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 consisting 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. Yetthere 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 anabstraction 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 a Riemannian manifold by introducing two complex parameters into an acoustic system,then characterizing its special topological properties both theoretically and experimentally. Interestingly, 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 points 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 forrelated studiesand 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 is 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.