The mathematics of exotic phases: fractons

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The goal of this project is to develop a rigorous and general mathematical framework for fractonic phases of matter. Fractons are exotic quasiparticles which have restricted mobility, and have potential applications in e.g., quantum error correcting. There are several proposals on the realisation of such exotic phases in actual quantum materials.The starting point is 2D gapped ground states with topological order. Such states support anyonic quasiparticles with exotic exchange statistics. Mathematically, anyons are described by what is a called a fusion category. This rich algebraic structure appears in many different contexts in mathematics. Here, it encodes all interesting physical properties of the anyons and can be found by studying (superselection) sectors of the theory. These correspond to different irreducible representations of the observable algebra. This structure is invariant under enlarging the system (if we only add product states), or gently changing the Hamiltonian whilst preserving the spectral gap.Foliated fracton order (FFO) in 3D is a generalisation of this: instead of adding product states, we are allowed to add 2D topologically ordered systems. Hence a stack of uncoupled 2D topologically ordered states is trivial from this point of view. Many non-trivial examples with fracton quasiparticles have been found. These examples have a "foliation" structure, where the models are built up from 1D or 2D "leaves" coupled in non-trivial ways.The goal of this project is to develop the mathematical tools to look at classes (or phases) of FFOs at the same time, based on the sector theory for 2D topological order. Our approach, rooted in operator algebra theory, allows us to work directly in the thermodynamic limit. Guided by various examples, we will develop a rigorous version of the notion of FFO. Fracton models have infinitely many superselection sectors, but many can be considered equivalent. This leads to the notion of a quotient superselection sector. One of the main tasks will be to translate this notion into the operator algebraic setting. This will pave the way for a theory of fusion for fractons.A second goal of the project is to systematically develop the "gauging" of a global symmetry in this framework. This is not only useful to describe various FFOs, but is also relevant for the study of symmetry enriched topological (SET) phases in 2D. Hence we can describe many examples in a unified framework.