arXiv subjects

Michael I. Weinstein

Publications and source records attributed to Michael I. Weinstein.

At least 37 records · Page 2Linked to original sources

arXiv2024

Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators

We study a semilinear hyperbolic system of PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model introduced by Hadad, Vitelli and Alu (HVA) in \cite{HVA17}. We classify the system's traveling waves, and study their stability properties. We focus on traveling pulse solutions (``solitons'') on a nontrivial background and moving domain wall solutions (kinks); both arise as heteroclinic connections between spatially uniform equilibrium of a reduced dynamical system. We present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and spectral stability of moving domain walls. Our stability results are in terms of weighted $H^1$ norms of the perturbation, which capture the phenomenon of {\it convective stabilization}; as time advances, the traveling wave ``outruns'' the \underline{growing} disturbance excited by an initial perturbation; the non-trivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.

nlin.PS↗

arXiv2023

Nonlocal PDEs and quantum optics: band structure of periodic atomic sytems

We continue our study of the quantum optics of a single photon interacting with a system of two level atoms. In this work we investigate the case of a periodic arrangement of atoms. We provide a general structure theorem characterizing the band functions of this problem, which comprise the spectrum of the associated Hamiltonian. Additionally, we study atomic densities arising as periodically arranged scaled inclusions. For this family of examples, we obtain explicit asymptotic formulas for the band functions.

math-ph↗

arXiv2023

Free boundary problem for a gas bubble in a liquid, and exponential stability of the manifold of spherically symmetric equilibria

We consider the dynamics of a gas bubble immersed in an incompressible fluid of fixed temperature, and focus on the relaxation of an expanding and contracting spherically symmetric bubble due to thermal effects. We study two models, both systems of PDEs with an evolving free boundary: the full mathematical model as well as an approximate model, arising for example in the study of sonoluminescence. For fixed physical parameters (surface tension of the gas-liquid interface, liquid viscosity, thermal conductivity of the gas, etc.), both models share a family of spherically symmetric equilibria, smoothly parametrized by the mass of the gas bubble. Our main result concerns the approximate model. We prove the nonlinear asymptotic stability of the manifold of equilibria with respect to small spherically symmetric perturbations. The rate of convergence is exponential in time. To prove this result we first prove a weak form of nonlinear asymptotic stability -- with no explicit rate of time-decay -- using the energy dissipation law, and then, via a center manifold analysis, bootstrap the weak time-decay to exponential time-decay. We also study the uniqueness of the family of spherically symmetric equilibria within each model. The family of spherically symmetric equilibria captures all regular spherically symmetric equilibria of the approximate system. However within the full model, this family is embedded in a larger family of spherically symmetric solutions. For the approximate system, we prove that all equilibrium bubbles are spherically symmetric, by an application of Alexandrov's theorem on closed surfaces of constant mean curvature.

math.AP↗

arXiv2023

Direct Observation of Landau Levels in Silicon Photonic Crystals

We experimentally observe photonic Landau levels that arise due to a strain-induced pseudomagnetic field in a silicon photonic crystal slab. The Landau levels are dispersive (i.e., they are not flat bands) due to the distortion of the unit cell by the strain. We employ an additional strain which induces a pseudoelectric potential to flatten them.

physics.optics↗

arXiv2023

Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid

We study the radial relaxation dynamics toward equilibrium and time-periodic pulsating spherically symmetric gas bubbles in an incompressible liquid due to thermal effects. The asymptotic model ([A. Prosperetti, J. Fluid Mech., 1991] and [Z. Biro and J. J. L. Velazquez, SIAM J. Math. Anal., 2000]) is one where the pressure within the gas bubble is spatially uniform and satisfies an ideal gas law, relating the pressure, density and temperature of the gas. The temperature of the surrounding liquid is taken to be constant and the behavior of the liquid pressure at infinity is prescribed to be constant or periodic in time. In arXiv:2207.04079, for the case where the liquid pressure at infinity is a positive constant, we proved the existence of a one-parameter manifold of spherical equilibria, parameterized by the bubble mass, and further proved that it is a nonlinearly and exponentially asymptotically stable center manifold. In the present article, we first refine the exponential time-decay estimates, via a study of the linearized dynamics subject to the constraint of fixed mass. We obtain, in particular, estimates for the exponential decay rate constant, which highlight the interplay between the effects of thermal diffusivity and the liquid viscosity. We then study the nonlinear radial dynamics of the bubble-fluid system subject to a pressure field at infinity which is a small-amplitude and time-periodic perturbation about a positive constant. We prove that nonlinearly and exponentially asymptotically stable time-periodically pulsating solutions of the nonlinear (asymptotic) model exist for all sufficiently small forcing amplitudes. The existence of such states is formulated as a fixed point problem for the Poincaré return map, and the existence of a fixed point makes use of our (constant mass constrained) exponential time-decay estimates of the linearized problem.

