Magic continuum in multi-moiré twisted trilayer graphene

Moiré lattices provide a highly tunable platform for exploring the interplay between electronic correlations and band topology. Introducing a second moiré pattern extends this paradigm: interference between the two moiré patterns produces a supermoiré modulation, opening a route to further tailor electronic properties. Twisted trilayer graphene generally exemplifies such a system: two distinct moiré patterns arise from the relative twists between adjacent graphene layers. Here, we report the observation of correlated phenomena across a wide range of twisted trilayer graphene devices whose twist angles lie along two continuous lines in the twist-angle parameter space. Depending on the degree of lattice relaxation, twisted trilayer graphene falls into two classes: moiré polycrystals, composed of periodic domains with locally commensurate moiré order, and moiré quasicrystals, characterized by smoothly varying local moiré configurations. In helically twisted moiré polycrystals, we observe an anomalous Hall effect, consistent with topological bands arising from domains with broken $xy$-inversion symmetry. In contrast, superconductivity appears generically in our moiré quasicrystals. A subset of these systems exhibits signatures of spatially modulated superconductivity, which we attribute to the supermoiré structure. Our findings uncover the organizing principles of the observed correlated phases in twisted trilayer graphene, highlight the critical roles of the supermoiré modulation and lattice relaxation, and suggest a broader framework in which magic conditions arise not as isolated points but as extended manifolds within the multi-dimensional twist-angle space of complex moiré materials.