Peierls substitution for magnetic Bloch bands

We consider the Schrödinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, φ(εx) and A(εx), for εll 1. For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrödinger operator to a corresponding almost invariant subspace. At leading order, our effective Hamiltonian can be interpreted as the Peierls substitution Hamiltonian widely used in physics for non-magnetic Bloch bands. However, while for non-magnetic Bloch bands the corresponding result is well understood, for magnetic Bloch bands it is not clear how to even define a Peierls substitution Hamiltonian beyond a formal expression. The source of the difficulty is a topological obstruction: magnetic Bloch bundles are generically not trivializable. As a consequence, Peierls substitution Hamiltonians for magnetic Bloch bands turn out to be pseudodifferential operators acting on sections of non-trivial vector bundles over a two-torus, the reduced Brillouin zone. Part of our contribution is the construction of a suitable Weyl calculus for such pseudos. As an application of our results we construct a new family of canonical one-band Hamiltonians H^B_θ,q for magnetic Bloch bands with Chern number θin mathbb{Z} that generalizes the Hofstadter model H^B_{rm Hof} = H^B_0,1 for a single non-magnetic Bloch band. It turns out that H^B_θ,q is isospectral to H^q2B_{rm Hof} for any θ and all spectra agree with the Hofstadter spectrum depicted in his famous black and white butterfly. However, the resulting Chern numbers of subbands, corresponding to Hall conductivities, depend on θ and q, and thus the models lead to different colored butterflies.