Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling
Hui Jiang(江慧)†, Enhong Cheng(成恩宏)†, Ziyu Zhou(周子榆), and Li-Jun Lang(郎利君)‡
Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
Abstract We study the nonlinear perturbation of a high-order exceptional point (EP) of the order equal to the system site number L in a Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity. Notably, we find a class of discrete breathers that aggregate to one boundary, here named as skin discrete breathers (SDBs). The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP. Specifically, the response of nonlinear energy to the perturbation is given by Em ∝ Γαm, where αm=3m-1 is the power with m=1,...,L labeling the nonlinear energy bands. This is in sharp contrast to the L-th root of a linear perturbation in general. These SDBs decay in a double-exponential manner, unlike the edge states or skin modes in linear systems, which decay exponentially. Furthermore, these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity. They are also stable, as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis. As nonreciprocal nonlinear models may be experimentally realized in various platforms, such as the classical platform of optical waveguides, where Kerr nonlinearity is naturally present, and the quantum platform of optical lattices with Bose-Einstein condensates, our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity, particularly on high-order EPs, and benchmark the relevant simulations.
(Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems)
Fund: Project supported by the National Key Research and Development Program of China (Grant No.2022YFA1405304), the Key-Area Research and Development Program of Guangdong Province, China (Grant No.2019B030330001), and the Guangdong Provincial Key Laboratory (Grant No.2020B1212060066).
Corresponding Authors:
Li-Jun Lang
E-mail: ljlang@scnu.edu.cn
Cite this article:
Hui Jiang(江慧), Enhong Cheng(成恩宏), Ziyu Zhou(周子榆), and Li-Jun Lang(郎利君) Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling 2023 Chin. Phys. B 32 084203
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