arXiv subjects

Bryn Davies

Publications and source records attributed to Bryn Davies.

At least 19 recordsLinked to original sources

arXiv2026

Target localization, identification and sensing using latent symmetries

We show that an array of scatterers which has been designed to have latent ("hidden") symmetries can be used as a sensor. We use the capacitance matrix as a canonical model for three-dimensional hybridisation and study how the introduction of an "intruder'' scatterer breaks the latent symmetries. By analysing the degree to which each symmetry is broken, we identify the radius of the intruder and localize its position. This can be achieved using a dictionary-based approach, however Bayesian inference or an artificial neural network (multi-layer perceptron) perform better in the presence of measurement noise. To our knowledge, this is the first time latent symmetries have been exploited successfully for sensing problems. It is also the first time latent symmetries have been observed in a three-dimensional open system that cannot be approximated by a sparse graph.

cs.LG↗

arXiv2026

Non-singular hotspots between closely spaced high-index nanoparticles

We study the concentration of the field between two nearly touching high-index dielectric resonators in three dimensions. The model is a scalar Helmholtz transmission problem in the resonant regime, wherein the wavelength inside the resonators of the same order as their typical diameter. The material contrast enters only a lower-order term of the equation and not its principal part. As a consequence, the gradient of the field stays bounded independently of the distance separating the particles, and does not blow up. Nevertheless, simulations and experiments show that the gradient still concentrates in the gap: as the particles approach, it grows like the inverse of their separation over an intermediate range of distances and then saturates once they are very close. We explain this pre-asymptotic effect through a weak-coupling regime, in which the resonant modes of the pair are perturbations of the modes of each isolated particle. When such a mode keeps a nonzero contrast between the two facing boundaries, a mean value argument across the gap accounts for the growth of the gradient; this requires strengthening the standard spectral perturbation estimates from an average to a pointwise control. The growth stops once the interaction between the particles is no longer weak and the perturbation theory fails. Numerical experiments confirm the transition from amplification to saturation.

math.AP↗

arXiv2026

High-frequency spectral asymptotics and homogenization for quasiperiodic operators

We study the spectral asymptotics of elliptic operators with quasiperiodic coefficients by exploiting projections from higher-dimensional periodic functions. Using the framework of two-scale convergence adapted to cut-and-project quasiperiodic structures we establish that, in the low-frequency (homogenization) regime, the spectrum converges to that of a homogenized operator with effective coefficients determined by a cell problem on the higher-dimensional torus. In the high-frequency regime, we introduce a rescaling approach that transforms the problem to an expanding domain with asymptotically frozen coefficients. In the critical scaling, the rescaled spectrum converges to the union of the Bloch spectra arising from the quasiperiodic bulk and a boundary layer spectrum consisting of eigenfunctions concentrated near the boundary of the macroscopic domain. This boundary spectrum can be characterised as a subset of the spectrum of a family of half-space operators with frozen macroscopic coefficients. For any non-critical scaling, the rescaled spectrum fills the positive real line.

math.AP↗

arXiv2026

Van Hove singularities in the density of states of a chaotic dynamical system

We show that the statistics of chaotic systems can be predicted by constructing an associated sequence of periodic differential operators and computing their densities of states. For such operators, the density of states is well understood and can be computed straightforwardly, often yielding explicit formulas. As a case study, we investigate a nonlinear recursion relation that maps naturally onto a family of periodic operators generated by a Fibonacci tiling rule. This correspondence enables us to derive an explicit formula for the limiting statistics of the chaotic system and to demonstrate that the clustering near to critical values is equivalent to the van Hove singularities in the operators' densities of states.

nlin.CD↗

arXiv2025

Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices

Using a generalised Floquet-Bloch theory, we present a mathematical method to construct eigenvectors for non-Hermitian Toeplitz operators. We extend the method to both banded Toeplitz operators and those with algebraically decaying, fully dense off-diagonal structure. We present sharp decay estimates for the amplitude of bulk eigenmodes as well as eigenmodes associated with defect eigenfrequencies inside the spectral band gap. The validity of those results is illustrated numerically and we show that banded approximations give poor reconstructions of the dense operators, due to the slow algebraic decay. We apply the insights gained to model the non-Hermitian skin effect in a three-dimensional system of subwavelength resonators, where the corresponding operator exhibits only algebraic decay of off-diagonal entries. We use our approach to demonstrate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localisation in the bulk.

math.AP↗

arXiv2025

Energy estimates for wave amplification in quasiperiodic Fibonacci time-modulated media

Fibonacci time quasicrystals can be approximated by temporal supercells to reveal a fractal collection of $k$ gaps, in which wave energy is amplified exponentially. These estimates are validated by the observation of "super" $k$ gaps that are independent of the duration of the temporal supercell. This approach predicts the regions of parametric amplification and provides accurate estimates of the energy growth rate.

physics.optics↗

arXiv2025

Spectral theoretic characterisation of Markov chain convergence

In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure of the Markov chain. This approach also leads to a direct proof of convergence to the invariant measure, along with explicit convergence rates. We show how our method can be applied to a class of related Markov chains including the logistic map, the tent map and Chebyshev maps of arbitrary order.

math.DS↗

arXiv2025

Universal estimates for the density of states for aperiodic block subwavelength resonator systems

We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks $M\to \infty$, the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

math-ph↗

arXiv2025

Hofstadter butterflies in phononic structures: commensurate spectra, wave localization and metal-insulator transitions

