BIT's progress in the analysis theory of generalized and strongly nonlinear topological invariant and anomalous edge modes

News resource & Photographer: School of Physics

Editor: News Agency of BIT

Translator: Duan Yuhan, News Agency of BIT


Recently, the research group of Prof. Yao Yugui and Researcher Zhou Di from the School of Physics published an article entitled “Topological invariant and anomalous edge modes of strongly nonlinear systems” in Nature Communications. Through mathematical derivation, the team studied a generalized and strongly nonlinear system, proposed a reasonable definition of Berry phase in nonlinear waves and strictly proved its quantization conditions. Through theoretical calculation and numerical simulation, the authors verified the stability of nonlinear topological modes, and promoted the “bulk-boundary correspondence” to a generalized nonlinear system. Researcher Zhou Di from BIT is the first author of the paper. Researcher Zhou Di and Professor Yao Yugui are the co-corresponding authors of the paper. The collaborators of the thesis include Professor D. Zeb Rocklin from Georgia University of Technology and Professor Michael J. Leamy.

The establishment of topological band theory has led to the burgeoning field of “topological phases of matter”. Electronic systems with topological properties will show the extraordinary transportability and stability that nature materials do not have. Compared with quantum systems, the physical properties of classical systems are more intuitive and easier to be applied in real life. These “topological metamaterials” constructed based on the physical properties of force, electricity, sound, and light form the emerging frontiers of the current physical research. The research of these metamaterials provides a new platform for the manipulation and control of the classical topology and the exploration and understanding of topological physics.

In recent years, in a series of classical topological systems represented by nonlinear optics, people have discovered the novelty of many nonlinear materials. For example, the topological frequency of amplitude regulation, topological phase transitions caused by amplitude, resonance synchronization of topological signals, and nonlinear self-induced topological state and so on. However, most of the current studies are limited to the nonlinear systems of "Kerr-like" and weak nonlinear systems. In fact, most nonlinear systems in nature are very complicated, and the nonlinear modes of them lack in-depth awareness, which leads to their topological invariants and topological phases cannot be quantitatively characterized. In daily life, complex nonlinear interactions include nonlinear circuits, elastic mechanics, fluid dynamics, biological evolutionary processes, and nonlinear optical issues that have been considered in secondary harmonic. In order to understand the topological properties caused by these broad nonlinear effects, we need to build a more universal topology theory. It has important scientific significance for the design of topological metamaterials and the research of nonlinear novel properties.

By studying generalized nonlinear modes, this article proposes the concept of Berry phase in the nonlinear oscillation mode, and deduces the mathematical form of the Berry phase. The author proves that under reflection symmetry constraints, the Barry phase can be quantized to portray the topological phase of generalized nonlinear systems. The work further promoted the self-induced topological state to a strong nonlinear area. Different from the linear topological modes, and the amplitude of these topological modes is attenuated from the boundary to a zero platform. The amplitude of the platform is determined by the stable fixed point of the system's nonlinear effect.

Through this work, the author proves that the concept of "topological protection" is universal in nonlinear systems, not limited to Kerr nonlinear and weak nonlinear systems. This article provides a theoretical foundation for the design of topological metamaterials and the study of nonlinear topology. Related work was published in Nature Communications 13, 3379 (2022) .

The work was supported by projects including the National Natural Science Foundation of China (11734003 and 12061131002), National Key R & D Plan (2020yFA0308800), and the Strategic Pioneer Science and Technology Special funding (XDB30000000) of the Chinese Academy of Sciences and so on.

Under the excitation of external signals with different amplitude, the same nonlinear system will show completely different topological properties. Figure 1 and 2 depict two nonlinear topological systems respectively. As shown in Figure 1, driven by the external signal with small amplitude, the system is in a nonlinear topological phase. Driven by the large amplitude, the system is in topological mediocrity . As shown in Figure 2, driven by a small amplitude, the system is in topological mediocrity. Under the excitation of large amplitude, the system is in the topological phase.


Figure 1. Small-amplitude topology, large-amplitude mediocrity nonlinear systems. As shown in (a, b), under the small amplitude excitation signal, the system shows the topological edge response. As shown in (c, d), under the large amplitude excitation signal, the system shows phantom response .


Figure 2. Small-amplitude mediocrity, large-amplitude topology nonlinear systems. As shown in (a, b), in the small amplitude excitation signal, the system shows the phantom response. As shown in (c, d), under the large amplitude incentive signal, the system shows the topological edge response.

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