arXiv subjects

Solomon Quinn

Publications and source records attributed to Solomon Quinn.

10 recordsLinked to original sources

arXiv2026

Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.

math-ph↗

arXiv2025

Complex scaling for open waveguides

In this work we analyze the complex scaling method applied to the problem of time-harmonic scalar wave propagation in junctions between `leaky,' or open dielectric waveguides. In [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674, arXiv:2411.11204], it was shown that under suitable assumptions the problem can be reduced to a system of Fredholm second-kind integral equations on an infinite interface, transverse to the waveguides. Here, we show that the kernels appearing in the integral equation admit a rapidly decaying analytic continuation on certain natural totally real submanifolds of $\mathbb{C}^2.$ We then show that for suitable, physically-meaningful, boundary data the resulting solutions to the integral equations themselves admit analytic continuation and satisfy related asymptotic estimates. By deforming the integral equation to a suitable contour, the decay in the kernels, density, and data enable straightforward discretization and truncation, with an error that decays exponentially in the truncation length. We illustrate our results with several representative numerical examples.

math.NA↗

arXiv2025

Higher-order continuum models for twisted bilayer graphene

The first-order continuum PDE model proposed by Bistritzer and MacDonald in \cite{bistritzer2011moire} accurately describes the single-particle electronic properties of twisted bilayer graphene (TBG) at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer-MacDonald model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis from \cite{watson2023bistritzer}, which rigorously established the validity of the (first-order) BM model.

math-ph↗

arXiv2024

Dirac cones and magic angles in the Bistritzer--MacDonald TBG Hamiltonian

We demonstrate the generic existence of Dirac cones in the full Bistritzer--MacDonald Hamiltonian for twisted bilayer graphene. Its complementary set, when Dirac cones are absent, is the set of magic angles. We show the stability of magic angles obtained in the chiral limit by demonstrating that the perfectly flat bands transform into quadratic band crossings when perturbing away from the chiral limit. Moreover, using the invariance of Euler number, we show that at magic angles there are more band crossings beyond these quadratic band crossings. This is the first result showing the existence of magic angles for the full Bistritzer--MacDonald Hamiltonian and solves Open Problem No.2 proposed in the recent survey arXiv:2310.20642.

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↗

arXiv2022

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 Schr\"odinger 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↗

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

Mathematical models of topologically protected transport in twisted bilayer graphene

Twisted bilayer graphene gives rise to large moir\'{e} 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

Optimal method for reconstructing polychromatic maps from broadband observations with an asymmetric antenna pattern

Broadband time-ordered data obtained from telescopes with a wavelength-dependent, asymmetric beam pattern can be used to extract maps at multiple wavelengths from a single scan. This technique is especially useful when collecting data on cosmic phenomena such as the Cosmic Microwave Background (CMB) radiation, as it provides the ability to separate the CMB signal from foreground contaminants. We develop a method to determine the optimal linear combinations of wavelengths (``colors'') that can be reconstructed for a given telescope design and the number of colors that are measurable with high signal-to-noise ratio. The optimal colors are found as eigenvectors of a matrix derived from the inverse noise covariance matrix. When the telescope is able to scan the sky isotropically, it is useful to transform to a spherical harmonic basis, in which this matrix has a particularly simple form. We propose using the optimal colors determined from the isotropic case even when the actual scanning pattern is not isotropic (e.g., covers only part of the sky). We perform simulations showing that maps in multiple colors can be reconstructed accurately from both full-sky and partial-sky scans. Although the original motivation for this research comes from mapping the CMB, this method of polychromatic map-making will have broader applications throughout astrophysics.

astro-ph.IM↗

arXiv2021

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↗