Supermoir\'{e} domain-resolved effective Hamiltonians and valley topology in helical multilayer graphene

Abstract

Extending moir\'{e} graphene beyond twisted bilayers, helical trilayer graphene has shown topological bands and correlated states with reshaped moir\'{e} periodicity. Here we develop a theoretical framework for helical multilayer graphene to investigate its supermoir\'{e} relaxation and low-energy electronic structure. Using real-space lattice calculations, we find that relaxation reconstructs the system into locally periodic single-moir\'{e} domains, which provide the basis for a continuum description. Within each reconstructed domain, downfolding the first-shell model yields effective Hamiltonians near the Dirac points that reveal how the low-energy spectrum decomposes into folded Dirac sectors. We further evaluate the valley Chern numbers encoded in these effective Hamiltonians, obtaining domain-dependent and gate-tunable topological responses consistent with the lattice calculations. Our results establish a domain-resolved organizing principle for thicker helical graphene stacks, in which folded Dirac sectors partition the low-energy spectrum, while local stacking families determine the corresponding band character and topological response.