math.AP↗

arXiv2023

Radiative decay of edge states in Floquet media

We consider the effect of time-periodic forcing on a one-dimensional Schr{ö}dinger equation with a topologically protected defect (edge) mode. The unforced system models a domain-wall or dislocation defect in a periodic structure, and it supports a defect mode which bifurcates from the Dirac point (linear band crossing) of the underlying bulk medium. We study the robustness of this state against time-periodic forcing of the type that arises in the study of Floquet Topological Insulators in condensed matter, photonics, and cold-atoms systems. Our numerical simulations demonstrate that under time-periodic forcing of sufficiently high frequency, the defect state undergoes radiative leakage of its energy away from the interface into the bulk; the time-decay is exponential on a time-scale proportional to the inverse square of the forcing amplitude. The envelope dynamics of our Floquet system are approximately governed, on long time scales, by an effective (homogenized) periodically-forced Dirac equation. Multiple scale analysis of the effective envelope dynamics yields an expansion of the radiating solution, which shows excellent agreement with our numerical simulations.

math.AP↗

arXiv2022

Is the continuum SSH model topological?

The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on the values $0$ and $1$ labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is therefore an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation to lattices which depend on the tight-binding asymptotic parameter.

math-ph↗

arXiv2022

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along their boundary. This property is quantified by topological indices associated with either the bulk or the edge system. In the field of condensed matter physics, tight binding (discrete) approximate models, parametrized by hopping coefficients, have been used successfully to capture the topological behavior of TIs in many settings. However, whether such tight binding models capture the same topological features as the underlying continuum models of quantum physics has been an open question. We resolve this question in the context of the archetypal example of topological behavior in materials, the integer quantum Hall effect. We study a class of continuum Hamiltonians, $H^λ$, which govern electron motion in a two-dimensional crystal under the influence of a perpendicular magnetic field. No assumption is made on translation invariance of the crystal. We prove, in the regime where both the magnetic field strength and depth of the crystal potential are sufficiently large, $λ\gg1$, that the low-lying energy spectrum and eigenstates (and corresponding large time dynamics) of $H^λ$ are well-described by a scale-free discrete Hamiltonian, $H^{\rm TB}$; we show norm resolvent convergence. The relevant topological index is the Hall conductivity, which is expressible as a Fredholm index. We prove that for large $λ$ the topological indices of $H^λ$ and $H^{\rm TB}$ agree. This is proved separately for bulk and edge geometries. Our results justify the principle of using discrete models in the study of topological matter.

math-ph↗

arXiv2022

Lower Bound on Quantum Tunneling for Strong Magnetic Fields

We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the eigenvalue splitting--is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping probability between the wells, a key parameter in tight binding models of solid state physics.

