Bloch Eigenvalue Research Articles

Fine Structure of Flat Bands in a Chiral Model of Magic Angles

AbstractWe analyse symmetries of Bloch eigenfunctions at magic angles for the Tarnopolsky–Kruchkov–Vishwanath chiral model of the twisted bilayer graphene (TBG) following the framework introduced by Becker–Embree–Wittsten–Zworski. We show that vanishing of the first Bloch eigenvalue away from the Dirac points implies its vanishing at all momenta, that is, the existence of a flat band. We also show how the multiplicity of the flat band is related to the nodal set of the Bloch eigenfunctions. We conclude with two numerical observations about the structure of flat bands.

On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients

On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients

Experimental and numerical investigation on effects of viscoelastic rubber and local resonator mass on broadband vibration attenuation in acoustic-metamaterial plates

Vibration attenuation in acoustic-metamaterial plates assembled from a periodic arrangement of unit cells with a cavity containing local resonator system is presented. Each cell incorporates a base aluminum plate with vertical rectangle-shaped cavities containing a viscoelastic membrane supporting a mass forming a local resonator system. These acoustic-metamaterial structures exhibit stop-band behavior due to Bragg scattering and local resonance. Floquet–Bloch approach and eigenvalue analysis is used to identify the stop bands for metamaterial unit cells. The dispersion analysis predictions are validated experimentally by studying the vibration responses of different acoustic-metamaterial plates excited by an electrodynamic shaker over a frequency range of 8−4000 Hz. The attenuation regions observed in the finite element simulation results have been compared to that of the experiments. The obtained results show the potential of FE simulation to predict the metamaterial plate attenuation with reasonably good accuracy. The viscoelastic material properties also affect the attenuation region, as observed while comparing experimental results for different viscoelastic materials. These results show the effectiveness of acoustic-metamaterial plates to provide broadband vibration attenuation.

Design of metasurfaces in 3D for omnidirectional band gaps: The scalar wave case

We discuss a comprehensive process for designing the material properties of three-dimensional periodic structures that are tasked to exhibit user-defined omnidirectional band gaps. To this end, we build upon the framework first posited in Goh and Kallivokas (2019) for two-dimensional metamaterials, and aim at engineering the dispersive properties of a metamaterial’s 3D unit cell using an inverse medium approach. The inversion is driven by forcing the vanishing of trial group velocities at the target band gap frequencies, subject to the satisfaction of the Floquet–Bloch eigenvalue problem defined over the unit cell’s IBZ. Omnidirectionality is controlled by enforcing the vanishing of the group velocities along the IBZ’s high-symmetry lines. Using an adjoint method, the unit cell’s properties are iteratively updated until the design goal is attained. Numerical examples for acoustic metamaterials and metasurfaces demonstrate the methodology.

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

<p style='text-indent:20px;'>Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.</p>

On the Schrödinger operator with a periodic PT-symmetric matrix potential

In this article we obtain asymptotic formulas for the Bloch eigenvalues of the operator generated by a system of Schrodinger equations with periodic PT-symmetric complex-valued coefficients. Then using these formulas we classify the spectrum of this operator and find a condition on the coefficients for which the spectrum contains a half line.

Combined effects of homogenization and singular perturbations: A bloch wave approach

<p style='text-indent:20px;'>In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.</p>

High-order homogenization in optimal control by the Bloch wave method

This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient Aε. The small parameter ε > 0 denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error O(εM), where M ∈ ℕ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities, leading to our main theoretical result on the high-oder approximation.

Wave propagation in mass embedded and pre-stressed hexagonal lattices

This paper investigates the elastic wave propagation, mode veering, and in-plane vibration of pre-stressed hexagonal lattice embedded in an elastic medium and composed of axially loaded Timoshenko beams with attached point masses. The frequency band structure of the lattice system is obtained by solving the corresponding eigenvalue problem based on the Bloch theorem and the finite element method. The parametric study is performed by investigating the effects of the pre-stress magnitude, stiffness of elastic medium, and attached point masses on the band structure of a lattice unit cell. For simulating the free vibration behavior of the proposed lattices with different topologies, the Hurty-Craig-Bampton method is introduced to reduce the number of degrees of freedom. Based on the reduced finite element model, the natural frequencies are determined for various boundary conditions. The additional interface reduction technique, called system-level reduction, has been observed to achieve accurate results compared to that of the full model. Numerical experiments demonstrated a significant influence of the additional masses, pre-stress, and stiffness of elastic medium on Bloch waves and eigenvalues of the proposed lattice systems. The effects of different parameters on the emergence of mode veering phenomenon and band gaps are investigated in detail.

Bloch wave homogenisation of quasiperiodic media

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

A High-Frequency Homogenization Approach Near the Dirac Points in Bubbly Honeycomb Crystals

In (Ammari et al. in SIAM J Math Anal. arXiv:1811.03905), the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape is shown. The aim of this paper is to prove that, near the Dirac points, the Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is highly oscillating. The slowly oscillating components of the eigenmodes satisfy a system of Dirac equations. Our results in this paper prove for the first time a near-zero effective refractive index near the Dirac points for the plane-wave envelopes of the Bloch eigenfunctions in a sub-wavelength metamaterial. They are illustrated by a variety of numerical examples. We also compare and contrast the behaviour of the Bloch eigenfunctions in the honeycomb crystal with that of their counterparts in a bubbly square crystal, near the corner of the Brillouin zone, where the maximum of the first Bloch eigenvalue is attained.

