# Topological systems

## David Carpentier, Pierre Delplace, Andrei Fedorenko

The topic of *Topological Systems* refers to a broad class of physical systems, both quantum and classical, that exhibit unique and robust properties due to their underlying topology. Topology is a branch of mathematics that deals with properties that are preserved under continuous deformations, such as stretching or bending, but not under more drastic transformations like tearing or gluing. In the context of *Topological Systems*, the topology of the electronic wavefunctions or other excitations plays a crucial role. It characterizes the arrangement of different states of a system that determines its behavior.

The famous examples include the topological phases in *Quantum Matter*, such as topological insulators, the quantum Hall effect and Dirac and Weyl semimetals. Topological materials often possess insulating properties in the bulk but host conducting states on their surfaces or edges, which are protected by certain topological invariants against impurities or disorder. However, there are also other types of topological systems, whether they are engineered in the laboratory or occur naturally, where the role of topology may differ from its usual role in *Quantum Matter*.

We combine various numerical and analytical techniques (adiabatic theory, semiclassical methods, field theory) and draws from mathematics (differential geometry, index theorems) to describe the topological aspects of a wide range of physical systems. We are particularly interested in study the interplay between topological properties and other aspects, such as disorder, nonlinearity, or non-Hermiticity in these systems. In the quantum domain, for example, we are interested in Weyl semimetals, artificial gauge fields with cold atoms, quantum dynamics induced by topological couplings between different degrees of freedom, and the emergence of analog gravitational effects in certain topological materials. In the classical domain, we explore new topological states in mechanics, photonics, acoustics, as well as in fluids, with applications in geophysics and astrophysics involving wave phenomena.