arXiv subjects

Alexis Drouot

Publications and source records attributed to Alexis Drouot.

At least 19 recordsLinked to original sources

arXiv2026

Geometric bulk-edge correspondence for $\mathbb{Z}_2$-topological insulators

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.

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arXiv2025

Edge spectrum for truncated $\mathbb{Z}_2$-insulators

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a $\mathbb{Z}_2$-index: the Fu-Kane-Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in an earlier work by two of the authors for quantum Hall systems. It boils down to showing that the $\mathbb{Z}_2$-index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the $\mathbb{Z}_2$-index.

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arXiv2024

Absolutely continuous spectrum for truncated topological insulators

We show that if a topological insulator is truncated along a curve that separates the plane in two sufficiently large regions, then the edge system admits absolutely continuous spectrum. Our approach combines a recent version of the bulk-edge correspondence along curves that separates geometry and intrinsic conductance [DZ24], with a result about absolutely continuous spectrum for straight edges [BW22].

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arXiv2024

Band spectrum singularities for Schr\"odinger operators

In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{\"o}dinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda \subset \mathbb{R}^n$ and respects the symmetries of $\Lambda$. Our analysis combines the theory of holomorphic families of operators of type (A) with the seminal work of Fefferman--Weinstein \cite{feffer12}. It allows us to extend results on the existence of spectral degeneracies past a perturbative regime. As an application, we describe the generic structure of some singularities in the band spectrum of Schr\"odinger operators invariant under the three-dimensional simple, body-centered and face-centered cubic lattices.

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arXiv2024

The bulk-edge correspondence for curved interfaces

The bulk-edge correspondence is a condensed matter theorem that relates the conductance of a Hall insulator in a half-plane to that of its (straight) boundary. In this work, we extend this result to domains with curved boundaries. Under mild geometric assumptions, we prove that the edge conductance of a topological insulator sample is an integer multiple of its Hall conductance. This integer counts the algebraic number of times that the interface (suitably oriented) enters the measurement set. This result provides a rigorous proof of a well-known experimental observation: arbitrarily truncated topological insulators support edge currents, regardless of the shape of their boundary.

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arXiv2023

Topological edge spectrum along curved interfaces

We prove that that if the boundary of a topological insulator divides the plane in two regions containing arbitrarily large balls, then it acts as a conductor. Conversely, we show that topological insulators that fit within strips do not need to admit conducting boundary modes.

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arXiv2022

Topological insulators in semiclassical regime

We study solutions of $2 \times 2$ systems $(h D_t + \mathcal{D}) \Psi_t = 0$ on $\mathbb{R}^2$ in the semiclassical regime $h \rightarrow 0$. Our Dirac operator $\mathcal{D}$ is a standard model for interfaces between topological insulators: it represents a semimetal at null energy, and distinct topological phases at positive / negative energies. Its semiclassical symbol has two opposite eigenvalues, intersecting conically along the curve $\Gamma$ of positions / momenta where the material behaves like a semimetal. We prove that wavepackets solving $(h D_t + \mathcal{D}) \Psi_t = 0$, initially concentrated in phase space on $\Gamma$, split in two parts. The first part travels coherently along $\Gamma$ with predetermined direction and speed. It is the dynamical manifestation of the famous "edge state". The second part immediately collapses. Our approach consists of: (i) a Fourier integral operator reduction; (ii) a careful WKB analysis of a canonical model; (iii) a reconstruction procedure. It yields concrete formulas for the speed and profile of the traveling modes. As applications, we analytically describe dynamical edge states for models of magnetic, curved, and strained topological insulators.

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arXiv2022

Magnetic slowdown of topological edge states

We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic-free case, the wavepackets propagate coherently at speed one, here they experience slowdown, dispersion, and Aharonov - Bohm effects. Several numerical simulations illustrate our results.

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arXiv2021

Edge state dynamics along curved interfaces

We study the propagation of wavepackets along weakly curved interfaces between topologically distinct media. Our Hamiltonian is an adiabatic modulation of Dirac operators omnipresent in the topological insulators literature. Using explicit formulas for straight edges, we construct a family of solutions that propagates, for long times, unidirectionally and dispersion-free along the curved edge. We illustrate our results through various numerical simulations.

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arXiv2020

Ubiquity of conical points in topological insulators

We show that generically, the degeneracies of a family of Hermitian matrices depending on three parameters have a conical structure. Our result applies to the study of topological phases of matter. It implies that adiabatic deformations of two-dimensional topological insulators come generically with Dirac-like propagating currents, whose total conductivity equals the chiral number of conical points.

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arXiv2019

Microlocal analysis of the bulk-edge correspondence

The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It suggests novel perspectives for the analysis of topologically protected transport.

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arXiv2019

The bulk-edge correspondence for continuous honeycomb lattices

We study bulk/edge aspects of continuous honeycomb lattices in a magnetic field. We compute the bulk index of Bloch eigenbundles: it equals $2$ or $-2$, with sign depending on nearby Dirac points and on the magnetic field. We then prove the existence of two topologically protected waves propagating along line defects. This shows the bulk/edge correspondence for our class of Hamiltonians.

