step edge. Remarkably, the edge state exhibits modulations along the edge that match wavelength of the CDW projected onto the edge. The wavelength of the edge state modulations is energy-independent, while the phase of the edge state smoothly shifts by π as a function of energy when crossing the CDW energy gap. This results in a continuous connection of fully gapped valence and conduction band states, which are originally separated by π phase in the bulk, via the energy-phase relation of the boundary mode. Such mechanism is similar to the topological spectral flow, where the gapless edge modes link the gapped bulk modes in energy-momentum, while here, the connectivity occurs in energy-phase. The existence of the edge state within insulating CDW gap along with its non-trivial energy-phase relation hint to a topological nature of the CDW state.