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Nonlinear quantum systems at dissociation: the example of graphene

Abstract : This thesis is devoted to the mathematical study of electronic properties of matter. The systems, both molecular and crystalline, are described by nonlinear models coming from quantum mechanics. Then, we consider the dissociation regime, that is when the distances between the nuclei are large. In Chapter 1, we study the diatomic Hartree model, both in dimension two and three, and we precisely estimate the quantum tunneling between the first two eigenfunctions. In Chapter 2, we show that if a non-degeneracy condition is satisfied then the reduced Hartree-Fock model of graphene presents conical singularities, called Dirac points. In addition, we show that the Fermi level coincides with the energy of these cones. In this direction, we derive conditions under which the dispersion relation of periodic Schrödinger operator is given, to leading order and in the dissociation regime, by the corresponding tight-binding model.
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https://tel.archives-ouvertes.fr/tel-03726340
Contributor : Jean Cazalis Connect in order to contact the contributor
Submitted on : Thursday, August 18, 2022 - 7:49:00 PM
Last modification on : Tuesday, August 23, 2022 - 3:54:33 AM

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  • HAL Id : tel-03726340, version 2

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Jean Cazalis. Nonlinear quantum systems at dissociation: the example of graphene. Mathematical Physics [math-ph]. Université PSL (Paris Sciences & Lettres), 2022. English. ⟨NNT : ⟩. ⟨tel-03726340v2⟩

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