  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. In 1976
Douglas Hofstadter theoretically considered the intersection of these two
problems and discovered that 2D electrons subjected to both a magnetic field
and a periodic electrostatic potential exhibit a self-similar recursive energy
spectrum. Known as Hofstadter's butterfly, this complex spectrum results from a
delicate interplay between the characteristic lengths associated with the two
quantizing fields, and represents one of the first quantum fractals discovered
in physics. In the decades since, experimental attempts to study this effect
have been limited by difficulties in reconciling the two length scales. Typical
crystalline systems (<1 nm periodicity) require impossibly large magnetic
fields to reach the commensurability condition, while in artificially
engineered structures (>100 nm), the corresponding fields are too small to
completely overcome disorder. Here we demonstrate that moire superlattices
arising in bilayer graphene coupled to hexagonal boron nitride provide a nearly
ideal-sized periodic modulation, enabling unprecedented experimental access to
the fractal spectrum. We confirm that quantum Hall effect features associated
with the fractal gaps are described by two integer topological quantum numbers,
and report evidence of their recursive structure. Observation of Hofstadter's
spectrum in graphene provides the further opportunity to investigate emergent
behaviour within a fractal energy landscape in a system with tunable internal
degrees of freedom.
