Peierls substitution for magnetic Bloch bands

Abstract

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, $\varphi (\epsilon x)$ and $A(\epsilon x)$, for $\epsilon \ll 1$. 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 $H_{\theta ,q}^B$ for magnetic Bloch bands with Chern number $\theta \in \mathbb{Z}$ that generalizes the Hofstadter model $H_{\mathrm{Hof}}^B=H_{0,1}^B$ for a single non-magnetic Bloch band. It turns out that $H_{\theta ,q}^B$ is isospectral to $H_{\mathrm{Hof}}^{q^{2}B}$ for any $\theta$ 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 $\theta$ and $q$, and thus the models lead to different colored butterflies.

39 pages, 4 figures. Final version to appear in Analysis & PDE

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