arXiv subjects

Guillaume Bal

Publications and source records attributed to Guillaume Bal.

At least 37 records · Page 2Linked to original sources

arXiv2024

Integral formulation of Klein-Gordon singular waveguides

We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.

math-ph↗

arXiv2024

Scattering theory of topologically protected edge transport

This paper develops a scattering theory for the asymmetric transport observed at interfaces separating two-dimensional topological insulators. Starting from the spectral decomposition of an unperturbed interface Hamiltonian, we present a limiting absorption principle and construct a generalized eigenfunction expansion for perturbed systems. We then relate a physical observable quantifying the transport asymmetry to the scattering matrix associated to the generalized eigenfunctions. In particular, we show that the observable is concretely expressed as a difference of transmission coefficients and is stable against perturbations. We apply the theory to systems of perturbed Dirac equations with asymptotically linear domain wall.

math.SP↗

arXiv2024

Topological Equatorial Waves and Violation (or not) of the Bulk Edge Correspondence

Atmospheric and oceanic mass transport near the equator display a well-studied asymmetry characterized by two modes moving eastward. This asymmetric edge transport is characteristic of interfaces separating two-dimensional topological insulators. The northern and southern hemispheres are insulating because of the presence of a Coriolis force parameter that vanishes only in the vicinity of the equator. A central tenet of topological insulators, the bulk edge correspondence, relates the quantized edge asymmetry to bulk properties of the insulating phases, which makes it independent of the Coriolis force profile near the equator. We show that for a natural differential Hamiltonian model of the atmospheric and oceanic transport, the bulk-edge correspondence does not always apply. In fact, an arbitrary quantized asymmetry can be obtained for specific, discontinuous, such profiles. The results are based on a careful analysis of the spectral flow of the branches of absolutely continuous spectrum of a shallow-water Hamiltonian. Numerical simulations validate our theoretical findings.

math.AP↗

arXiv2023

Approximations of interface topological invariants

This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy, and show that it is approximately quantized even in the presence of perturbations.

math-ph↗

arXiv2023

Integral formulation of Dirac singular waveguides

This paper concerns a boundary integral formulation for the two-dimensional massive Dirac equation. The mass term is assumed to jump across a one-dimensional interface, which models a transition between two insulating materials. This jump induces surface waves that propagate outward along the interface but decay exponentially in the transverse direction. After providing a derivation of our integral equation, we prove that it has a unique solution for almost all choices of parameters using holomorphic perturbation theory. We then extend these results to a Dirac equation with two interfaces. Finally, we implement a fast numerical method for solving our boundary integral equations and present several numerical examples of solutions and scattering effects.

math-ph↗

arXiv2023

Semiclassical propagation along curved domain walls

We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set $Γ$ of a domain wall. The main applications we consider are a topologically non-trivial Dirac model and a similar but topologically trivial Klein-Gordon equation. The static domain wall models an interface separating two insulating media in possibly different topological phases. We propose a semiclassical oscillatory representation of the wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along $Γ$ and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary localized and smooth initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets.

math.AP↗

arXiv2023

Topological Anderson Insulators by homogenization theory

A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A Topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This paper proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the systems is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transport indeed appears at an interface separating perturbed and unperturbed phases. The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.

math.AP↗

arXiv2023

$\mathbb{Z}_2$ classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the $\mathbb{Z}_2$ classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

math-ph↗

arXiv2023

Optimizing dual-energy CT technique for iodine-based contrast-to-noise ratio

Purpose: This study proposes a systematic method for determining the optimal x-ray tube settings/energy windows and fluence for minimal noise and maximum CNR in material density images obtained from DECT scans by fixing the subject size and the total radiation dose. Methods: The noise propagation in the process of sinogram and image reconstruction from DECT measurements is analyzed. Analytic estimates for the sinogram and monochromatic image pixel variances and the CNR as functions of tube potentials, fluence, and virtual monochromatic image (VMI) energy are derived, and then used in a phantom experiment as an objective function for optimizing the tube settings to minimize the image noise and maximize the CNR. Results: A non-trivial example that shows the existence of singular solutions to the inversion of sinograms-to-DECT measurements map was presented. Additionally, the optimal VMI energy for maximal CNR was determined. The optimal energy VMI was found to be the least noisy monochromatic image synthesized from the iodine and water density images, and it was shown that using more general weights in combining the two images linearly does not improve image quality. When the x-ray beam filter material was fixed at 2mm of Aluminum and the photon fluence for low and high kV scans were considered equal, the tube potential pair of 60/120 kV led to the maximal CNR in the VMI formed at energy 55 KeV. Conclusions: Optimizing DECT scan parameters to maximize the CNR can be done in a systematic way. Also choosing the parameters that maximize the Jacobian determinant over the sinogram domain would lead to more stable reconstructions due to the reduced amplification of the measurement noise. Since the values of the Jacobian determinant depend strongly on the imaging task, careful consideration of all of the relevant factors is needed when implementing the proposed framework.

