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Helical twisted trilayer graphene exhibits zero-energy flat bands with large degeneracy in the chiral limit. The flat bands emerge at a discrete set of magic twist angles and feature properties intrinsically distinct from those realized in twisted bilayer graphene. Their degeneracy and the associated band Chern numbers depend on the parity of the magic angles. Two degenerate flat bands with Chern numbers $C_A=2$ and $C_B=-1$ arise at odd magic angles, whereas even magic angles display four flat bands, with Chern number $C_{A/B}=\pm1$, together with a Dirac cone crossing at zero energy. All bands are sublattice polarized. We demonstrate the structure behind these flat bands and obtain analytical expressions for the wavefunctions in all cases. Each magic angle is identified with the vanishing of a zero-mode wavefunction at high-symmetry position and momentum. The whole analytical structure results from whether the vanishing is linear or quadratic for the, respectively, odd and even magic angle. The $C_{3z}$ and $C_{2y}T$ symmetries are shown to play a key role in establishing the flat bands. In contrast, the particle-hole symmetry is not essential, except from gapping out the crossing Dirac cone at even magic angles., Comment: 14 pages, 6 figures

We study trilayer graphene arranged in a staircase stacking configuration with equal consecutive twist angle. On top of the moir\'e cristalline pattern, a supermoir\'e long-wavelength modulation emerges that we treat adiabatically. For each valley, we find that the two central bands are topological with Chern numbers $C=\pm 1$ forming a Chern mosaic at the supermoir\'e scale. The Chern domains are centered around the high-symmetry stacking points ABA or BAB and they are separated by gapless lines connecting the AAA points, where the spectrum is fully connected. In the chiral limit and at a magic angle of $\theta \sim 1.69^\circ$, we prove that the central bands are exactly flat with ideal quantum curvature at ABA and BAB. Furthermore, we decompose them analytically as a superposition of an intrinsic color-entangled state with $\pm 2$ and a Landau level state with Chern number $\mp 1$. To connect with experimental configurations, we also explore the non-chiral limit with finite corrugation and find that the topological Chern mosaic pattern is indeed robust and the central bands are still well separated from remote bands., Comment: 7+7 pages, 5 figures

Stacking three monolayers of graphene with a twist generally produces two moir\'e patterns. A moir\'e of moir\'e structure then emerges at larger distance where the three layers periodically realign. We devise here an effective low-energy theory to describe the spectrum at distances larger than the moir\'e lengthscale. In each valley of the underlying graphene, the theory comprises one Dirac cone at the ${\bf \Gamma}_M$ point of the moir\'e Brillouin zone and two weakly gapped points at ${\bf K}_M$ and ${\bf K}'_M$. The velocities and small gaps exhibit a spatial dependence in the moir\'e-of-moir\'e unit cell, entailing a non-abelian connection potential which ensures gauge invariance. The resulting model is numerically solved and a fully connected spectrum is obtained, which is protected by the combination of time-reversal and twofold-rotation symmetries., Comment: 17 pages, 15 figures