Stacking and gate tunable topological flat bands, gaps and anisotropic strip patterns in twisted trilayer graphene

Trilayer graphene with a twisted middle layer has recently emerged as a new platform exhibiting correlated phases and superconductivity near its magic angle. A detailed characterization of its electronic structure in the parameter space of twist angle $\theta$, interlayer potential difference $\Delta$, and top-bottom layer stacking $\tau$ reveals that flat bands with large Coulomb energy vs bandwidth $U/W > 1$ are expected within a range of $\pm 0.2^{\circ}$ near $\theta \simeq1.5^{\circ}$ and $\theta \simeq1.2^{\circ}$ for $\tau_{\rm AA}$ top-bottom layer stacking, between a wider $1^{\circ} \sim 1.7^{\circ}$ range for $\tau_{\rm AB}$ stacking, whose bands often have finite valley Chern numbers thanks to the opening of primary and secondary band gaps in the presence of a finite $\Delta$, and below $\theta \lesssim 0.6^{\circ}$ for all $\tau$ considered. The largest $U/W$ ratios are expected at the magic angle $\sim 1.5^{\circ}$ when $|\Delta| \sim 0$~meV for AA, and slightly below near $\sim 1.4^{\circ}$ for finite $|\Delta| \sim 25$~meV for AB stackings, and near $\theta \sim 0.4^{\circ}$ for both stackings. When ${\tau}$ is the saddle point stacking vector between AB and BA we observe pronounced anisotropic local density of states (LDOS) strip patterns with broken triangular rotational symmetry. We present optical conductivity calculations that reflect the changes in the electronic structure introduced by the stacking and gate tunable system parameters.