Universal properties of anyon braiding on one-dimensional wire networks

World lines of anyons that exchange in 2D form braids in spacetime. These braids are subject to certain universal topological relations coming from their continuous deformations. In 2D, such an approach leads to the well-known braiding relation also known as the Yang-Baxter relation. In my talk, I will show how to define counterparts of braids and derive braiding relations for anyons constrained to move on planar wire networks. In particular, I will demonstrate that anyons on wire networks have fundamentally different braiding properties than anyons in 2D. My analysis reveals an unexpectedly wide variety of possible non-abelian braiding behaviours on networks. The character of braiding depends on the topological invariant called the connectedness of the network. As one of our most striking consequences, particles on modular networks can change their statistical properties when moving between different modules. However, sufficiently highly connected networks already reproduce braiding properties of 2D systems.

My talk is based on my recent work with Byung Hee An, arXiv:2007.01207 and arXiv:2006.15256.