Band symmetries and singularities in twisted multilayer graphene

The electronic spectra of rotationally faulted graphene bilayers are calculated using a continuum formulation for small fault angles that identifies two distinct electronic states of the coupled system. The low-energy spectra of one state features a Fermi velocity reduction, which ultimately leads to pairwise annihilation and regeneration of its low-energy Dirac nodes. The physics in the complementary state is controlled by pseudospin selection rules that prevent a Fermi velocity renormalization and produce second generation symmetry-protected Dirac singularities in the spectrum. These results are compared with previous theoretical analyses and with experimental data.

Figure 1

Lattice structures of twisted graphene bilayers rotated away from $AA$ stacking by angles $\theta ={3.89}^{\circ }$ (top) and its commensuration partner[14] with $\theta ={56.11}^{\circ }$ (bottom). The insets show schematically the dispersions of two nearby Dirac cones in these structures in the absence of their interlayer coupling.

Figure 2

Lattice structure for a segment of twisted bilayer at rotation angle ${3.89}^{\circ }$, with superlattice translation vectors $T_1$ and $T_2$. The points labeled $\alpha$, $\beta$, and $\gamma$ are high symmetry points in the unit cell. The inset[23] illustrates three hopping processes in the interlayer Hamiltonian.

Figure 3

Electronic spectra for twisted bilayers using the interaction parameters (a) ${\widehat{t}}_0=\tilde{c}(\mathtt{I}+{\sigma }_x),\tilde{c}=0.21$, (b) ${\widehat{t}}_0=\tilde{c}{\sigma }_x,\tilde{c}=0.55$, and (c) ${\widehat{t}}_0=\tilde{c}\mathtt{I},\tilde{c}=0.55$. $Q_{\mathrm{par}}$ and $Q_{\mathrm{perp}}$ are momenta in units of the offset $|\Delta \vec{K}|$ and the ordinate is the scaled energy $E/E_{\theta }=E/\left(\hslash v_F\Delta K\right)$. In (b), Dirac cones with opposite helicity are coupled; in (c), Dirac cones with the same helicity are coupled. The insets give the locations of singular points in the spectra describing the annihilation and regeneration of Dirac nodes (red diamond) in the compensated case (b) and the appearance of a singular point of degeneracy ($C$) for the uncompensated case (c). The point $C$ is a second generation Dirac point singularity in the coupled spectrum.