Spectrum of 2D Bloch electrons in a periodic magnetic field : algebraic approach

Algebraic methods recently introduced for 2D Bloch electrons in a uniform magnetic field are extended to the case of periodic magnetic fields. Using a semiclassical approach, we investigate the case where the magnetic unit cell is commensurate with the lattice unit cell. In general and according to the value Φ of the average flux through the magnetic unit cell, two distinct cases take place. The first one corresponds to finite values of Φ, where the usual structure of Landau levels is recovered (non commutative case). In the second case where Φ = 0, a non trivial band structure is obtained (commutative case). Our results are illustrated by simple examples. In particular we show that, under certain conditions, the mechanism of stabilization of the Fermi sea by the gaps (with one quantum flux per fermion) holds in the general case of periodic magnetic fields.