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arXiv2018

Characterization of edge states in perturbed honeycomb structures

This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by work of Haldane-Raghu , we continue the study of a linear PDE initiated in papers of Fefferman-Lee-Thorp-Weinstein. This PDE is induced by a continuous honeycomb Schrodinger operator with a line defect. This operator exhibits remarkable connections between topology and spectral theory. It has essential spectral gaps about the Dirac point energies of the honeycomb background. In a perturbative regime, Fefferman-Lee-Thorp-Weinstein construct edge states: time-harmonic waves propagating along the interface, localized transversely. At leading order, these edge states are adiabatic modulations of the Dirac point Bloch modes. Their envelops solve a Dirac equation that emerges from a multiscale procedure. We develop a scattering-oriented approach that derives all possible edge states, at arbitrary precision. The key component is a resolvent estimate connecting the Schrodinger operator to the emerging Dirac equation. We discuss topological implications via the computation of the spectral flow, or edge index.

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arXiv2018

The bulk-edge correspondence for continuous dislocated systems

We study topological aspects of defect modes for a family of operators $\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$ on $L^2(\mathbb{R})$. $\mathscr{P}(t)$ is a periodic Schr\"odinger operator $P_0$ perturbed by a dislocated potential. This potential is periodic on the left and on the right, but acquires a phase defect $t$ from $-\infty$ relative to $+\infty$. When $t=\pi$ and the dislocation is small and adiabatic, Fefferman, Lee-Thorp and Weinstein showed in previous work that Dirac points of $P_0$ (degeneracies in the band spectrum of $P_0$) bifurcate to defect modes of $\mathscr{P}(\pi)$. We show that these modes are topologically protected at the level of the family $\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$. This means that local perturbations cannot remove these states for all values of $t$, even outside the small adiabatic regime studied by Fefferman$-$Lee-Thorp$-$Weinstein. We define two topological quantities: an edge index (the signed number of eigenvalues crossing an energy gap as $t$ runs from $0$ to $2\pi$) and a bulk index (the Chern number of a Bloch eigenbundle for the periodic operator near $+\infty$). We prove that these indexes are equal to the same odd winding number. This shows the bulk-edge correspondence for the family $\{\mathscr{P}(t)\}_{t \in [0,2\pi]}$. We express this winding number in terms of the asymptotic shape of the dislocation and of the Dirac point Bloch modes. We illustrate the topological depth of our model via the computation of the bulk/edge index on a few examples.

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arXiv2018

Defect modes for dislocated periodic media

We study defect modes in a one-dimensional periodic medium with a dislocation. The model is a periodic Schrodinger operator on $\mathbb{R}$, perturbed by an adiabatic dislocation of amplitude $\delta\ll 1$. If the periodic background admits a Dirac point $-$ a linear crossing of dispersion curves $-$ then the dislocated operator acquires a gap in its essential spectrum. For this model (and its 2-dimensional honeycomb analog) Fefferman, Lee-Thorp and Weinstein constructed in previous work defect modes with energies within the gap. The bifurcation of defect modes is associated with the discrete eigenmodes of an effective Dirac operator. We improve upon this result: we show that all the defect modes of the dislocated operator arise from the eigenmodes of the Dirac operator. As a byproduct, we derive full expansions of the eigenpairs in powers of $\delta$. The self-contained proof relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials developed by the first author. This work significantly advances the understanding of the topological stability of certain defect states, particularly the bulk-edge correspondence for continuous dislocated systems.

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arXiv2017

Resonances for random highly oscillatory potentials

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of $-\Delta_{\mathbb{R}^d}+V_N$ as $N \rightarrow \infty$. We identify a stochastic and a deterministic regime for the speed of convergence. The type of regime depends whether the low frequencies effects due to large deviations overcome the (deterministic) constructive interference between highly oscillatory terms.

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arXiv2016

Bound states for rapidly oscillatory Schr\"odinger operators in dimension 2

We study the eigenvalues of Schr\"odinger operators on $\mathbb{R}^2$ with rapidly oscillatory potential $V(x) = W(x,x/\varepsilon)$, where $W(x,y) \in C^\infty_0(\mathbb{R}^2 \times \mathbb{T}^2)$ satisfies $\int_{\mathbb{T}^2} W(x,y) dy =0$. We show that for $\varepsilon$ small enough, such operators have a unique negative eigenvalue, that is exponentially close to $0$.

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arXiv2016

Pollicott-Ruelle resonances via kinetic Brownian motion

The kinetic Brownian motion on the cosphere bundle of a Riemannian manifold $\mathbb{M}$ is a stochastic process that models the geodesic equation perturbed by a random white force of size $\varepsilon$. When $\mathbb{M}$ is compact with negative curvature we show that the $L^2$-spectrum of the infinitesimal generator of this process converges to the Pollicott--Ruelle resonances of $\mathbb{M}$ as $\varepsilon \rightarrow 0$.

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