It is well known that, given a purely magnetic Landau Hamiltonian with a constant magnetic field which generates a magnetic flux per unit area, then any spectral island consisting of infinitely degenerate Landau levels carries an integrated density of states . 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 Bloch–Landau operator which also has a bounded -periodic electric potential. Assume that has a spectral island which remains isolated from the rest of the spectrum as long as lies in a compact interval . Then on such intervals, where the constant while . The integer is the Chern marker of the spectral projection onto the spectral island . This result also implies that the Fermi projection on , albeit continuous in in the strong topology, is nowhere continuous in the norm topology if either or 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.
Cite this article
Horia Cornean, Domenico Monaco, Massimo Moscolari, Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3679–3705DOI 10.4171/JEMS/1079