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 ' per unit area, then any spectral island σ_b consisting of M infinitely degenerate Landau levels carries an integrated density of states I_b D M'. 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 Z²-periodic electric potential. Assume that H_b has a spectral island σ_b which remains isolated from the rest of the spectrum as long as ' lies in a compact interval Œ'₁; '₂]. Then I_b D c₀ C c1' on such intervals, where the constant c₀ 2 Q while c₁ 2 Z. The integer c₁ is the Chern marker of the spectral projection onto the spectral island σ_b. This result also implies that the Fermi projection on σ_b, albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c₁ ¤ 0 or c1 D 0 and ' 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.