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Bryn Davies

Publications and source records attributed to Bryn Davies.

At least 37 records · Page 2Linked to original sources
arXiv2023

The Non-Hermitian Skin Effect With Three-Dimensional Long-Range Coupling

We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix $\mathcal{C}_N^γ$, which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range-k approximation to keep the long-range interaction to a certain extent, thus obtaining a k-banded gauge capacitance matrix $\mathcal{C}_{N,k}^γ$ . Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of $\mathcal{C}_{N,k}^γ$ and demonstrate that they approximate those of the gauge capacitance matrix $\mathcal{C}_N^γ$ well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes. Moreover, we also illustrate numerically the behaviour of the eigenmodes and the stability of the non-Hermitian skin effect with respect to disorder in a variety of three-dimensional structures.

math.AP↗
arXiv2023

Topologically protected modes in dispersive materials: the case of undamped systems

This work extends the theory of topological protection to dispersive systems. This theory has emerged from the field of topological insulators and has been established for continuum models in both classical and quantum settings. It predicts the existence of localised interface modes based on associated topological indices and shows that, when such modes exist, they benefit from enhanced robustness with respect to imperfections. This makes topologically protected modes an ideal starting point for building wave guiding devices. However, in many practical applications such as optics or locally resonant meta-structures, materials are dispersive in the operating frequency range. In this case, the associated spectral theory is less straightforward. This work shows that the existing theory of topological protection can be extended to dispersive settings. We consider time-harmonic waves in one-dimensional systems with no damping.

math.AP↗
arXiv2023

Landscape of wave focusing and localisation at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localisation at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterisation in terms of the generalised capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalised capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localisation in random and non-periodic systems of subwavelength resonators.

math.AP↗
arXiv2023

Functional analytic methods for discrete approximations of subwavelength resonator systems

We survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalised capacitance matrix, which appears by perturbing the elements of the kernel of the limiting operator. Using this formulation, we are able to describe subwavelength resonance and related phenomena. In particular, we demonstrate large-scale effective parameters with exotic values. We also show that these systems can exhibit localised and guided waves on very small length scales. Using the concept of topologically protected edge modes, such localisation can be made robust against structural imperfections.

math.AP↗
arXiv2023

On the problem of comparing graded metamaterials

We use a simple effective model, obtained through the application of high-frequency homogenisation, to tackle the fundamental question of how the choice of gradient function affects the performance of a graded metamaterial. This approach provides a unified framework for comparing gradient functions efficiently and in a way that allows us to draw conclusions that apply to a range of different wave regimes. We consider the specific problem of single-frequency localisation, for which the appropriate effective model is a one-dimensional Schrodinger equation. Our analytic results both corroborate those of existing studies (which use either expensive full-field wave simulations or black-box numerical optimisation algorithms) and extend them to other metamaterial regimes. Based on our analysis, we are able to propose a design strategy for optimising monotonically graded metamaterials and offer an explanation for the lack of a universal optimal gradient function.

physics.class-ph↗
arXiv2023

Convergence rates for defect modes in large finite resonator arrays

We show that defect modes in infinite systems of resonators have corresponding modes in finite systems which converge as the size of the system increases. We study the generalized capacitance matrix as a model for three-dimensional coupled resonators with long-range interactions and consider defect modes that are induced by compact perturbations. If such a mode exists, then there are elements of the discrete spectrum of the corresponding truncated finite system that converge to each element of the pure point spectrum. The rate of convergence depends on the dimension of the lattice. When the dimension of the lattice is equal to that of the physical space, the convergence is exponential. Conversely, when the dimension of the lattice is less than that of the physical space, the convergence is only algebraic, because of long-range interactions arising due to coupling with the far field.

