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Peierls substitution for magnetic Bloch bands

Stefan Teufel
(Universität Tübingen)
Tirsdag, 7 oktober, 2014, at 15:15-16:00, in Koll. G4 (1532-222)
We consider the one-particle 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, $\phi(\epsilon x)$ 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, both on a heuristic and on a rigorous level, 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: In contrast to the non-magnetic case, 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.

As an application of our results we construct a family of canonical one-band Hamiltonians $H_\theta$ for magnetic Bloch bands with Chern number $\theta\in {\mathbb Z}$ that generalizes the Hofstadter model $H_{\theta=0}$ for a single non-magnetic Bloch band. It turns out that the spectrum of $H_\theta$ is independent of $\theta$ and thus agrees 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 thus the models lead to different colored butterflies.
This is joint work with Silvia Freund.
Kontaktperson: Søren Fournais