Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians

It is well known that, given a 2d purely magnetic Landau Hamiltonian with a constant magnetic field b which generates a magnetic flux phi per unit area, then any spectral island sigma(b) consisting of M infinitely degenerate Landau levels carries an integrated density of states l(b) = M phi. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any 2d Bloch-Landau operator H-b which also has a bounded Z(2)-periodic electric potential. Assume that H-b has a spectral island sigma(b) which remains isolated from the rest of the spectrum as long as phi lies in a compact interval [phi(1), phi(2)]. Then l(b) = c(0) + c(1)phi on such intervals, where the constant C-0 is an element of Q while c(1) is an element of Z. The integer c(1) is the Chern marker of the spectral projection onto the spectral island sigma(b). This result also implies that the Fermi projection on sigma(b), albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c(1) not equal 0 or c(1) = 0 and phi is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.