Interacting Topological Matter in Synthetic Dimensions

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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.