EAPSI: Investigating the Stability of Particles in Topological Phases of Matter

科学中的一个基本挑战是描述物质的状态。物质的相是物质的一组状态，它们具有非常均匀的物理性质。常见的相包括液体、气体和固体，它们通过不同的温度来区分。在80年代？2005年，科学家在二维电子气体结构中发现了物质的新相，即分数量子霍尔态。这些状态拥有具有有趣统计行为的粒子，称为任意子。特别是，两个任意子的交换导致了一个复相的状态倍增。这些新的物质状态属于拓扑有序相。本项目研究了任意子的结构在拓扑相内是稳定的猜想。这项研究将与东京大学数学科学系教授Yasuyuki Kawahigashi博士合作进行。Kawahigashi博士是算子代数理论及其在数学物理中的应用方面的专家。这次合作使我们能够进一步探索拓扑相的算子代数视角，并在该主题中追求严谨的结果。这项研究将推动实现通用量子计算的努力。我们首先研究在热力学极限下精确可解的哈密顿晶格模型，其中任意子结构是完全已知的。这些模型包括Kitaev？表面代码和莱文和温？S弦网模型。每个任子都与可观测代数的超选择部分有关。对超选择扇区的分析允许人们恢复完整的任意子结构，包括粒子融合和编织。在物理允许的扰动下，我们将研究超选择扇区和粒子融合和编织的稳定性。任意子的稳定性是容错量子计算即拓扑量子计算中一个重要程序的前提。这个项目的结果为通过拓扑有序状态继续寻找通用量子计算机奠定了基础。NSF EAPSI奖是与日本科学促进会合作资助的。

A fundamental challenge in science is to characterize states of matter. A phase of matter is a family of states of matter which have very uniform physical properties. Common phases include liquid, gas and solid, which are distinguished by varying temperature. In the 1980?s, scientist discovered new phases of matter in two dimensional electron gas structures, namely fractional quantum hall states. These states possessed particles with interesting statistical behavior, called anyons. In particular, an exchange of two anyons resulted in a multiplication of the state by a complex phase. These new states of matter belong to topologically ordered phases. This project investigates the conjecture that the structure of anyons is stable within a topological phase. This research will be conducted in collaboration with Dr. Yasuyuki Kawahigashi, professor in the Department of Mathematical Science at the University of Tokyo. Dr. Kawahigashi is an expert in the theory of operator algebras and their applications to mathematical physics. The collaboration allows us to further explore the operator algebraic perspective of topological phases and pursue rigorous results in the subject. This research will advance efforts to realize universal quantum computing.We begin by studying exactly solvable Hamiltonian lattice models, in the thermodynamic limit, for which the anyon structure is completely known. These models include Kitaev?s surface codes and Levin and Wen?s string-net models. Each anyon is related to a superselection sector of the algebra of observables. Analysis of the superselection sectors allows one to recover the complete anyon structure, including particle fusion and braiding. Under physically allowable perturbations, we will study the stability of the superselection sectors and particle fusion and braiding. The stability of anyons is the premise of a major program in fault-tolerant quantum computation, namely topological quantum computation. The results of this project form the basis for the continued search for a universal quantum computer through topologically ordered states. This NSF EAPSI award is funded in collaboration with the Japan Society for the Promotion of Science.