Research

2025

1. arXiv

   Subwavelength Localisation in Disordered Systems

   Habib Ammari, Silvio Barandun, and Alexander Uhlmann

   May 2025

   Abs DOI

   We elucidate the different mechanisms of wave localisation in disordered finite systems of subwavelength resonators, where the disorder is in the spatial arrangement of the resonators. To do so, we employ the capacitance matrix formalism and develop a variety of tools to understand localisation in this setting. Namely, we adapt the Thouless criterion of localisation to quantify the various mechanisms of localisation. We also employ a propagation matrix approach to prove a Saxon-Hutner type theorem ensuring the existence of band gaps for disordered systems and to predict the corresponding Lyapunov exponents. We further investigate the stabilities of bandgaps and localised eigenmodes with respect to fluctuations in the system by introducing a method of defect mode prediction based on the discrete Green function and illuminating the phenomena of Anderson localisation and level repulsion under global disorder.

2. arXiv

   Universal Estimates for the Density of States for Aperiodic Block Subwavelength Resonator Systems

   Habib Ammari, Silvio Barandun, Bryn Davies, Erik Orvehed Hiltunen, and Alexander Uhlmann

   May 2025

   Abs DOI

   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.

3. RSPA

   Generalized Brillouin Zone for Non-Reciprocal Systems

   Habib Ammari, Silvio Barandun, Ping Liu, and Alexander Uhlmann

   Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Mar 2025

   Abs DOI

   Non-reciprocal physical systems, such as waveguides with imaginary gauge potentials or mass-spring chains with active elements, present new mathematical challenges beyond those of ordinary Hermitian structures. In a phenomenon known as the non-Hermitian skin effect, eigenmodes condensate at one edge of the structure and decay exponentially in space. Traditional Floquet–Bloch theory, relying on real-valued quasi-periodicities, fails to capture this decay and thus gives incomplete predictions of spectral behaviour. In this paper, we develop a rigorous theory to address this shortcoming. By extending the Brillouin zone into the complex plane, we introduce the notion of generalized Brillouin zone. We show that this complex formulation reflects the unidirectional decay inherent in non-reciprocal systems and provides an accurate framework for spectral analysis. Our main results are proven in the general context of k-Toeplitz matrices and operators, which model a wide variety of both finite and semi-infinite or infinite non-Hermitian periodic structures. We demonstrate that our generalized Floquet–Bloch formalism correctly identifies spectra and restores spectral convergences for large finite lattices.

2024

1. arXiv

   Truncated Floquet-Bloch Transform for Computing the Spectral Properties of Large Finite Systems of Resonators

   Habib Ammari, Silvio Barandun, and Alexander Uhlmann

   Oct 2024

   Abs DOI

   The truncated Floquet-Bloch transform can be used to characterise the spectral properties of finite periodic and aperiodic large systems of resonators. This paper aims to provide for the first time the mathematical foundations of this transform.

2. arXiv

   Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials

   Habib Ammari, Silvio Barandun, Ping Liu, and Alexander Uhlmann

   May 2024

   Abs DOI

   The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity - an imaginary gauge potential and on-site gain and loss - we prove that all but two eigenmodes of the system decouple when going through an exceptional point. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and non-reciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics where the tight binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.