物理系统的维度通常是指系统的几何维度。实际上，通过引入合成维度，可以使物理系统的维度可以大于其几何维度。此外，引入合成维度还可以很大程度上简化样品结构。合成维度可以通过引入合成参数空间或形成晶格来构造。相比于几何维度空间，合成维度空间具有动态可控性，且可用于构造更高维空间。合成维度和几何维度的结合是研究基础高维度物理的理想平台，并有潜在的应用前景。现阶段合成维度在波功能材料中的研究才刚刚兴起，研究只实现了二维/三维量子霍尔效应、Weyl半金属这些相对简单的系统。本项目拟将聚焦于合成维度这个科研前沿，将合成维度在研究高维、复杂系统上的优势运用到电磁波功能材料的设计中。具体地，我们将系统地比较各种构造合成维度方法的优劣点，并选择合适的方法构造各种高维、复杂拓扑结构；我们也将研究这些系统的体与边界对应关系以及体和面的输运性质，并探索潜在应用。

The dimension of a physical system is usually defined in terms of its geometrical dimension. Actually, with the synthetic dimension, the dimension of a system can even be larger than its geometrical dimension. Moreover, synthetic dimensions can also help to simplify the structure of experimental samples. Synthetic dimensions can be created by either forming a lattice or taking advantage of the parameter space. Compared with the geometrical dimension, synthetic dimensions can be easily controlled and utilized to form higher dimensional systems, say, a four-dimensional space. Thus, the synthetic dimension provides an ideal platform for studying fundamental higher dimensional physics and also leads to potential applications. The study of synthetic dimensions in wave functional material is a new frontier. So far, there have been only a few realizations of 2D/3D quantum Hall systems, and Weyl semimetal systems using the synthetic dimension. In this proposal, we will take advantage of the merits of synthetic dimensions and focus on complex, higher dimensional systems within wave functional materials. To be more specific, this proposal will firstly systemically compare various ways of creating synthetic dimensions, then combined with geometrical dimensions to construct complex higher dimensional topologies. After that, we will study the bulk-boundary correspondence in these systems and also the transport property of both the bulk and the boundary. Thereafter, we will try to find potential application scenario by taking advantage of the unique properties of the synthetic dimensions.