Hierarchical symmetry breaking in Moir\'e graphene domain-wall networks

Abstract

Moir\'e network formation in graphene bilayers breaks stacking symmetry, giving rise to domain walls that host topologically protected one-dimensional states. Here we show that these systems undergo an additional symmetry breaking at the level of the domain-wall network geometry, leading to the spontaneous emergence of chiral network configurations that are not determined by topology alone. Using atomistic structural relaxation and electronic-structure calculations, we show that TDW networks adopt chiral geometries through lattice relaxation. Via developing a comprehensive phase diagram defined by strain and interlayer flexibility, we discover three equilibrium network morphologies: straight, mono-chiral, and dual-chiral. Chiral networks arise from the global minimization of TDW energy under moir\'e geometric constraints. Tight-binding calculations show that straight networks host junction-centred states, whereas chiral networks shift spectral weight toward asymmetric edge modes. While topologically protected states naturally emerge at AB/BA domain boundaries in moir\'e bilayers, we demonstrated that the localization of boundary states is network-symmetry dependent. Our results show that symmetry breaking at both the stacking and network levels provides a new way to understand and control low-energy electronic states in moir\'e bilayers.