math-ph↗

arXiv2021

High contrast elliptic operators in honeycomb structures

We study the band structure of self-adjoint elliptic operators $\mathbb{A}_g= -\nabla \cdot σ_{g} \nabla$, where $σ_g$ has the symmetries of a honeycomb tiling of $\mathbb{R}^2$. We focus on the case where $σ_{g}$ is a real-valued scalar: $σ_{g}=1$ within identical, disjoint "inclusions", centered at vertices of a honeycomb lattice, and $σ_{g}=g \gg1 $ (high contrast) in the complement of the inclusion set (bulk). Such operators govern, e.g. transverse electric (TE) modes in photonic crystal media consisting of high dielectric constant inclusions (semi-conductor pillars) within a homogeneous lower contrast bulk (air), a configuration used in many physical studies. Our approach, which is based on monotonicity properties of the associated energy form, extends to a class of high contrast elliptic operators that model heterogeneous and anisotropic honeycomb media. Our results concern the global behavior of dispersion surfaces, and the existence of conical crossings (Dirac points) occurring in the lowest two energy bands as well as in bands arbitrarily high in the spectrum. Dirac points are the source of important phenomena in fundamental and applied physics, e.g. graphene and its artificial analogues, and topological insulators. The key hypotheses are the non-vanishing of the Dirac (Fermi) velocity $v_D(g)$, verified numerically, and a spectral isolation condition, verified analytically in many configurations. Asymptotic expansions, to any order in $g^{-1}$, of Dirac point eigenpairs and $v_D(g)$ are derived with error bounds. Our study illuminates differences between the high contrast behavior of $\mathbb{A}_g$ and the corresponding strong binding regime for Schroedinger operators.

math-ph↗

arXiv2021

Effective gaps in continuous Floquet Hamiltonians

We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$iψ_t (t,x) = H^\varepsilon(t)ψ=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)ψ,\quad H^0=-Δ+V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) by an effective time-periodic Dirac equation with a gap in its quasi-energy spectrum. This quasi-energy gap is typical of many reduced models of time-periodic (Floquet) materials and plays a role in conclusions drawn about the full system: conduction vs. insulation, topological vs. non-topological bands. Much is unknown about nature of the quasi-energy spectrum of original time-periodic Schroedinger equation, and it is believed that no such quasi-energy gap occurs. In this paper, we explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schroedinger dynamics. We introduce the notion of an {\it effective quasi-energy gap}, and establish its existence in the Schroedinger model. In the current setting, an effective quasi-energy gap is an interval of quasi-energies which does not support modes with large spectral projection onto band-limited Dirac wave-packets. The notion of effective quasi-energy gap is a physically relevant relaxation of the strict notion of quasi-energy spectral gap; if a system is tuned to drive or measure at momenta and energies near the Dirac point of $H^0$, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.

math.AP↗

arXiv2020

Spectral band degeneracies of $\fracπ{2}-$rotationally invariant periodic Schrödinger operators

The dynamics of waves in periodic media is determined by the band structure of the underlying periodic Hamiltonian. Symmetries of the Hamiltonian can give rise to novel properties of the band structure. Here we consider a class of periodic Schrödinger operators, $H_V=-Δ+V$, where $V$ is periodic with respect to the lattice of translates $Λ=\mathbb{Z}^2$. The potential is also assumed to be real-valued, sufficiently regular and such that, with respect to some origin of coordinates, inversion symmetric (even) and invariant under $π/2$ rotation. The present results are the $\mathbb{Z}^2-$ analogue of results obtained for conical degenerate points (Dirac points) in honeycomb structures. Our proofs make use of the framework developed by Fefferman-Weinstein and Fefferman-Lee-Thorp-Weinstein.

math.AP↗

arXiv2019

Dirac operators and domain walls

We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying ``mass'' term. It is well-known that when the mass function has the form of a kink, or \emph{domain wall}, transitioning between strictly positive and strictly negative asymptotic mass, $\pmκ_\infty$, at $\pm\infty$, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding \emph{zero mode}, an exponentially localized eigenfunction. We prove that when the mass function has the form of \emph{two} domain walls separated by a sufficiently large distance $2 δ$, the Dirac operator has two real simple eigenvalues of opposite sign and of order $e^{- 2 |κ_\infty| δ}$. The associated eigenfunctions are, up to $L^2$ error of order $e^{- 2 |κ_\infty| δ}$, linear combinations of shifted copies of the single domain wall zero mode. For the case of three domain walls, there are two non-zero simple eigenvalues as above and a simple eigenvalue at energy zero. Our methods are based on a Lyapunov-Schmidt reduction strategy and we outline their natural extension to the case of $n$ domain walls for which the minimal distance between domain walls is sufficiently large. The class of Dirac operators we consider controls the bifurcation of topologically protected ``edge states'' from Dirac points (linear band crossings) for classes of Schrödinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.

