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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...

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Documents 81V10 | enregistrements trouvés : 27

Peierls substitution for magnetic Bloch bands Teufel, Stefan | CIRM H Nouveau

Spectral and Scattering Theory for Quantum Magnetic Systems Marseille # July 7-11, 2008 Briet, Philippe ; Germinet, François ; Raikov, Georgi | American Mathematical Society 2009 Nouveau

Advances in quantum computation:representation theory, quantum field theory, category theory, mathematical physics, and quantum information theoryTyler # september 20-23, 2007 Mahdavi, Kazem ; Koslover, Deborah | American Mathematical Society 2009 Nouveau

Quantum control :mathematical and numerical challenges#CRM workshop#Oct. 6-11 Bandrauk, André D. ; Delfour, Michel C. ; Le Bris, Claude | American Mathematical Society 2003 Nouveau

Advances in differential equations and mathematical physics :1997 georgia tech-uab international conference on ... held in georgia institue of technology #March 23-29 Carlen, Eric ; Harrell, Evans M. ; Loss, Michael | American Mathematical Society 1998 Nouveau

Hiérarchie de modèles en optique quantique:de Maxwell-Bloch à Schrödinger non-linéaire Bidégaray-Fesquet, Brigitte | Springer;Société de Mathématiques Appliquées et Industrielles 2006 Nouveau

Dynamics of charged particles and their radiation field Spohn, Herbert | Cambridge University Press 2004 Nouveau

Quantum field theory:a tourist guide for mathematicians Folland, Gerald B. | American Mathematical Society 2008 Nouveau

Advanced quantum mechanics Dyson, Freeman | World Scientific 2007 Nouveau

Path integrals and quantum anomalies Fujikawa, Kazuo ; Suzuki, Hiroshi | Clarendon Press 2004 Nouveau

Wedenie v kvantovuyu elektrodinamiku [Introduction à l'électrodynamique quantique] Sokolov, A.A. | Etat de littérature Physico-Mathématique 1958 Nouveau

Kvantovaya teoriya polya [Théorie quantique des champs] Umezawa, H. ; Matveev, A.N. ; Sokolov, A.A. | Littérature étrangère 1958 Nouveau

Les idées modernes sur la constitution de la matièreconférence faites en 1912 Bauer, Edmond ; Blanc, A. ; Bloch, E. | Gauthier-Villars 1913 Nouveau

Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations Flato, Moshe ; Simon, Jacques C. H. ; Taflin , Erik | American Mathematical Society 1997 Nouveau

What is integrability ? Zakharov, Vladimir | Springer-Verlag 1991 Nouveau

Mathematical problems of relativistic physics Segal, Irving E. ; Mackey, George W. | American Mathematical Society 1986 Nouveau

Physique theorique,vol. 4electodynamique quantique Landau, L. ; Lifchitz, E. | Edition Librairie Du Globe;Editions Mir 1989 Nouveau

Quantum electrodynamics Akhiezer, A. I. ; Berestetskii, V. B. ; Volkokk, G. M. | Interscience Publishers 1965 Nouveau

Mécanique ondulatoire du photon et théorie quantique des champs De Broglie, Louis | Gauthier-Villars 1957 Nouveau

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Spectral and Scattering Theory for Quantum Magnetic Systems
Marseille # July 7-11, 2008 Briet, Philippe ; Germinet, François ; Raikov, Georgi | American Mathematical Society 2009 Nouveau

Advances in quantum computation:
representation theory, quantum field theory, category theory, mathematical physics, and quantum information theory
Tyler # september 20-23, 2007 Mahdavi, Kazem ; Koslover, Deborah | American Mathematical Society 2009 Nouveau

Quantum control :
mathematical and numerical challenges#CRM workshop#Oct. 6-11 Bandrauk, André D. ; Delfour, Michel C. ; Le Bris, Claude | American Mathematical Society 2003 Nouveau

Advances in differential equations and mathematical physics :
1997 georgia tech-uab international conference on ... held in georgia institue of technology #March 23-29 Carlen, Eric ; Harrell, Evans M. ; Loss, Michael | American Mathematical Society 1998 Nouveau

Hiérarchie de modèles en optique quantique:
de Maxwell-Bloch à Schrödinger non-linéaire Bidégaray-Fesquet, Brigitte | Springer;Société de Mathématiques Appliquées et Industrielles 2006 Nouveau

Quantum field theory:
a tourist guide for mathematicians Folland, Gerald B. | American Mathematical Society 2008 Nouveau

Les idées modernes sur la constitution de la matière
conférence faites en 1912 Bauer, Edmond ; Blanc, A. ; Bloch, E. | Gauthier-Villars 1913 Nouveau

Physique theorique,vol. 4
electodynamique quantique Landau, L. ; Lifchitz, E. | Edition Librairie Du Globe;Editions Mir 1989 Nouveau