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The non-Hermitian skin effect with three-dimensional long-range coupling

by H. Ammari and S. Barandun and J. Cao and B. Davies and E.O. Hiltunen and P. Liu

(Report number 2023-39)

Abstract
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, 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. Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of the k-banded gauge capacitance matrix and demonstrate that they approximate those of the gauge capacitance matrix 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.

Keywords: Non-Hermitian systems, non-Hermitian skin effect, subwavelength resonators, imaginary gauge potential, stability, Toeplitz matrix

@Techreport{ABCDHL23_1076,
  author = {H. Ammari and S. Barandun and J. Cao and B. Davies and E.O. Hiltunen and P. Liu},
  title = {The non-Hermitian skin effect with three-dimensional long-range coupling},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-39.pdf },
  year = {2023}
}

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