Noninteracting electrons in 2D in presence of a magnetic field and a periodic potential?

This the famous Hofstadter problem. (Douglas Hofstadter is the author of Godel-Escher-Bach).

Statistical Physics, Part 2 by Landau & Livshits contains a chapter on "The symmetry of electron states in a lattice in a magnetic field" (Translation might be unprecise, as I am using my Russian edition.) Basically, the presence of magnetic field breaks the translational symmetry and leads to some interesting quirks, but for weak magnetic fields on still can use Peierls substitution in the effective mass Hamiltonian.

This is an admittedly old resource, but it could serve as a good starting point for serious analysis.