Band structure analysis for 2D acoustic phononic structure using isogeometric boundary element method

In this paper, a computational methodology for the band structure analysis of 2D acoustic phononic structures based on the isogeometric boundary element method(IGABEM) is studied. IGABEM not only improves the accuracy of numerical model, but also retains the geometric accuracy and makes the pre-processing procedure relatively simpler for models designed through using computer-aided design(CAD) softwares. B-splines, which are widely employed in computer graphics and CAD industry, are used as basis functions for 2D phononic unit cells. The Bloch periodic boundary condition is directly specified to the control points on the virtual boundary of a unit cell. The Bloch eigenvalue problems derived with IGABEM show strong nonlinear features resulted from the adoption of fundamental solutions. To overcome the nonlinearity, a contour integral projection method with Gerschgorin disk theory-based eigenspace identification is introduced to the extraction of the Bloch eigenvalues for the description of dispersion curves. Numerical examples with models from different CAD softwares are investigated to demonstrate the accuracy and effectiveness of the proposed method.

Inverse band gap design of elastic metamaterials for P and SV wave control

We describe a systematic approach for engineering the dispersive properties of elastic metamaterials in order to band-gap, at user-defined frequency ranges, the propagation of elastic waves, accounting simultaneously for both P and SV waves. To this end, we focus on the inverse design of a metamaterial’s unit cell, and cast the design problem as an optimization problem, driven by the desired band gap, and constrained only by the cell’s dispersive characteristics; the latter are expressed in terms of the Floquet–Bloch eigenvalue problem. Numerical results in the time-domain, using metamaterial assemblies with only narrow periodicity, demonstrate that the engineered metamaterial attains the desired behavior. Extensions to wave steering, shielding, and other wave-control applications, follow naturally.

Spectral analysis of the Schrödinger operator with a PT-symmetric periodic optical potential

In this paper, we give a description of the spectral analysis of the Schrödinger operator L(q) with the potential q(x) = 4 cos2 x + 4iV sin 2x for all V > 1/2. First, we consider the Bloch eigenvalues and spectrum of L(q). Then, we investigate spectral singularities and essential spectral singularities (ESS). We prove that there exists a sequence V2 < V3 < ⋯ such that the operator L(q) has no ESS and has ESS, respectively, if and only if V ≠ Vk and V = Vk for k ≥ 2, where Vk → ∞ as k → ∞, V2 is the second critical point. Using ESS, we classify the spectral expansion in term of the points Vk for k ≥ 2. Finally, we discuss the critical points Vk, formulate some conjectures, and describe the changes in the spectrum of L(q) when V changes from 1/2 to ∞.

Generic simplicity of spectral edges and applications to homogenization

We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.

A reduced Bloch operator finite element method for fast calculation of elastic complex band structures

This article presents an efficient reduced formulation of the Bloch Operator Finite Element method to calculate complex band structures of periodic waveguides. The use of a Bloch operator formulation allows building and solving a Bloch eigenvalue problem along a generic wave direction, thus being not limited to the unit cell Irreducible Brillouin Zone (IBZ) edges, so that band gap directionality and material absorption in elastic and damped waveguides can be fully disclosed. The proposed Reduced-Order Modeling (ROM) exploits a small set of Bloch modes, extracted at relevant frequency locations along one or more wave directions and post-processed with a Singular Value Decomposition, to reduce the dimensions of the eigenvalue problem. The performances of the proposed numerical technique are evaluated in terms of accuracy and computational saving by analyzing a linear elastic and a damped bi-periodic stubbed plate. Results demonstrate that the reduced formulation yields accurate predictions of propagative, evanescent and complex wave solutions with a reduction in computational time of more than one order of magnitude with respect to the full model calculations. Complex band structures can thus be efficiently computed over the whole IBZ.

On a class of nonself-adjoint multidimensional periodic Schrödinger operators

We investigate the Schrödinger operator $L(q)$ in $L_{2}\left( \mathbb{R}^{d}\right) \ (d\geq1)$ with the complex-valued potential $q$ that is periodic with respect to a lattice $\Omega.$ Besides, it is assumed that the Fourier coefficients $q_{\gamma}$ of $q$ with respect to the orthogonal system $\{e^{i\left\langle \gamma x\right\rangle }:\gamma\in\Gamma\}$ vanish if $\gamma$ belongs to a half-space, where $\Gamma$ is the lattice dual to $\Omega.$ We prove that the Bloch eigenvalues are $\mid\gamma+t\mid^{2}$ for $\gamma\in\Gamma,$ where $t$ is a quasimomentum and find explicit formulas for \ the Bloch functions. Moreover, we investigate the multiplicity of the Bloch eigenvalue and consider necessary and sufficient conditions on the potential which provide some root functions to be eigenfunctions. Besides, in case $d=1$ we investigate in detail the root functions of the periodic and antiperiodic boundary value problems.

Peierls' substitution for low lying spectral energy windows

We consider a 2d periodic magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a “Landau type” structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolatedwhile its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic [Formula: see text]-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by [Formula: see text] exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in [Formula: see text]. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on [Formula: see text] and is [Formula: see text]-periodic; if [Formula: see text], the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

Bloch Waves in Bubbly Crystal Near the First Band Gap: A High-Frequency Homogenization Approach

This paper is concerned with the high-frequency homogenization of bubbly phononic crystals. It is a follow-up of the works [H. Ammari et al., Sub-wavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263 (2017), 5610--5629] which shows the existence of a sub-wavelength band gap. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. In this paper, we prove that the first Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone. Moreover, by computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. They rigorously justify, in the nondilute case, the observed super-focusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue and confirm the band gap opening near and above this critical frequency.