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Documents  Teufel, Stefan | enregistrements trouvés : 4

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Research talks;Mathematical Physics

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, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , 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 $H_{\theta=0}$ 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.
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, $\phi(\varepsilon x)$ and $A(\varepsilon x)$ , 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 co...

81Q20 ; 81V10 ; 82D20

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- ix; 148 p.
ISBN 978-3-0348-0465-3

Trends in mathematics

Localisation : Colloque 1er étage (TUBI)

microlocal analysis # physique mathématique # analyse globale

35-XX ; 58-XX ; 00B25 ; 35-06 ; 58-06 ; 35Qxx ; 58JXX ; 81Qxx

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- v; 83 p.
ISBN 978-0-8218-9489-7

Memoirs of the american mathematical society , 1083

Localisation : Collection 1er étage

opérateur Hamiltonien # théorie quantique # valeur propre # mécanique # équation de Schrödinger # contrainte # dynamique effective # limite adiabatique

81Q15 ; 58J37 ; 81Q70

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- 234 p.
ISBN 978-3-540-40723-2

Lecture notes in mathematics , 1821

Localisation : Collection 1er étage

perturbation adiabatique # dynamique quantique # opérateur pseudodifférentiel # EDP de la mécanique quantique # équation du transport # dynamique moléculaire # dynamique des élections

81-02 ; 81Q15 ; 47G30 ; 35Q40

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