经典动力学的拓扑效应和其中若干问题的研究

We will study topological properties of various classical systems. The classical vibrational lattices consisting of mass points and springs can also have nontrivial topological phases. We will study the models such as: the Chern insulating-like models in classical two-dimensional systems induced by forced vibrations or by dissipative forces; the effective four-dimensional systems having non-zero second kind Chern number; the various three- and one-dimensional topological insulating-like and metallic-like models; and so on. From these models, we will further investigate the physical problems including the topological phase transition at finite temperatures; the effects of disorder and dissipation on the phase transition; the floquet topological phases and their stability; and the manifestations of floating-up mechanism caused by disorder in two-dimensional classical Chern models. Based on the unique properties of these classical systems, such as the Dirac-like dispersion relations and the goldstone mode free of localization by disorder, the study on them will largely enrich our knowledge of topological systems. These models have many common points with phonon and magnon models. So many results of the studies on them can be directly applied in the relevant systems. Our systems can also provide a platform to study some unsolved problems of electronic topological systems, such as the phase transition at finite temperatures. Our investigations can also obtain several applicable effects, such as the unidirectional propagations of vibrational waves and the focusing of vibrational waves. The topologically protected modes obtained near the zero frequency also have very important application prospect.

本项目研究各类经典系统的拓扑性质。由质点和弹簧等组成的宏观晶格可拥有非平庸的拓扑相。我们将研究由外界强迫振动或耗散力诱导出的经典二维Chern类绝缘体，拥有非零第二类Chern数的等效四维系统，三维和一维的拓扑类绝缘体或类金属等多种模型。由上述模型出发探讨一系列重要的物理问题，包括：有限温的拓扑相变、无序和耗散对这个相变的影响、拓扑floquet系统的稳定性、无序引起的上浮（floating up）机制在经典二维系统中的表现等。此类经典系统具有许多独特的性质，例如类Dirac型的色散不会出现在普通声子系统中，对它们的研究将丰富我们对拓扑系统的认识。由于这类模型与声子模型以及磁子模型有共通之处，所以很多研究结果可直接用于相应的系统。这类系统还提供了一个平台以研究拓扑系统中一些尚未解决的问题，同时还可获得零频附近振动波的单向传播、振动波的聚焦等效应，在民用和军事方面有潜在的应用前景。

我们研究了经典系统中与拓扑有关的物理性质。通过对机械振动波等线性系统的研究，我们主要关注了两方面的物理问题：第一类问题是已知量子拓扑模型的经典类比或实现，我们引入的主动响应模型，可以在经典振动模型中模拟几乎所有的量子拓扑模型，如floquet系统中的高阶反常拓扑态，这为拓扑模型的实验研究提供了新的一类实现平台。第二类问题是在经典模型中超越已知的拓扑理论，我们发现了一类单带的非厄密拓扑模型，此带的拓扑性质可以追溯到另一个处在无穷远的能带，它们通过exceptional point有效的联系在一起。与拓扑超导线类似，会有分数的模式出现在边界上，但由于没有对称性来保护，所以这些分数模式并非majorana模，更为有趣的是，这类系统还有边体对应，边界态的存在必然会在无穷远的体带上引入一个新的态，它将通过分数的exceptional point与边界态联系在一起。