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Louis Garrigue

Publications and source records attributed to Louis Garrigue.

12 recordsLinked to original sources
arXiv2026

Some effective operators for graphene monolayer superlattices, from variational perturbation theory

Our goal is to provide precise effective operators for monolayer graphene at Fermi energy. We consider the microscopic potential created by a lattice, and add a macroscopic potential with the same periodicity but varying at a scale $\varepsilon^{-1} \in \mathbb{N}$, creating a superlattice. Our approach consists in coupling the variational approximation, perturbation theory together with a multiscale method. At the effective level the usual massless Dirac operator is replaced by other operators, and we provide simulations in the case of graphene.

math-ph↗
arXiv2024

On reduced basis methods for eigenvalue problems, and on its coupling with perturbation theory

In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact eigenmodes and the approximated ones and also consider degenerate cases in the analysis. When the operator depends on a parameter, we apply the bounds assuming that the reduced space contains the derivatives of the eigenfunction with respect to the parameter. Finally, we provide some numerical examples that reflect the analytical results.

math-ph↗
arXiv2023

Coefficients of almost-degenerate density matrix perturbation theory for eigenvalue problems

We investigate almost-degenerate perturbation theory of eigenvalue problems, using spectral projectors, also named density matrices. When several eigenvalues are close to each other, the coefficients of the perturbative series become singular because inverses of differences between eigenvalues arise as some factors. We remove those artificial singularities in the expressions of the coefficients of the series, allowing eigenvalue gaps to be arbitrarily small and even vanishing in the resulting formulas.

math-ph↗
arXiv2023

A multipoint perturbation formula for eigenvalue problems

Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several nearby operators are known, standard perturbation theory cannot simultaneously use all this knowledge to provide a better approximation. We derive a resolvent formula enabling such an approximation result, and provide numerical examples for which this method is more competitive than standard perturbation theory.

math-ph↗
arXiv2022

Building Kohn-Sham potentials for ground and excited states

We analyze the inverse problem of Density Functional Theory using a regularized variational method. First, we show that given $k$ and a target density $ρ$, there exist potentials having $k^{\text{th}}$ bound mixed states which densities are arbitrarily close to $ρ$. The state can be chosen pure in dimension $d=1$ and without interactions, and we provide numerical and theoretical evidence consistently leading us to conjecture that the same pure representability result holds for $d=2$, but that the set of pure-state $v$-representable densities is not dense for $d=3$. Finally, we present an inversion algorithm taking into account degeneracies, removing the generic blocking behavior of standard ones.

math-ph↗
arXiv2022

A simple derivation of moir\'e-scale continuous models for twisted bilayer graphene

We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tight-binding model. The so-obtained model is similar to that of the Bistritzer-MacDonald (BM) model but contains additional terms. Its parameters can be easily computed from Kohn-Sham calculations on single-layer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters $w_{\rm AA}$ and $w_{\rm AB}$ of the BM model from first-principles. The resulting numerical values, namely $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV for the experimental interlayer mean distance are in good agreement with the empirical values $w_{\rm AA}= w_{\rm AB}=110$ meV obtained by fitting to experimental data. We also show that if the BM parameters are set to $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV, the BM model is an accurate approximation of our reduced model.

cond-mat.mes-hall↗
arXiv2021

Second-order homogenization of periodic Schrödinger operators with highly oscillating potentials

We consider the homogenization at second-order in $\varepsilon$ of $\mathbb{L}$-periodic Schrödinger operators with rapidly oscillating potentials of the form $H^\varepsilon =-Δ+ \varepsilon^{-1} v(x,\varepsilon^{-1}x ) + W(x)$ on $L^2(\mathbb{R}^d)$, where $\mathbb{L}$ is a Bravais lattice of $\mathbb{R}^d$, $v$ is $(\mathbb{L} \times \mathbb{L})$-periodic, $W$ is $\mathbb{L}$-periodic, and $\varepsilon \in \mathbb{N}^{-1}$. We treat both the linear equation with fixed right-hand side and the eigenvalue problem, as well as the case of physical observables such as the integrated density of states. We illustrate numerically that these corrections to the homogenized solution can significantly improve the first-order ones, even when $\varepsilon$ is not small.

math-ph↗
arXiv2021

Some properties of the potential-to-ground state map in quantum mechanics

We analyze the map from potentials to the ground state in static many-body quantum mechanics. We first prove that the space of binding potentials is path-connected. Then we show that the map is locally weak-strong continuous and that its differential is compact. In particular, this implies the ill-posedness of the Kohn-Sham inverse problem.

math-ph↗
arXiv2020

Hohenberg-Kohn theorems for interactions, spin and temperature

We prove Hohenberg-Kohn theorems for several models of quantum mechanics. First, we show that the pair correlation function of any ground state contains the information of the interactions and of the external potentials. Then, in the presence of the Zeeman interaction, a strong constraint on external fields is derived for systems having the same ground state densities and magnetizations. Moreover, we provide a counterexample in a setting involving non-local potentials. Next, we prove that the density and the entropy of a ground state contain the information of both the imposed external potential and temperature. Eventually, we conclude that at positive temperature, Hohenberg-Kohn theorems generically hold.

math-ph↗
arXiv2019

Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem

We prove the strong unique continuation property for many-body Schrödinger operators with an external potential and an interaction potential both in $L^p_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$, independently of the number of particles. With the same assumptions, we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.

math.AP↗