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Nonlinear non-Hermitian skin effect

C. Yuce

TL;DR

This work investigates how nonlocal edge effects intrinsic to the non-Hermitian skin effect extend into nonlinear regimes. By analyzing a nonreciprocal discrete nonlinear Schrödinger equation and a non-Hermitian Ablowitz-Ladik model, it demonstrates that nonlinearities can produce fractal and continuum bands and edge-localized stationary states, with nonlinear exceptional points arising or disappearing depending on lattice size and boundary conditions. Key findings include the emergence of fractal bands and edge localization in NRDNLS, the existence of a nonlinear exceptional point in the AL model at zero nonreciprocity that disappears in the semi-infinite limit, and a critical nonlinear strength beyond which NHSE vanishes. The results highlight substantial differences between linear and nonlinear NHSE, as well as the importance of system size and boundary conditions for accurately predicting spectral and localization properties.

Abstract

Distant boundaries in linear non-Hermitian lattices can dramatically change energy eigenvalues and corresponding eigenstates in a nonlocal way. This effect is known as non-Hermitian skin effect (NHSE). Combining non-Hermitian skin effect with nonlinear effects can give rise to a host of novel phenomenas, which may be used for nonlinear structure designs. Here we study nonlinear non-Hermitian skin effect and explore nonlocal and substantial effects of edges on stationary nonlinear solutions. We show that fractal and continuum bands arise in a long lattice governed by a nonreciprocal discrete nonlinear Schrodinger equation. We show that stationary solutions are localized at the edge in the continuum band. We consider a non-Hermitian Ablowitz-Ladik model and show that nonlinear exceptional point disappears if the lattice is infinitely long.

Nonlinear non-Hermitian skin effect

TL;DR

This work investigates how nonlocal edge effects intrinsic to the non-Hermitian skin effect extend into nonlinear regimes. By analyzing a nonreciprocal discrete nonlinear Schrödinger equation and a non-Hermitian Ablowitz-Ladik model, it demonstrates that nonlinearities can produce fractal and continuum bands and edge-localized stationary states, with nonlinear exceptional points arising or disappearing depending on lattice size and boundary conditions. Key findings include the emergence of fractal bands and edge localization in NRDNLS, the existence of a nonlinear exceptional point in the AL model at zero nonreciprocity that disappears in the semi-infinite limit, and a critical nonlinear strength beyond which NHSE vanishes. The results highlight substantial differences between linear and nonlinear NHSE, as well as the importance of system size and boundary conditions for accurately predicting spectral and localization properties.

Abstract

Distant boundaries in linear non-Hermitian lattices can dramatically change energy eigenvalues and corresponding eigenstates in a nonlocal way. This effect is known as non-Hermitian skin effect (NHSE). Combining non-Hermitian skin effect with nonlinear effects can give rise to a host of novel phenomenas, which may be used for nonlinear structure designs. Here we study nonlinear non-Hermitian skin effect and explore nonlocal and substantial effects of edges on stationary nonlinear solutions. We show that fractal and continuum bands arise in a long lattice governed by a nonreciprocal discrete nonlinear Schrodinger equation. We show that stationary solutions are localized at the edge in the continuum band. We consider a non-Hermitian Ablowitz-Ladik model and show that nonlinear exceptional point disappears if the lattice is infinitely long.

Paper Structure

This paper contains 4 sections, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nonreciprocal discrete nonlinear Schrodinger equation
  3. A non-Hermitian Ablowitz-Ladik model
  4. Conclusion

Figures (1)

  • Figure 1: The plot (a) shows a fractal frequency spectrum when there are $7$ lattice sites at ${g=1}$. The plot (b) shows the spectrum in a very narrower region depicted in the small red rectangle in the bottom left corner of the plot (a). We see a similar structure if we zoom in the plot. The fractal band is not continuous and every zoom is similar to the global plot.