arXiv subjects

Sylvain Zalczer

Publications and source records attributed to Sylvain Zalczer.

5 recordsLinked to original sources

arXiv2024

Localization for quasi-one-dimensional Dirac operators

We consider a random family of Dirac operators on $N$ parallel real lines, modelling for example a graphene nanoribbon. We establish a localization criterion involving properties on the group generated by transfer matrices. In particular, we consider not only the case where this group is the symplectic group but also a strict subgroup of it. We establish under quite general hypotheses that the sum of the Lyapunov exponents and the integrated density of states are Hölder continuous. Moreover, for a set of concrete cases where the potentials are on Pauli matrices, we compute the transfer matrices and prove either localization or delocalization, depending on the potential and on the parity of $N$.

math-ph↗

arXiv2023

Regularity of the density of states for random Dirac operators

We consider the random Dirac operators for which we have proved Anderson localization in arXiv:1812.01868. We use the Wegner estimate we have got in that paper to prove Lipschitz regularity of the density of states. Since usual methods for Schrödinger operators do not work in this case, we give a new one based on the Helffer-Sjöstrand formula.

math-ph↗

arXiv2023

On boundedness of isomerization paths for non- and semirelativistic molecules

This article focuses on isomerizations of molecules, i.e. chemical reactions during which a molecule is transformed into another one with the same atoms in a different spatial configuration. We consider the special case in which the system breaks into two submolecules whose internal geometry is solid during the whole procedure. We prove, under some conditions, that the distance between the two submolecules stays bounded during the entire reaction. To this end, we provide an asymptotic expansion of the interaction energy between two molecules, including multipolar interactions and the van der Waals attraction. In addition to this static result, we proceed to a quasistatic analysis to investigate the variation of the energy when the nuclei move. This paper generalizes a recent work by M. Lewin and the first author in two directions. The first one is that we relax the assumption that the ground state eigenspaces of the submolecules have to fulfill. The second one is that we allow semirelativistic kinetic energy as well.

math-ph↗

arXiv2021

Localization for one-dimensional Anderson-Dirac models

We prove spectral and dynamical localization for a one-dimensional Dirac operator to which is added an ergodic random potential, with a discussion on the different types of potential. We use scattering properties to prove the positivity of the Lyapunov exponent through Fürstenberg theorem. We get then the Hölder regularity of the integrated density of states through a new version of Thouless formula, and thus the Wegner estimate necessary for the multiscale analysis.

math-ph↗

arXiv2018

Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator $H_0 := D_S + V_0$ is the sum of a Dirac-like operator $D_S$ plus a periodic matrix valued potential $V_0$, and is assumed to have an open gap. The random potential $V_ω$ is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family $H_ω:= H_0 + V_ω$ is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

math-ph↗