The mathematics of exotic phases: fractons

这个项目的目标是为物质的分相建立一个严格而通用的数学框架。Fracton是一种迁移率受限的奇异准粒子，在量子纠错等方面具有潜在的应用前景。关于在实际量子材料中实现这种奇特的位相，人们提出了几种建议，其起点是具有拓扑顺序的2D带隙基态。这样的态支持具有奇异交换统计的任意子准粒子。从数学上讲，任意子是由所谓的融合范畴来描述的。这种丰富的代数结构出现在数学中的许多不同环境中。在这里，它编码了任意子的所有有趣的物理性质，并可以通过研究(超选)理论的部分来找到。这些对应于可见代数的不同的不可约表示。这种结构在放大系统(如果我们只添加乘积态)下是不变的，或者在保持光谱间隙的情况下轻微改变哈密顿量。3D中的分层分数阶(FFO)是这样的推广：我们被允许添加2D拓扑有序系统，而不是添加乘积态。因此，从这个观点来看，一堆未耦合的2D拓扑有序状态是微不足道的。已经发现了许多关于分数子准粒子的不平凡的例子。这些例子具有“叶状”结构，其中模型是由一维或二维“叶”以非平凡的方式耦合而成的。该项目的目标是开发数学工具来同时观察FFOs的类别(或阶段)，基于2D拓扑序的扇形理论。我们的方法植根于算符代数理论，允许我们直接在热力学极限下工作。在各种例子的指导下，我们将开发出FFO概念的严格版本。Fracton模型有无限多个超选扇区，但许多可以被认为是等价的。这导致了商超选部门的概念。主要任务之一将是将这一概念转化为算子代数设置。这将为分形学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”。这不仅对描述各种FFOs是有用的，而且对于研究2D中对称性丰富的拓扑(集)相也是有意义的。因此，我们可以在一个统一的框架中描述许多示例。

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.