We present a new simple and easy-to-implement one-dimensional phononic system whose spectrum exactly corresponds to the Hofstadter butterfly when a parameter is modulated. The system consists of masses that are coupled by linear springs and are mounted on flexural beams whose cross section (and, hence, stiffness) is modulated. We show that this system is the simplest version possible to achieve the Hofstadter butterfly exactly; in particular, the local resonances due to the beams are an essential component for this. We examine the various approaches to producing spectral butterflies, including Bloch spectra for rational parameter choices, resonances of finite-sized systems and transmission coefficients of sections of finite length. For finite-size systems, we study the localisation of the modes by calculating the inverse participation ratio, and detect a phase transition characterised by a critical value of the stiffness modulation amplitude, where the state of the system changes from mainly extended to localised, corresponding to a metal-insulator phase transition. The obtained results offer a practical strategy to realize experimentally a system with similar dynamical properties. The transmission coefficient for sections of finite length is benchmarked through the comparison with Bloch spectra of the same finite-sized systems. The numerical results for the transmission spectra confirms the evidence of a phase transition in the dynamical state of the system. Our approach opens significant new perspectives in order to design mechanical systems able to support phase transitions in their vibrational properties.

physics.app-ph↗

arXiv2025

Edge modes in modulated metamaterials based on the three-gap theorem

We present a new paradigm for generating complex structured materials based on the three-gap theorem that unifies and generalises several key concepts in the study of localised edge states. Our model has both the discretised coupling strengths of the SSH model and a modulation parameter that can be used to characterise the spectral flow of edge modes and produce images reminiscent of the Hofstadter butterfly. By defining a localisation factor associated to each eigenmode, we are able to establish conditions for the existence of localised edge states in finite systems. This allows us to compare their eigenfrequencies with the spectra of the corresponding infinitely periodic problem and characterise the rich pattern of localised edge modes appearing and disappearing (in the sense of becoming delocalised) as the parameters of our three-gap algorithm are varied.

physics.app-ph↗

arXiv2024

Convergence of supercell and superspace methods for computing spectra of quasiperiodic operators

We study the convergence of two of the most widely used and intuitive approaches for computing the spectra of differential operators with quasiperiodic coefficients: the supercell method and the superspace method. In both cases, Floquet-Bloch theory for periodic operators can be used to compute approximations to the spectrum. We illustrate our results with examples of Schrödinger and Helmholtz operators.

math.SP↗

arXiv2024

Topological interface modes in systems with damping

We extend the theory of topological localised interface modes to systems with damping. The spectral problem is formulated as a root-finding problem for the interface impedance function and Rouché's theorem is used to track the zeros when damping is introduced. We show that the localised eigenfrequencies, corresponding to interface modes, remain for non-zero dampings. Using the transfer matrix method, we explicitly characterise the decay rate of the interface mode.

math.AP↗

arXiv2024

Mixing Skyrmions and Merons in Topological Quasicrystals of Evanescent Optical Field

Photonic skyrmion and meron lattices are structured light fields with topologically protected textures, analogous to magnetic skyrmions and merons. Here, we report the theoretical existence of mixed skyrmion and meron quasicrystals in an evanescent optical field. Topological quasiperiodic tilings of even and odd point group symmetries are demonstrated in both the electric field and spin angular momentum. These quasicrystals contain both skyrmions and merons of Néel-type topology. Interestingly, the quasiperiodic tilings are in agreement with the observations of quasiperiodic arrangements of carbon nanoparticles in water driven by ultrasound, and pave the way towards engineering hybrid topological states of light which may have potential applications in optical manipulation, metrology and information processing.

physics.optics↗

arXiv2024

Coupled harmonics due to time-modulated point scatterers

We consider the resonance and scattering properties of a composite medium containing scatterers whose properties are modulated in time. When excited with an incident wave of a single frequency, the scattered field consists of a family of coupled harmonics at frequencies differing by the frequency of temporal modulation. Similarly, the temporal modulation induces coupling between the resonance frequencies, leading to exceptional points at certain modulation amplitudes. Moreover, the lack of energy conservation causes scattering coefficients to blow up when (complex) resonances cross the real axis. We have developed an integral operator approach to characterize the scattering problem and, for high-contrast scatterers, we present small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case. We conclude the paper with a boundary integral formulation of the time-modulated problem, which gives an efficient numerical approach and corroborates the asymptotic formulas.

cond-mat.mes-hall↗

arXiv2024

A two-scale effective model for defect-induced localization transitions in non-Hermitian systems

We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with non-reciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and derive a simple and intuitive effective model for the transition that occurs when the defect is sufficiently large that one of the modes is localized at the defect site, rather than at the edge of the system.

cond-mat.mes-hall↗

arXiv2024

Exponentially localised interface eigenmodes in finite chains of resonators

This paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localisation of their associated eigenmode. Then we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localised interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterise quantitatively the localised interface modes in systems of finitely many resonators.

math-ph↗

arXiv2024

Hierarchical Band Gaps in Complex Periodic Systems

Complex periodic structures inherit spectral properties from the constituent parts of their unit cells, chiefly their spectral band gaps. Exploiting this intuitive principle, which is made precise in this work, means spectral features of periodic systems with very large unit cells can be predicted without numerical simulation. We study a class of difference equations with periodic coefficients and show that they inherit spectral gaps from their constituent elements. This result shows that if a frequency falls in a band gap for every constituent element then it must be in a band gap for the combined complex periodic structure. This theory and its instantaneous utility is demonstrated in a series of vibro-acoustic and mechanical examples.

math.CA↗

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↗