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Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators

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

(Report number 2023-32)

Abstract
The aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer systems and (iv) to prove localisation of the interface modes between two dimer structures with non-Hermitian gauge potentials of opposite signs based on new estimates of the decay of the entries of the eigenvectors of block matrices with mirrored blocks.

Keywords: Block Toeplitz matrix, tridiagonal 2-Toeplitz matrix, non-Hermitian skin effect, dimer system, complex gauge potential, gauge capacitance matrix, condensation of the eigenmodes, interface eigenmodes, topological invariant, subwavelength physics.

@Techreport{ABL23_1069,
  author = {H. Ammari and S. Barandun and P. Liu},
  title = {Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-32.pdf },
  year = {2023}
}

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