Rhombohedral multilayer graphene is a physical realization of the chiral two-dimensional electron gas that can host zero-line modes (ZLMs), also known as kink states, when the local ap opened by inversion symmetry breaking potential changes sign in real space. Here we study how the variations in the local stacking coordination of multilayer graphene affects the formation of the ZLMs. Our analysis indicates that the valley Hall effect develops whenever an interlayer potential difference is able to open up a band gap in stacking faulted multilayer graphene, and that ZLMs can appear at the domain walls separating two distinct regions with imperfect rhombohedral stacking configurations. Based on a tight-binding formulation with distant hopping terms between carbon atoms, we first show that topologically distinct domains characterized by the valley Chern number are separated by a metallic region connecting AA and AA$$ stacking line in the layer translation vector space. We find that gapless states appear at the interface between the two stacking faulted domains with different layer translation or with opposite perpendicular electric field if their valley Chern numbers are different.
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