math-ph↗
arXiv2023

Stability of the non-Hermitian skin effect

This paper shows that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of an imaginary gauge potential, which leads to a skin effect that is manifested by the system's eigenmodes accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate the competition between the two different localisation effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localisation. We show that, as the strength of the disorder increases, more and more eigenmodes become localised in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalised also to tight-binding models in condensed matter theory.

math-ph↗
arXiv2023

Anderson localization in the subwavelength regime

In this paper, we use recent breakthroughs in the study of coupled subwavelength resonator systems to reveal new insight into the mechanisms responsible for the fundamental features of Anderson localization. The occurrence strong localization in random media has proved difficult to understand, particularly in physically derived multi-dimensional models and systems with long-range interactions. We show here that the scattering of time-harmonic waves by high-contrast resonators with randomly chosen material parameters reproduces the characteristic features of Anderson localization. In particular, we show that the hybridization of subwavelength resonant modes is responsible for both the repulsion of energy levels as well as the widely observed phase transition, at which point eigenmode symmetries swap and very strong localization is possible. We derive results from first principles, using asymptotic expansions in terms of the material contrast parameter and obtain a characterization of the localized modes in terms of generalized capacitance matrices. This model captures the long-range interactions of the wave-scattering system and provides a concise framework to explain the exotic phenomena that are observed.

cond-mat.dis-nn↗
arXiv2023

Mathematical foundations of the non-Hermitian skin effect

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

math.AP↗
arXiv2023

The effect of singularities and damping on the spectra of photonic crystals

Understanding the dispersive properties of photonic crystals is a fundamental and well-studied problem. However, the introduction of singular permittivities and damping complicates the otherwise straightforward theory. In this paper, we study photonic crystals with a Drude-Lorentz model for the permittivity, motivated by halide perovskites. We demonstrate how the introduction of singularities and damping affects the spectral band structure and show how to interpret the notion of a "band gap" in this setting. We present explicit solutions for a one-dimensional model and show how integral operators can be used to handle multi-dimensional systems.

math.AP↗
arXiv2023

Spectral convergence in large finite resonator arrays: the essential spectrum and band structure

We show that resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states for the finite system converge in distribution to the (continuous) density of states of the infinite system. We achieve this by proving a weak convergence of the finite capacitance matrix to corresponding (translationally invariant) Toeplitz matrix of the infinite structure. With this characterization at hand, we use the truncated Floquet transform to introduce a notion of spectral band structure for finite materials. This principle is also applicable to structures that are not translationally invariant and have interfaces. We demonstrate this by considering examples of perturbed systems with defect modes, such as an analogue of the well-known interface Su-Schrieffer-Heeger (SSH) model.

math-ph↗
arXiv2023

Graded quasiperiodic metamaterials perform fractal rainbow trapping

The rainbow trapping phenomenon of graded metamaterials can be combined with the fractal spectra of quasiperiodic waveguides to give a metamaterial that performs fractal rainbow trapping. This is achieved through a graded cut-and-project algorithm that yields a projected geometry for which the effective projection angle is graded along its length. As a result, the fractal structure of local band gaps varies with position, leading to broadband fractal rainbow trapping. We demonstrate this principle by designing an acoustic waveguide, which is characterised using theory, simulation and experiments.

physics.class-ph↗
arXiv2023

Asymptotic characterisation of localised defect modes: Su-Schrieffer-Heeger and related models

Motivated by topologically protected states in wave physics, we study localised eigenmodes in one-dimensional periodic media with defects. The Su-Schrieffer-Heeger model (the canonical example of a one-dimensional system with topologically protected localised defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterise the existence of localised eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenisation and transfer matrix analysis, with good agreement between the two methods.

