Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling

doi:10.1088/1674-1056/accb47

中国物理B ›› 2023, Vol. 32 ›› Issue (8): 84203-084203.doi: 10.1088/1674-1056/accb47

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

收稿日期:2023-01-27 修回日期:2023-03-28 接受日期:2023-04-07 发布日期:2023-07-26
通讯作者: Li-Jun Lang E-mail:ljlang@scnu.edu.cn
基金资助:
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).

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

Received:2023-01-27 Revised:2023-03-28 Accepted:2023-04-07 Published:2023-07-26
Contact: Li-Jun Lang E-mail:ljlang@scnu.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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 E_m ∝ Γ^α_m, where α_m=3^m-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.

关键词: skin discrete breather, hierarchical power-law scaling, double-exponential decay, non-Hermitian skin effect, high-order exceptional point, Kerr nonlinearity

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 E_m ∝ Γ^α_m, where α_m=3^m-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.

Key words: skin discrete breather, hierarchical power-law scaling, double-exponential decay, non-Hermitian skin effect, high-order exceptional point, Kerr nonlinearity

中图分类号: (Nonlinear optics)

42.65.-k

05.90.+m (Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems)

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[J]. 中国物理B, 2023, 32(8): 84203-084203.

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Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling

Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling

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