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We study the electronic structure of alternating-twist tetralayer graphene, especially near its magic angle $\theta = 1.75^\circ$, for different AA, AB, and SP sliding geometries at their middle interface that divides two twisted bilayer graphenes. This sliding dependence is shown for the bandwidths, band gaps, and $K$-valley Chern numbers of the lowest-energy valence and conduction bands as a function of twist angle and interlayer potential difference. Our analysis reveals that the AA sliding is most favorable for narrow bands and gaps, and the AB sliding is most prone to developing finite valley Chern numbers. We further analyze the linear longitudinal optical absorptions as a function of photon energy and the absorption map in the moir\'{e} Brillouin zone for specific transition energies. A self-consistent Hartree calculation reveals that the AA system's electronic structure is the most sensitive to variations in carrier density., Comment: 15 pages, 11 figures

We theoretically study the energy and optical absorption spectra of alternating twist multilayer graphene (ATMG) under a perpendicular electric field. We obtain analytically the low-energy effective Hamiltonian of ATMG up to pentalayer in the presence of the interlayer bias by means of first-order degenerate-state perturbation theory, and present general rules for constructing the effective Hamiltonian for an arbitrary number of layers. Our analytical results agree to an excellent degree of accuracy with the numerical calculations for twist angles $\theta \gtrsim 2.2^{\circ}$ that are larger than the typical range of magic angles. We also calculate the optical conductivity of ATMG and determine its characteristic optical spectrum, which is tunable by the interlayer bias. When the interlayer potential difference is applied between consecutive layers of ATMG, the Dirac cones at the two moir\'{e} Brillouin zone corners $\bar{K}$ and $\bar{K}'$ acquire different Fermi velocities, generally smaller than that of monolayer graphene, and the cones split proportionally in energy resulting in a step-like feature in the optical conductivity., Comment: 11 pages, 11 figures, 2 tables