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Documents  Germinet, François | enregistrements trouvés : 7

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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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- XII-186 p.
ISBN 978-0-8218-4744-2

Contemporary mathematics , 0500

Localisation : Collection 1er étage

théorie quantique # physique mathématique # diffusion quantique # opérateur de Schrödinger # opérateur de Dirac # opérateur de Pauli # opérateur magnétique # équation de Ginsburg-Landau # superconductivité # bouteille magnétique # condensat de Bose-Einstein # équation de Gross-Pitaevski # inégalité de Lieb-Thirring magnétique # stabilité de la matière

81Q10 ; 81V10 ; 35J10 ; 82B44 ; 60H25 ; 47B80 ; 81Q70 ; 35P20 ; 35P25

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Research talks;Mathematical Physics

One of the important aspects of many-body quantum mechanics of electrons is the analysis of two-body density matrices. While the characterization of one-body density matrices is well known and simple to state, that of two-body matrices is far from simple - indeed, it is not fully known. In this talk I will present joint work with Eric Carlen in which we study the possible entropy of such matrices. We find, inter alia, that minimum entropy is achieved for Slater determinant N-body parent functions. Thus, from the entropic point of view, Slater determinants play the same role as condensates play for bosons. One of the important aspects of many-body quantum mechanics of electrons is the analysis of two-body density matrices. While the characterization of one-body density matrices is well known and simple to state, that of two-body matrices is far from simple - indeed, it is not fully known. In this talk I will present joint work with Eric Carlen in which we study the possible entropy of such matrices. We find, inter alia, that minimum entropy is ...

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Research talks;Mathematical Physics;Topology

According to a widely accepted terminology, a topological insulator is a (independent) Fermion system which has surface modes that are not exposed to Anderson localization. This stability results from topological constraints given by non-trivial invariants like non-commutative Chern numbers and higher winding numbers, but sometimes also more subtle Z2 invariants associated to adequate Fredholm operators with symmetries. Prime examples are quantum Hall systems, but the talk also considers chiral and BdG systems as well as time-reversal symmetric systems with Z2 invariants. According to a widely accepted terminology, a topological insulator is a (independent) Fermion system which has surface modes that are not exposed to Anderson localization. This stability results from topological constraints given by non-trivial invariants like non-commutative Chern numbers and higher winding numbers, but sometimes also more subtle Z2 invariants associated to adequate Fredholm operators with symmetries. Prime examples are ...

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Research talks;Mathematical Physics

A quantum phase transition is commonly referred to as a point in a family of gapped Hamiltonians where the spectral gap closes. In the absence of a general perturbation theory for quantum spin systems in the thermodynamic limit, I will discuss necessary, and sufficient, conditions for a transition, and present explicit constructions of paths of uniformly gapped Hamiltonians in one dimension.

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Research talks;Mathematical Physics

We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins $S= 1/2$. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. The proof combines a bosonic representation of the model introduced by Holstein and Primakoff with probabilistic estimates, localization bounds and functional inequalities.
Joint work with Michele Correggi and Alessandro Giuliani
We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins $S= 1/2$. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. The proof combines a bosonic representation of the model introduced by Holstein and Primakoff with probabilistic estimates, localization bounds and functional ...

82D05 ; 82D40 ; 82D45 ; 82B10

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- 256 p.
ISBN 978-0-8218-4241-6

Contemporary mathematics , 0447

Localisation : Collection 1er étage

théorie spectrale # valeurs propres pour opérateurs différentielles # équation pour la physique mathématique # opérateurs pseudo-différentiels # opérateurs linéaire # théorie quantique

35Pxx ; 35Qxx ; 35Sxx ; 47Axx ; 81-XX

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