arXiv subjects

Michael Hott

Publications and source records attributed to Michael Hott.

10 recordsLinked to original sources

arXiv2024

Mathematical foundations of phonons in incommensurate materials

In some models, periodic configurations can be shown to be stable under, both, global $\ell^2$ or local perturbations. This is not the case for aperiodic media. The specific class of aperiodic media we are interested, in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. In periodic media, phonons are generalized eigenvectors for a stability operator acting on $\ell^2$, coming from a mechanical energy. The goal of our analysis is to provide phonons in the given class of aperiodic media with meaning. As rigorously established for the 1D Frenkel-Kontorova model and previously applied by one of the authors, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, for which we previously derived a continuous energy density functional. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an $\ell^2$-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under $\ell^1$-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.

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arXiv2024

Learning the local density of states of a bilayer moir\'e material in one dimension

Recent work of three of the authors showed that the operator which maps the local density of states of a one-dimensional untwisted bilayer material to the local density of states of the same bilayer material at non-zero twist, known as the twist operator, can be learned by a neural network. In this work, we first provide a mathematical formulation of that work, making the relevant models and operator learning problem precise. We then prove that the operator learning problem is well-posed for a family of one-dimensional models. To do this, we first prove existence and regularity of the twist operator by solving an inverse problem. We then invoke the universal approximation theorem for operators to prove existence of a neural network capable of approximating the twist operator.

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arXiv2024

Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas

Our previous work [37] presented a rigorous derivation of quantum Boltzmann equations near a Bose-Einstein condensate (BEC). Here, we extend it with a complete characterization of the leading order fluctuation dynamics. For this purpose, we correct the latter via an appropriate Bogoliubov rotation, in partial analogy to the approach by Grillakis-Machedon et al. [59], in addition to the Weyl transformation applied in [37]. Based on the analysis of the third order expansion of the BEC wave function, and the second order expansions of the pair-correlations, we show that through a renormalization strategy, various contributions to the effective Hamiltonian can be iteratively eliminated by an appropriate choice of the Weyl and Bogoliubov transformations. This leads to a separation of renormalized Hartree-Fock-Bogoliubov (HFB) equations and quantum Boltzmann equations. A multitude of terms that were included in the error term in [37] are now identified as contributions to the HFB renormalization terms. Thereby, the error bound in the work at hand is improved significantly. To the given order, it is now sharp, and matches the order or magnitude expected from scaling considerations. Consequently, we extend the time of validity to $t\sim (\log N)^2$ compared to $t\sim (\log N/\log \log N)^2$ before. We expect our approach to be extensible to smaller orders in $\frac1N$.

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arXiv2023

From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit

We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy-Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen-Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition.

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arXiv2023

Upper bounds on homogeneous fractional Gagliardo-Nirenberg-Sobolev constants via parabolic estimates

Common proofs of the Gagliardo-Nirenberg-Sobolev (GNS) do not provide explicit bounds on the involved constants, unless a sharp constant is being determined. GNS inequalities naturally occur in error estimates for numerical approximations. In particular, bounds on GNS constants allow us to provide explicit a priori estimates. We provide an algorithm that determines upper bounds on the non-endpoint homogeneous GNS inequalities in terms of explicit upper bounds for Young's convolution inequality and parabolic estimates. Our method is based on the heat-kernel representation of the inverse Laplacian, from which we deduce interpolation estimates.

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arXiv2021

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $\Lambda=(L\mathbb{T})^3$ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density $N$ surrounded by thermal fluctuations with density $1$, we assume that the system is described by a mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It is known that the BEC and the HFB fluctuations both evolve at microscopic time scales $t\sim1$. Given a quasifree initial state, we determine the time evolution of the centered correlation functions $\langle a\rangle$, $\langle aa\rangle-\langle a\rangle^2$, $\langle a^+a\rangle-|\langle a\rangle|^2$ at mesoscopic time scales $t\sim\lambda^{-2}$, where $0<\lambda\ll1$ denotes the size of the HFB interaction. For large but finite $N$, we consider both the case of fixed system size $L\sim1$, and the case $L\sim \lambda^{-2-}$. In the case $L\sim1$, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in $\mathbb{Q}$; this phenomenon is reminiscent of the Talbot effect. For the case $L\sim \lambda^{-2-}$, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $\lambda\sim \Big(\frac{\log \log N}{\log N}\Big)^{\alpha}$, for different values of $\alpha>0$.

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arXiv2018

Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms

This is a work extending the results of \cite{AH} and \cite{AHH}. We want to show convergence of the Schr\"odinger equation towards the Hartree equation with more natural assumptions. We first consider both the defocusing and the focusing semi-relativistic Hartree equation. We show that the tools of \cite{P} are essentially sufficient for deriving the Hartree equation in those cases. Next, we extend this result to the case of fractional Hartree equations with possibly more singular potentials than the Coulomb potential. Finally, we show that, in the non-relativistic case, one can derive the Hartree equation assuming only $L^2$-data in the case of potentials that are more than or as regular as the Coulomb potential. We also derive the Hartree equation for more singular potentials in this case. This work is inspired by talks given at the conference 'MCQM 2018' at Sapienza/Rome.

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arXiv2017

Derivation of the Hartree equation for compound Bose gases in the mean field limit

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field $A\in L^2_{\text{loc}}(\R^3;\R^3)$ and the semi-relativistic case.

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arXiv2016

A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the k-th marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see \cite{MS}, \cite{Lu}. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H^1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional $a_N$ introduced by Pickl in \cite{Pi}.

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arXiv2016

The Lieb--Thomas strategy for strongly coupled fermionic multipolarons with general external fields

In this article, we prove that the ground-state energy of a fermionic Fr\"ohlich multipolaron can be approximated, in the strong electron-phonon coupling limit, by the ground-state energy of a corresponding fermionic Pekar-Tomasevich multipolaron, even in the presence of external electric and magnetic fields. Our analysis builds upon Lieb and Thomas' approach \cite{liebthomas}, which was originally developed for a single polaron without external fields, and Wellig's generalization to multipolarons \cite{wellig} with (specialized) external fields. Our main new contributions are twofold. First, we take into account the fermionic statistics of the multipolaron by employing a localization method from \cite{liebloss}. Second, we relax an assumption in \cite{wellig} on the external electric and magnetic fields, which is not easily verifiable unless the fields are periodic. Instead, we allow for general fields that only ensure self-adjointness of the Fr\"ohlich Hamiltonian. In particular, our work demonstrates the robustness of the Lieb--Thomas strategy when extended to fermionic multipolarons and general external potentials.

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