Unfolding energy spectra of multi-periodicity materials

We propose a new unfolding scheme to analyze energy spectra of complex large-scale systems which are inherently of multi-periodicity. Considering twisted bilayer graphene (tBLG) as an example, we first show that the conventional unfolding scheme in the past using a single primitive-cell representation causes serious problems in analyses of the energy spectra. We then introduce our multi-space representation scheme in the unfolding method and clarify its validity for tBLG. Velocity renormalization of Dirac electrons in tBLG is elucidated in the present unfolding scheme.