Topological states in collective dynamics

Abstract : States of matter (such as solid, liquid, etc) are characterized by different types of order associated with local invariances under different transformation groups. Recently, a new notion of topological order, popularized by the 2016 physics nobel prize awarded to Haldane, Kosterlitz and Thouless, has emerged. It refers to the global rigidity of the system arising in some circumstances from topological constraints. Topologically ordered states are extremely robust i.e. « topologically protected » against localized perturbations. Collective dynamics occurs when a system of self-propelled particles organizes itself into a coherent motion, such as a flock, a vortex, etc. Recently, the question of realizing topologically protected collective states has been raised. In this work, we consider a system of self-propelled solid bodies interacting through local full body alignment up to some noise. In the large-scale limit, this system can be described by hydrodynamic equations with topologically non-trivial explicit solutions. At the particle level, these solutions persist for a certain time but eventually, for some of them, decay towards a topologically trivial state, due to the noise induced by the stochastic nature of the particle system. We numerically analyse these topological phase transitions and investigate to what extent topologically non-trivial states are 'protected' against perturbations. To our knowledge, it is the first time that a hydrodynamic model guides the design of topologically non-trivial states of a particle system and allows for their quantitative analysis and understanding. In passing, we will raise fascinating mathematical questions underpinning the analysis of collective dynamics systems.  

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