The paper by R. Biströter and A.H. MacDonald explores the electronic structure of twisted double-layer graphene, focusing on the formation of moiré bands due to the periodic overlap of two periodically structured layers with a relative twist. The authors construct a low-energy continuum model Hamiltonian that includes single-layer Dirac Hamiltonians and a tunneling term to describe interlayer hopping. They derive a continuum model for the tunneling term, assuming a smooth function of separation between π-orbitals. The model captures the local stacking sequence of misaligned layers and predicts moiré band structures at arbitrary twist angles. Key findings include: - The Dirac velocity at the Dirac point oscillates with the twist angle, vanishing at a series of "magic angles" where the lowest moiré band flattens. - The Dirac-point density-of-states and counterflow conductivity are strongly enhanced at these magic angles. - The velocity vanishes at specific angles, leading to a flat moiré band and a sharp peak in the density-of-states. - The counterflow conductivity remains finite at the magic angles, despite the flattening of the lowest band, due to an increased density of carriers. The authors also provide analytical insights into the behavior of the moiré bands, particularly at small twist angles, and discuss the implications for transport properties in counterflow geometries. They acknowledge the importance of electron-electron interactions at magic twist angles, which could lead to novel states such as counterflow superfluidity or flat-band magnetism.The paper by R. Biströter and A.H. MacDonald explores the electronic structure of twisted double-layer graphene, focusing on the formation of moiré bands due to the periodic overlap of two periodically structured layers with a relative twist. The authors construct a low-energy continuum model Hamiltonian that includes single-layer Dirac Hamiltonians and a tunneling term to describe interlayer hopping. They derive a continuum model for the tunneling term, assuming a smooth function of separation between π-orbitals. The model captures the local stacking sequence of misaligned layers and predicts moiré band structures at arbitrary twist angles. Key findings include: - The Dirac velocity at the Dirac point oscillates with the twist angle, vanishing at a series of "magic angles" where the lowest moiré band flattens. - The Dirac-point density-of-states and counterflow conductivity are strongly enhanced at these magic angles. - The velocity vanishes at specific angles, leading to a flat moiré band and a sharp peak in the density-of-states. - The counterflow conductivity remains finite at the magic angles, despite the flattening of the lowest band, due to an increased density of carriers. The authors also provide analytical insights into the behavior of the moiré bands, particularly at small twist angles, and discuss the implications for transport properties in counterflow geometries. They acknowledge the importance of electron-electron interactions at magic twist angles, which could lead to novel states such as counterflow superfluidity or flat-band magnetism.