Topological Matter, Anyons and Many-Body Entanglement

物质的拓扑态代表了量子物质的一种奇异的组织形式，与传统的凝聚态物理范式相矛盾。在这些状态下，粒子遵循一个隐藏的全局模式进行排序，该模式与任何对称性的破坏无关，并且在遵循奇异统计的拓扑准粒子的存在下被揭示。我们对量子多体系统的微观自由度如何产生拓扑秩序的理解还远远不完整。尤其令人感兴趣的是可能形成的非阿贝尔拓扑相，其激发具有非阿贝尔编织特性，在量子信息处理中具有潜在的应用。这个项目的目标是探索非阿贝尔拓扑相产生的物理机制，使我们对非阿贝尔拓扑相起源的理论图景更加清晰，并为它们在物理系统中的实现提供指导。为此，我们将利用超冷原子系统控制方面的最新惊人进展，提出现实的和实验上可行的模型设置。该建议围绕三个挑战构建：i)确定产生物质拓扑态的新物理机制，ii)开发和研究能够产生这些相的理论模型，以及iii)开发理论工具来设计超冷原子气体，用于拓扑相的量子模拟和具有奇异编织统计的准粒子的操纵。我们的焦点将集中在非阿贝尔物质和非阿贝尔任意子的不同实例上，例如非阿贝尔陈绝缘体、类法费费态和准费米子。此外，我们将研究新对称类中的拓扑绝缘体和超流体，并探索它们如何作为物理路径来证明Rieman猜想。我们相信，分析理论方法和模型设置的结合将增强我们对拓扑序的理论理解，为驯服和驾驭量子物质铺平道路，对量子信息处理具有潜在的影响。

Topological states of matter represent an exotic organizational form of quantum matter that contradicts the traditional paradigms of condensed matter physics. In these states, particles are ordered following a hidden global pattern which is not associated with the breaking of any symmetry and is revealed in the presence of topological quasiparticles obeying exotic statistics. Our understanding of how topological order emerges from the microscopic degrees of freedom of a quantum many-body system is far from complete. Especially intriguing is the possible formation of non-Abelian topological phases, whose excitations display non-Abelian braiding properties with potential application for quantum information processing. The goal of this project is to explore physical mechanisms by which non-Abelian topological phases arise, sharpening our theoretical picture of their origin and serving as a guide for their realization in physical systems. To this end, we will take advantage of the most recent spectacular advances in the control of ultracold atomic systems to propose realistic and experimentally feasible model settings.The proposal is structured around three challenges: i) Identify novel physical mechanisms for the generation of topological states of matter, ii) Develop and investigate theoretical models able to produce these phases, and iii) Develop theoretical tools to engineer ultracold atomic gases for the quantum simulation of topological phases and the manipulation of quasiparticles with exotic braiding statistics. Our focus will be on different instances of non-Abelian matter and non-Abelian anyons, such as non-Abelian Chern insulators, Pfaffian-like states and parafermions. Moreover, we will investigate topological insulators and superfluids in novel symmetry classes and explore how they can serve as physical pathways to prove Rieman's conjecture.We believe that the combination of analytical theoretical approaches and model settings that will result from this research program will enhance our theoretical understanding of topological order, paving the way towards taming and harnessing quantum matter, with potential impact in quantum information processing.