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Mathematical foundations of the non-Hermitian skin effect

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

(Report number 2023-29)

Abstract
We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

Keywords: Non-Hermitian systems, skin effect, subwavelength resonators, imaginary gauge potential, generalised Brillouin zone, exceptional points, topological invariant, vorticity, phase transition, bulk boundary correspondence

@Techreport{ABCDH23_1066,
  author = {H. Ammari and S. Barandun and J. Cao and B. Davies and E.O. Hiltunen},
  title = {Mathematical foundations of the non-Hermitian skin effect},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-29},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-29.pdf },
  year = {2023}
}

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