math.AP↗

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 $δ\ll 1$. 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 $δ$. 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.

math.AP↗

arXiv2018

Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schroedinger's equation in one spatial dimension with a potential which is the sum of a periodic function $V$ and a smooth function $W$. We assume that the period of $V$ is much shorter than the scale of variation of $W$ and denote the ratio of these scales by $ε$. We consider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in the limit $ε\downarrow 0$) solutions which are spectrally localized near to a $\textit{crossing}$ of two Bloch band dispersion functions of the periodic operator $- \frac{1}{2} \partial_z^2 + V(z)$. We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is `excited' which has $\textit{opposite}$ group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a `Landau-Zener'-type model.

math-ph↗

arXiv2017

Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls-Nabarro Barrier

We study a class of discrete focusing nonlinear Schr{ö}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems. Our class of nonlocal interactions is defined in terms of coupling coefficients, $J_m$, where $m\in\mathbb{Z}$ is the lattice site index, with $J_m\simeq m^{-1-2s}, s\in[1,\infty)$ and $J_m\sim e^{-γ|m|},\ s=\infty,\ γ>0,$ (Kac-Baker). For $s\ge1$, the bifurcation is seeded by solutions of the (effective / homogenized) cubic focusing nonlinear Schr{ö}dinger equation (NLS). However, for $1/4<s<1$, the bifurcation is controlled by the fractional nonlinear Schr{ö}dinger equation, FNLS, with $(-Δ)^s$ replacing $-Δ$. The proof is based on a Lyapunov-Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann \& Silvestre.

nlin.PS↗

arXiv2016

Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (re-scaled) Planck's constant and the ratio of the lattice spacing to the scale of variation of the external potential. We consider the special case where both parameters are equal and denote this parameter $ε$. In the limit $ε\downarrow 0$, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to `Ehrenfest time' $t \sim \ln 1/ε$. To leading order, the center of mass and average quasi-momentum of these solutions evolve along trajectories generated by the classical Hamiltonian given by the sum of the Bloch band energy and the external potential. We then derive all corrections to the evolution of these observables proportional to $ε$. The corrections depend on the gauge-invariant Berry curvature of the Bloch band, and a coupling to the evolution of the wave-packet envelope which satisfies Schrödinger's equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled `particle-field' system may be derived from an `extended' $ε$-dependent Hamiltonian. It is known that such coupling of observables (discrete particle-like degrees of freedom) to the wave-envelope (continuum field-like degrees of freedom) can have a significant impact on the overall dynamics.

math-ph↗

arXiv2016

Edge states in honeycomb structures

An edge state is a time-harmonic solution of a conservative wave system, e.g. Schroedinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge". Topologically protected edge states are edge states which are stable against spatially localized (even strong) deformations of the edge. First studied in the context of the quantum Hall effect, protected edge states have attracted huge interest due to their role in the field of topological insulators. Theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. In this paper we consider a rich family of continuum PDE models for which we rigorously study regimes where topologically protected edge states exist. Our model is a class of Schroedinger operators on $\mathbb{R}^2$ with a background 2D honeycomb potential perturbed by an "edge-potential". The edge potential is a domain-wall interpolation, transverse to a prescribed "rational" edge, between two distinct periodic structures. General conditions are given for the bifurcation of a branch of topologically protected edge states from Dirac points of the background honeycomb structure. The bifurcation is seeded by the zero mode of a 1D effective Dirac operator. A key condition is a spectral no-fold condition for the prescribed edge. We then use this result to prove the existence of topologically protected edge states along zigzag edges of certain honeycomb structures. Our results are consistent with the physics literature and appear to be the first rigorous results on the existence of topologically protected edge states for continuum 2D PDE systems describing waves in a non-trivial periodic medium. We also show that the family of Hamiltonians we study contains cases where zigzag edge states exist, but which are not topologically protected.

math-ph↗