math.AP↗
arXiv2023

Super band gaps and periodic approximants of generalised Fibonacci tilings

We present mathematical theory for understanding the transmission spectra of heterogeneous materials formed by generalised Fibonacci tilings. Our results, firstly, characterise super band gaps, which are spectral gaps that exist for any periodic approximant of the quasicrystalline material. This theory, secondly, establishes the veracity of these periodic approximants, in the sense that they faithfully reproduce the main spectral gaps. We characterise super band gaps in terms of a growth condition on the traces of the associated transfer matrices. Our theory includes a large family of generalised Fibonacci tilings, including both precious mean and metal mean patterns. We demonstrate our fundamental results through the analysis of three different one-dimensional wave phenomena: compressional waves in a discrete mass-spring system, axial waves in structured rods and flexural waves in multi-supported beams. In all three cases, the theory is shown to give accurate predictions of the super band gaps, with negligible computational cost and with significantly greater precision than previous estimates.

math.CA↗
arXiv2022

A mathematical design strategy for highly dispersive resonator systems

Designing devices composed of many small resonators is a challenging problem that can easily incur significant computational cost. Can asymptotic techniques be used to overcome this often limiting factor? Integral methods and asymptotic techniques have been used to derive concise characterisations for scattering by resonators, but can these be generalised to systems of many dispersive resonators whose material parameters have highly non-linear frequency dependence? In this paper, we study halide perovskite resonators as a demonstrative example. We extend previous work to show how a finite number of coupled resonators can be modelled concisely in the limit of small radius. We also show how these results can be used as the basis for an inverse design strategy, to design resonator systems that resonate at specific frequencies.

math.AP↗
arXiv2022

Asymptotic links between signal processing, acoustic metamaterials and biology

Biomimicry is a powerful science that takes advantage of nature's remarkable ability to devise innovative solutions to challenging problems. In this work, we use asymptotic methods to develop the mathematical foundations for the exchange of design inspiration and features between biological hearing systems, signal processing algorithms and acoustic metamaterials. Our starting point is a concise asymptotic analysis of high-contrast acoustic metamaterials. We are able to fine tune this graded structure to mimic the biomechanical properties of the cochlea, at the same scale. We then turn our attention to developing a biomimetic signal processing algorithm. We use the response of the cochlea-like metamaterial as an initial filtering layer and then add additional biomimetic processing stages, designed to mimic the human auditory system's ability to recognise the global properties of natural sounds. This demonstrates the three-way exchange of ideas that, thanks to our analysis, is possible between signal processing, metamaterials and biology.

math.AP↗
arXiv2022

Symmetry-induced quasicrystalline waveguides

Introducing an axis of reflectional symmetry in a quasicrystal leads to the creation of localised edge modes that can be used to build waveguides. We develop theory that characterises reflection-induced localised modes in materials that are formed by recursive tiling rules. This general theory treats a one-dimensional continuous differential model and describes a broad class of both quasicrystalline and periodic materials. We present an analysis of a material based on the Fibonacci sequence, which has previously been shown to have exotic, Cantor-like spectra with very wide spectral gaps. Our approach provides a way to create localised edge modes at frequencies within these spectral gaps, giving strong and stable wave localisation. We also use our general framework to make a comparison with reflection-induced modes in periodic materials. These comparisons show that while quasicrystalline waveguides enjoy enhanced robustness over periodic materials in certain settings, the benefits are less clear if the decay rates are matched. This shows the need to carefully consider equivalent structures when making robustness comparisons and to draw conclusions on a case-by-case basis, depending on the specific application.

math-ph↗
arXiv2022

Bioinspired random projections for robust, sparse classification

Inspired by the use of random projections in biological sensing systems, we present a new algorithm for processing data in classification problems. This is based on observations of the human brain and the fruit fly's olfactory system and involves randomly projecting data into a space of greatly increased dimension before applying a cap operation to truncate the smaller entries. This leads to a simple algorithm that is very computationally efficient and can be used to either give a sparse representation with minimal loss in classification accuracy or give improved robustness, in the sense that classification accuracy is improved when noise is added to the data. This is demonstrated with numerical experiments, which supplement theoretical results demonstrating that the resulting signal transform is continuous and invertible, in an appropriate sense.

stat.ML↗