近日，日本东京大学的Kazuki Sone及其研究团队取得一项新进展。经过不懈努力，他们发现非线性陈数表征的非线性诱导拓扑相变。相关研究成果已于2024年4月11日在国际知名学术期刊《自然—物理学》上发表。
本文基于二维系统的非线性特征值问题，提出了陈数的非线性推广，并证明了在弱非线性区域外体边界对应的存在性。具体而言，研究人员发现了非线性诱导的拓扑相变，其中拓扑边缘模式的存在取决于振荡模式的振幅。本文提出并分析了非线性陈氏绝缘子的最小模型，得到了该模型的精确体解。该模型展现了非线性陈数的幅值依赖性，从而证实了体边界对应关系的非线性扩展。因此，这一研究结果揭示了真正非线性拓扑相的存在，与线性体系实现了绝热分离。
据悉，正如陈数对量子霍尔效应的表征所首次证明的那样，拓扑学为实现不受无序存在影响的凝聚态系统的鲁棒性提供了指导原则。体-边界对应原理确保了具有非零拓扑不变量的拓扑系统中会出现无间隙边界模式。虽然近期有研究提出了将拓扑概念拓展至非线性系统的可能性，但关于拓扑不变量的非线性对应物，目前尚缺乏深入的理解。
附：英文原文
Title: Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number
Author: Sone, Kazuki, Ezawa, Motohiko, Ashida, Yuto, Yoshioka, Nobuyuki, Sagawa, Takahiro
Issue&Volume: 2024-04-11
Abstract: As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the robust properties of condensed-matter systems immune to the existence of disorder. The bulk–boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.
DOI: 10.1038/s41567-024-02451-x
Source: https://www.nature.com/articles/s41567-024-02451-x