Silvio Alberto Barandun

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Barandun

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Silvio Alberto

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0000-0003-1499-4352

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Publications 1 - 10 of 23

Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
Ammari, Habib; Barandun, Silvio Alberto; De Bruijn, Yannick; et al. (2024)
SAM Research Report
We establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in [H. Ammari et al., arXiv:2307.13551]. We also numerically verify our theory for these systems.
Universal estimates for the density of states for aperiodic block subwavelength resonator systems
Ammari, Habib; Barandun, Silvio Alberto; Davies, Bryn; et al. (2025)
Journal of Physics A: Mathematical and Theoretical
We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states (DoS). We prove that for random block configurations, as the number of blocks M → ∞, the integrated DoS converges to a non-random, continuous function. We show both analytically and numerically that the DoS 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 DoS in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.
Truncated Floquet–Bloch transform for computing the spectral properties of large finite systems of resonators
Ammari, Habib; Barandun, Silvio Alberto; Uhlmann, Alexander (2025)
Transactions of the London Mathematical Society
In subwavelength physics, a challenging problem is to characterise the spectral properties of finite systems of subwavelength resonators. In particular, it is important to identify localised modes as well as bandgaps and associated mobility edges. It was shown numerically that the truncated Floquet–Bloch transform can be used to characterise the spectral properties of finite periodic and some aperiodic large systems of resonators. In this paper, for the first time, the mathematical foundations of this transform are provided. In particular, it is proved that the truncated Floquet–Bloch transform enables an accurate recovery of the structure's bandgap and defect localised eigenmodes.
Subwavelength localization in disordered systems
Ammari, Habib; Barandun, Silvio Alberto; Uhlmann , Alexander (2025)
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We elucidate the different mechanisms of wave localization 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 localization in this setting. Namely, we adapt the Thouless criterion of localization to quantify the various mechanisms of localization. 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 localized 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 localization and level repulsion under global disorder.
Tunable localisation in parity-time-symmetric resonator arrays with imaginary gauge potentials
Ammari , Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2025)
Forum of Mathematics, Sigma
The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in the 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 pass through exceptional points and decouple. 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 nonreciprocal 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.
Stability of the Non-Hermitian Skin Effect in One Dimension
Ammari, Habib; Barandun, Silvio Alberto; Davies, Bryn; et al. (2024)
SIAM Journal on Applied Mathematics
This paper shows both analytically and numerically 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 a directional damping term (motivated by an imaginary gauge potential), which leads to a skin effect that is manifested by the system's eigenmo des accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate numerically the competition between the two different localization effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localization. We show numerically that, as the strength of the disorder increases, more and more eigenmo des become localized in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalized also to tight-binding models in condensed matter theory.
Applications of Chebyshev polynomials and Toeplitz theory to topological metamaterials
Ammari, Habib; Barandun, Silvio Alberto; Liu, Ping (2025)
Reviews in Physics
We survey the use of Chebyshev polynomials and Toeplitz theory for the study of topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to quantitatively explain and characterise some spectacular properties of metamaterials. Our characterisations are based on translation invariance properties of the capacitance matrices associated to the different investigated systems of resonators together with properties of Chebyshev polynomials. The three-term recurrence relation satisfied by the Chebyshev polynomials is shown to be the key to the mathematical analysis of spectra of tridiagonal (perturbed) both Toeplitz (for monomer systems) and 2-Toeplitz (for dimer systems) capacitance matrices.
Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
Ammari, Habib; Barandun, Silvio Alberto; De Bruijn, Yannick; et al. (2025)
Journal of Physics A: Mathematical and Theoretical
We establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials. We also numerically verify our theory for these polymer systems.
Generalised Brillouin Zone for Non-Reciprocal Systems
Ammari, Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2024)
SAM Research Report
Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the k-Toeplitz matrix cases.
Generalized Brillouin zone for non-reciprocal systems
Ammari, Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2025)
Royal Society of London. Proceedings A
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.

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Silvio Alberto Barandun

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