Hofstadter butterfly

In a crystal, the periodic arrangement of atoms produces Bloch energy bands. A perpendicular magnetic field quantizes the motion of charged particles into Landau levels. When a two-dimensional electron system experiences both a periodic potential and a magnetic field, the two quantization schemes interact. As the magnetic flux through a lattice cell changes relative to the magnetic flux quantum, the allowed energies form a complicated spectrum. When plotted against flux and energy, this spectrum has a branching, self-similar appearance known as the Hofstadter butterfly, after physicist Douglas Hofstadter, who analyzed the problem in 1976.

The key challenge is the relative scale of the magnetic length and the lattice period. In an ordinary atomic crystal, the unit cell is so small that the magnetic fields needed to make a flux quantum comparable to the flux through a cell are impractically large. Artificial lattices with much larger periods can lower the required field, but disorder can obscure the fine energy structure. Moiré patterns in stacked two-dimensional crystals provide an intermediate length scale that can make the competing effects experimentally accessible.

A moiré superlattice can form when graphene is placed on hexagonal boron nitride with a suitable relative alignment. The resulting periodic modulation has a length scale of roughly tens of nanometres, larger than the atomic lattice spacing. In a 2013 experiment, researchers observed a Hofstadter spectrum in bilayer graphene coupled to hexagonal boron nitride. They identified quantum Hall features associated with fractal gaps and reported evidence of recursive structure. This demonstrated a way to examine the spectrum in a tunable electronic material rather than only as a theoretical prediction.

The spectrum's gaps are related to quantum Hall states, in which the Hall conductance takes quantized values. In the graphene moiré experiment, the observed fractal gaps were described using two integer topological quantum numbers. These labels connect the measured Hall response with the underlying hierarchy of magnetic subbands and gaps. The Hofstadter butterfly therefore links band structure, magnetic quantization and topological characterization.

The butterfly is a model example of how a simple periodic system can yield a rich spectrum when two length scales compete. Moiré systems have since become a flexible setting for exploring electronic phases and magnetic-field-dependent band structures. The original Hofstadter spectrum describes non-interacting electrons in an idealized periodic potential; real materials may also involve disorder, interactions, finite temperature and deviations from perfect periodicity. Experiments therefore reveal particular realizations of the model, not every feature of an ideal mathematical spectrum.