Interacting Topological Matter in Synthetic Dimensions

近年来，对“物质拓扑状态”的研究重塑了我们对物理学的理解，使我们能够发现、研究和分类许多新的物理效应。在数学中，拓扑学是一种对不同表面进行分类的方法；例如，甜甜圈属于有一个孔的表面族，而橙子属于没有孔的表面族。如果我们平滑地压扁一个橙子，它的形状会改变，但它不会变成甜甜圈，除非我们撕开一个洞，改变它的拓扑结构。在物理学中，类似的想法可以用来对量子粒子或波状现象的状态进行分类。就像可挤压的橙子一样，这些拓扑状态是鲁棒的（例如，不受无序或杂质的影响），只要变化不影响拓扑特性。这既提出了有趣的基本问题，也为未来的通信和计算技术提供了可能性，特别是一旦粒子之间的相互作用也包括在内。然而，设计物质的拓扑状态并不总是那么容易，特别是在光子或冷原子的系统中。这个问题的一个成功解决方案是基于强大而通用的“合成维度”方法。“合成维度”的本质是识别系统固有的某些属性（例如，例如，设备中光的频率，即颜色），然后仔细设计系统，使该属性可以随时间变化（例如，光可以改变其颜色）。在某种定义明确的意义上，这种属性的变化类似于粒子在真实空间中左右、上下、前后移动时空间坐标的变化。这激发了对该属性（例如光的颜色）的重新解释，就像系统中额外的“合成维度”一样。虽然“合成维度”最初可能看起来非常抽象，但这种方法实际上对工程拓扑物质有许多优点，正如最近在原子和光子实验中迅速普及所表明的那样。然而，到目前为止，这些实验大多只探索了单粒子（即非相互作用）物理，而大的开放性问题仍然存在，即在研究更有趣的相互作用拓扑物质的机制时，合成维度将有多大用处，其中必须包括粒子间的相互作用。这个理论项目将通过在合成维度上探索相互作用的拓扑物质来解决这个挑战。我们将研究和评估有前途的综合维方案中的相互作用效应，并提出在二维甚至更高维的四维拓扑晶格中实现相互作用状态的实验。特别是后者，是合成维度带来的令人兴奋的机会，因为这种方法提供了一种人工增加系统有效维度的方法，超出了我们的3D物理世界。在整个项目中，我们的目标是通过解决基本的开放性问题来推进知识，例如在不同系统和非平衡环境中的相互作用，并为该领域未来的重要实验奠定基础。

In recent years, the study of "topological states of matter" has reshaped our understanding of physics, allowing us to discover, study and classify many new physical effects. In mathematics, topology is a way to classify different surfaces; for example, doughnuts belong to a family of surfaces with one hole, while oranges belong to that with no holes. If we smoothly squish an orange, its shape changes but it does not become like a doughnut unless we tear a hole and change its topology. In physics, similar ideas can be used to classify the states of a quantum particle or of wave-like phenomena. Just like the squishable orange, these topological states are robust (e.g. unaffected by disorder or impurities), provided that changes do not affect the topological properties. This has raised both interesting fundamental questions and possibilities for future technologies in communications and computing, especially once interactions between particles are also included.However, engineering a topological state of matter is not always easy, particularly in systems of photons or cold atoms. One successful solution to this problem is based on the powerful and general approach of "synthetic dimensions". The essence of a "synthetic dimension" is to identify some property inherent to a system (such as, for example, the frequency, i.e. the colour, of light in a device), and then to carefully engineer the system such that this property can change over time (e.g. that the light can change its colour). In a certain well-defined sense, the change of this property is analogous to the change in the spatial coordinates of a particle as it moves left/right, up/down and forwards/backwards through real space. This motivates the re-interpretation of that property (e.g. the colour of light) as being like an extra "synthetic dimension" in the system. Although "synthetic dimensions" may initially seem to be very abstract, this approach actually has many advantages for engineering topological matter, as shown by a recent rapid rise in popularity in atomic and photonic experiments. However, so far, most of these experiments have only explored single-particle (i.e. non-interacting) physics, and big open questions remain as to how useful synthetic dimensions will be to investigate the more interesting regime of interacting topological matter, in which inter-particle interactions must be included. This theoretical project will tackle this challenge by exploring interacting topological matter in synthetic dimensions. We shall examine and evaluate interaction effects in promising synthetic-dimension schemes, and propose experiments to realise interacting states in 2D and even higher-dimensional 4D topological lattices. The latter, in particular, is an exciting opportunity opened up by synthetic dimensions, as this approach provides a way to artificially augment the effective dimensionality of a system beyond our 3D physical world. Throughout this project, we will aim to advance knowledge by addressing fundamental open questions, such as the role of interactions in different systems and in non-equilibrium settings, and by laying the groundwork for important future experiments in this field.