Physics And Mathematics of Anyons

Garth A. Baker, S. B. Mulay

Belén Paredes

Can we find a physical intuitive language to represent the complex mathematical properties of anyon models? Is there a hidden organization in the space of anyon models? Can we systematically construct complex anyon models from other, more simple, anyon models? This work is an attempt to understand and explore the anatomy of the space of anyon models. I develop a construction to systematically generate and interconnect anyon models. The construction is formulated in a very physical, visual, and intuitive manner. The complex mathematical properties of anyon models, usually expressed in the complex language of braided modular tensor categories, are described here using familiar simple physical concepts like bosons in lattices, Fock states, and tunneling Hamiltonians. The construction formalism is conceptually simple and appealing. As in a Lego construction, a set of building block anyon models are assembled to construct towers of other, more complex, anyon models. A principle of assembly is established that dictates how to articulate the building blocks, setting out the global blueprint for the constructed anyon model. By assembling sets of the simple Abelian ${\mathbb{Z}}_{n}$ anyon models, the construction succeeds in generating towers of well known nontrivial anyon models. Moreover, novel anyon models (nontabulated, to my knowledge) arise. Furthermore, by applying the construction principle to other building blocks, such as Ising or Fibonacci, further series of anyon models are generated. In an intriguing connection, the boson-lattice construction unveils an alluring duality between anyon models and curved space geometries. Bosons induce curvature in the lattice geometry. I disclose a geography in the space of anyon models, where one is engendered from another by deforming the geometry of space. Moreover, the potential of the boson-lattice construction can be further explored via the quantum simulation of the corresponding boson-lattice systems in the versatile accessible playground of atomic systems.

Zhenghan Wang

The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then, I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity.

Carlo A. Trugenberger

Kerstin Beer, Bondarenko, Dmytro, Alexander Hahn +8 more

In this paper we provide an overview of category theory, focussing on applications in physics. The route we follow is motivated by the final goal of understanding anyons and topological QFTs using category theory. This entails introducing modular tensor categories and fusion rings. Rather than providing an in-depth mathematical development we concentrate instead on presenting the "highlights for a physicist".