Experimental realization of unique non-Hermitian topological space
In our daily life, buildings are always erected uprightly, in which shops, offices, restaurants, etc. are placed in an orderly manner. In fact, these phenomena that we are familiar with are determined by geometry: on the scale of daily human life, space basically satisfies Euclidean geometry. More specifically, one of the salient features of Euclidean geometric space is that parallel lines never intersect. As we all know, the three-dimensional Euclidean space can be completely characterized by a three-dimensional coordinate system consists of three mutually perpendicular coordinate axes known as x, y, and z. However, not all objects can be embedded in Euclidean space. For example, the Penrose stairs seen in science fiction movies cannot be constructed in three-dimensional space. It turns out that there actually exist many types of geometric spaces. Take the earth for example, the earth itself is a sphere, but the geometry of this sphere does not satisfy Euclidean geometry. For instance, any two parallel meridians can intersect at the north or south poles, and the sum of internal angles of a triangle that is randomly cut from this sphere is not equal to 180 degrees. From these examples, it is apparent that the properties of geometric space also have a critical impact on the laws of physics.
As the abstraction of geometry, topology is also a branch of mathematics. (For example, the English letter "B" can be transformed into the Arabic numeral "8" by stretching, while the English letter "D" fails to be changed into "8" by stretching, due to the fact that the topological properties of "8" are similar to "B" but different from "D".) In that sense, topology is of extremely profound importance to physics. Moreover, the 2016 Nobel Prize in Physics was awarded to three scientists for their outstanding contributions to the application of topology in physics. In this study, we realized a geometric space called Riemannian manifold by introducing two complex parameters into an acoustic system, and then characterized its special topological properties both theoretically and experimentally. Particularly, the system can produce a unique singularity in a specific parameter space, which is also referred to as a higher-order exceptional point. The Riemannian manifold of higher-order exceptional point will bifurcate into multiple intersecting surfaces. As a consequence, the evolution process of the system around an exceptional point can reflect the geometric characteristics of these Riemann surfaces as well as the unique topological properties of the exceptional point. When the system evolves around an exceptional point, the eigenmodes of the system will exchange, resulting in fractional topological charges. More interestingly, there is a continuous curve composed of low-order singularities near the high-order singularities. Therefore the parameter space is cut as cracked glass and is no longer continuous, which results in different windings of the higher-order exceptional point, so that we can simultaneously get different topological charges for the same exceptional point. Through the actual measurement in the experiment, the experimental results are completely consistent with the theoretical results.
This project demonstrates both theoretically and experimentally a unique order-3 exceptional point with two different topological charges, which is of great significance in the research field of the topological properties of non-Hermitian systems. Additionally, this study not only paves a new way for the related researches and applications of non-Hermitian systems and their topological properties, but also provides new directions for the applications of optics and acoustics.
The research result is titled Exceptional Nexus with A Hybrid Topological Invariant and published online in the journal Science. [Science 370, 1077-1080 (2020).]
Figure 1. Acoustic experimental setup
Figure 2. Riemannian manifold of the system
Figure 3. Multiple exceptional arcs are connected to form an exceptional nexus under certain parameters
拓扑学正是对几何的抽象化的一个数学分支。（举个例子，英文字母“B”可以通过拉伸转化为阿拉伯数字“8”，而英文字母“D”却不能靠拉伸转化为阿拉伯数字“8”，因为“8”和“B”的拓扑特性相同，而与“D”不同。）因此，拓扑对物理有着非凡的深刻意义。2016年的诺贝尔物理学奖正是颁给了三位对拓扑在物理学中的应用做出开创性贡献的科学家。我们在该研究中，通过在声学系统引入复参数，实现了一种叫做黎曼流形的几何空间，并从理论和实验表征了其特殊的拓扑性质。值得注意的是，系统在参数空间中能产生一种特有的奇点——高阶奇异点(higher-order exceptional point)。高阶奇异点的黎曼流形会出现分岔成多个相互交叉的面。系统在绕奇异点的演化过程就能体现其这些黎曼面的几何特性，体现独特的拓扑性质。当系统围绕着奇异点演化时，本征模会发生交换，从而产生分数拓扑荷。更有意思的是，高阶奇异点附近存在着低阶奇异点构成的连续曲线。参数空间因此被切割，正好比开裂的玻璃不再连续。该特性造就了高阶奇异点不同的卷绕，因此我们能对同一个点同时得到不同的拓扑荷。通过实验的实际测量，得到的实验结果与理论结果完全吻合。
该成果以Exceptional Nexus with A Hybrid Topological Invariant为题，在线发表在《科学》（Science）期刊上。[Science 370, 1077-1080 (2020).]