The mathematics of exotic phases: fractons

该项目的目标是为物质的分形相建立一个严格和通用的数学框架。分形子是具有有限迁移率的奇异准粒子，并且在例如，量子纠错关于在实际量子材料中实现这种奇异相，有几种方案，其出发点是具有拓扑有序的二维带隙基态。这种状态支持具有奇异交换统计的任意准粒子。在数学上，任意子被称为融合范畴。这种丰富的代数结构出现在数学中的许多不同背景中。在这里，它编码了任意子的所有有趣的物理性质，可以通过研究（超选择）理论的部门来找到。这些对应于可观测代数的不同不可约表示。这种结构是不变的扩大系统（如果我们只添加产品的状态），或轻轻地改变哈密顿量，同时保持频谱gap.Foliated fracton order（FFO）在3D是一个推广：而不是添加产品的状态，我们被允许添加2D拓扑有序系统.因此，从这个角度来看，一堆非耦合的2D拓扑有序态是微不足道的。已经发现了许多具有分形准粒子的非平凡例子。这些例子具有“叶状”结构，其中模型是由以非平凡方式耦合的1D或2D“叶子”建立的。本项目的目标是开发数学工具，同时基于2D拓扑序的扇区理论来研究FFO的类（或相）。我们的方法，植根于算子代数理论，使我们能够直接在热力学极限。在各种例子的指导下，我们将开发FFO概念的严格版本。分形模型有无限多个超选择扇区，但许多可以被认为是等价的。这导致了商超选择扇区的概念。主要任务之一将是把这个概念转化为算子代数设置。这将为分形子的融合理论铺平道路。该项目的第二个目标是在这个框架内系统地发展全球对称性的“测量”。这不仅有助于描述各种FFO，而且还与2D中的对称性富集拓扑（SET）相的研究有关。因此，我们可以在一个统一的框架中描述许多例子。

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.