physics.med-ph↗

arXiv2023

Mathematical models of topologically protected transport in twisted bilayer graphene

Twisted bilayer graphene gives rise to large moiré patterns that form a triangular network upon mechanical relaxation. If gating is included, each triangular region has gapped electronic Dirac points that behave as bulk topological insulators with topological indices depending on valley index and the type of stacking. Since each triangle has two oppositely charged valleys, they remain topologically trivial. In this work, we address several questions related to the edge currents of this system by analysis and computation of continuum PDE models. Firstly, we derive the bulk invariants corresponding to a single valley, and then apply a bulk-interface correspondence to quantify asymmetric transport along the interface. Secondly, we introduce a valley-coupled continuum model to show how valleys are approximately decoupled in the presence of small perturbations using a multiscale expansion, and how valleys couple for larger defects. Thirdly, we present a method to prove for a large class of continuum (pseudo-)differential models that a quantized asymmetric current is preserved through a junction such as a triangular network vertex. We support all of these arguments with numerical simulations using spectral methods to compute relevant currents and wavepacket propagation.

math-ph↗

arXiv2022

Long time asymptotics of mixed-type Kimura diffusions

This paper concerns the long-time asymptotics of diffusions with degenerate coefficients at the domain's boundary. Degenerate diffusion operators with mixed linear and quadratic degeneracies find applications in the analysis of asymmetric transport at edges separating topological insulators. In one space dimension, we characterize all possible invariant measures for such a class of operators and in all cases show exponential convergence of the Green's kernel to such invariant measures. We generalize the results to a class of two-dimensional operators including those used in the analysis of topological insulators. Several numerical simulations illustrate our theoretical findings.

math.AP↗

arXiv2022

Asymmetric transport for magnetic Dirac equations

This paper concerns the asymmetric transport associated with a low-energy interface Dirac model of graphene-type materials subject to external magnetic and electric fields. We show that the relevant physical observable, an interface conductivity, is quantized and robust to a large class of perturbations. These include defects that decay along or away from the interface, and sufficiently small or localized changes in the external fields. An explicit formula for the interface conductivity is given by a spectral flow.

math-ph↗

arXiv2022

Asymmetric transport computations in Dirac models of topological insulators

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme. We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.

math-ph↗

arXiv2022

Topological charge conservation for continuous insulators

This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. A Fedosov-Hörmander formula implementing in Euclidean spaces an Atiyah-Singer index theorem allows for an explicit computation of the index in terms of the symbol of the Fredholm operator. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable degree of a naturally associated map. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. This asymmetry is captured by a line conductivity, a physical observable of the system. We prove that the line conductivity is quantized and given by the index of a second Fredholm operator of Toeplitz type. We also prove a topological charge conservation stating that the two aforementioned indices agree. This result generalizes to higher dimensions and higher-order topological insulators the bulk-edge correspondence of two-dimensional materials. We apply this procedure to evaluate the topological charge of several classical examples of (standard and higher-order) topological insulators and superconductors in one, two, and three spatial dimensions.

math-ph↗

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.

math-ph↗

arXiv2021

An inversion algorithm for P-functions with applications to Multi-energy CT

Multi-energy computed tomography (ME-CT) is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine the elemental composition of an object of interest. Mathematically, forward ME-CT measurements are modeled by a nonlinear integral transform. In this paper, local conditions for global invertibility of the ME-CT transform are studied, and explicit stability estimates quantifying the error propagation from measurements to reconstructions are provided. Motivated from the inverse problem of image reconstruction in ME-CT, an iterative inversion algorithm for the so-called P-functions is proposed. Numerical simulations for ME-CT, in two and three materials settings with an equal number of energy measurements, confirm the theoretical predictions.

physics.med-ph↗

arXiv2021

Fermi pencil beams and off-axis laser detection

This paper concerns the reconstruction of properties of narrow laser beams propagating in turbulent atmospheres. We consider the setting of off-axis measurements, based on light detection away from the main path of the beam. We first model light propagation in the beam itself by macroscopic approximations of radiative transfer equations that take the form of Fermi pencil beam or fractional Fermi pencil beam equations. Such models are effective in the small mean-free-path, large transport-mean-free path regime. The reconstruction of their constitutive parameters is also greatly simplified compared to the more accurate radiative transfer equations or (fractional) Fokker-Planck models. From off-axis measurements based on wide-angle single scattering off the beam, we propose a framework allowing us to reconstruct the main features of the beam, and in particular its direction of propagation and the location of the emitting source.

math.AP↗

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.

math-ph↗