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- W3100382102 abstract "We consider the Schrodinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $phi(epsilon x)$ and $A(epsilon x)$, for $epsilonll 1$. For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schrodinger 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 $H^B_{theta,q}$ for magnetic Bloch bands with Chern number $thetain mathbb{Z}$ that generalizes the Hofstadter model $H^B_{rm Hof} = H^B_{0,1}$ for a single non-magnetic Bloch band. It turns out that $H^B_{theta,q}$ is isospectral to $H^{q^2B}_{rm Hof}$ 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." @default.
- W3100382102 created "2020-11-23" @default.
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- W3100382102 date "2016-07-03" @default.
- W3100382102 modified "2023-09-25" @default.
- W3100382102 title "Peierls substitution for magnetic Bloch bands" @default.
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- W3100382102 doi "https://doi.org/10.2140/apde.2016.9.773" @default.
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