Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems

The Hofstadter butterfly is the energy spectrum of an infinite square lattice, plotted as a function of the magnetic field. We illustrate a method of calculating similar spectra for finite lattices in a magnetic field, using methods that consider the appropriate molecular orbitals, and find that the spectra resemble the Hofstadter butterfly. We relate the bonding and antibonding orbitals used to describe small systems to the Landau levels of the infinite system. This approach provides an unusual, but instructive, method of introducing the physics of Landau levels from the basic quantum mechanics of small systems.