van der Waals heterostructures constitute a new class of artificial materials formed by stacking atomically thin planar crystals. We demonstrated band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate. The spatially varying interlayer atomic registry results in both a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. In our samples, this interplay between short- and long-wavelength effects resulted in a band structure described by isolated superlattice minibands and an unexpectedly large band gap at charge neutrality. This picture is confirmed by our observation of fractional quantum Hall states at ± 5/3 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.

Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely. Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.

We review the electronic properties of bilayer graphene, beginning with a description of the tightbinding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra-and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects. CONTENTS

We develop an effective extended Hubbard model to describe the low-energy electronic properties of the twisted bilayer graphene. By using the Bloch states in the effective continuum model and with the aid of the maximally localized algorithm, we construct the Wannier orbitals and obtain an effective tight-binding model on the emergent honeycomb lattice. We found the Wannier state takes a peculiar three-peak form in which the amplitude maxima are located at the triangle corners surrounding the center. We estimate the direct Coulomb interaction and the exchange interaction between the Wannier states. At the filling of two electrons per super cell, in particular, we find an unexpected coincidence in the direct Coulomb energy between a charge-ordered state and a homogeneous state, which would possibly lead to an unconventional many-body state.

We theoretically study the optical absorption property of twisted bilayer graphenes with various stacking geometries, and demonstrate that the spectroscopic characteristics serve as a fingerprint to identify the rotation angle between two layers. We find that the absorption spectrum almost continuously evolves in changing the rotation angle, regardless of the lattice commensurability. The spectrum is characterized by series of peaks associated with the van Hove singularity, and the peak energies systematically shift with the rotation angle. We calculate the optical absorption in two different frameworks; the tight-binding model and the effective continuum model based on the Dirac equation. For small rotation angles less than 10 • , the effective model well reproduces the lowenergy band structure and the optical conductivity of the tight-binding model, and also explains the optical selection rule analytically in terms of the symmetry of the effective Hamiltonian.