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

by H. Ammari and S. Barandun and P. Liu and A. Uhlmann

(Report number 2024-16)

Abstract
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 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 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.

Keywords: Non-Hermitian systems, non-Hermitian skin effect, exceptional point degeneracy, subwavelength resonators, topological phase transition, broken symmetry, properties of Chebyshev polynomials, Toeplitz matrices and operators

@Techreport{ABLU24_1098,
  author = {H. Ammari and S. Barandun and P. Liu and A. Uhlmann},
  title = {Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-16.pdf },
  year = {2024}
}

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