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，与凝聚态物理学的传统范式相矛盾。在这些状态下，粒子是有序的，遵循一个隐藏的全局模式，这与任何对称性的破缺无关，并且在服从奇异统计的拓扑准粒子的存在下被揭示出来。我们对拓扑秩序如何从量子多体系统的微观自由度中产生的理解还远远没有完成。特别有趣的是可能形成的非阿贝尔拓扑相，其激发显示非阿贝尔编织的量子信息处理的潜在应用性质。该项目的目标是探索非阿贝尔拓扑相位产生的物理机制，使我们对它们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，我们将利用最近在超冷原子系统控制方面取得的惊人进展，提出现实的和实验上可行的模型设置。该提案围绕三个挑战展开：i）确定产生物质拓扑状态的新物理机制，ii）开发和研究能够产生这些相的理论模型，㈢开发理论工具，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。我们的重点将放在非阿贝尔物质和非阿贝尔任意子的不同实例上，例如非阿贝尔陈省身绝缘体，类普夫朗态和副费米子。此外，我们将研究拓扑绝缘体和超流体在新的对称类，并探讨它们如何作为物理路径来证明黎曼猜想。我们相信，分析理论方法和模型设置的结合，将导致从这个研究计划将提高我们的拓扑秩序的理论理解，铺平道路走向驯服和利用量子物质，对量子信息处理有潜在的影响。

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.