Studies in geometry and topology of translationally ordered materials.

平移有序材料的几何和拓扑研究。

• 批准号: 2444881
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2444881
• 关键词: Studies; geometry; topology; translationally; ordered

Despite many attempts, there is still no coherent theory of defects and goldstones theorem for translationally ordered materials, as viewed through the paradigm of spontaneous symmetry breaking.This project will study the geometry and topology of translationally ordered materials, using smectic-A liquid crystals as a prototype.The first part of the project will be an examination of defects in two dimensional translationally order materials through understanding the decomposition of paths in the groundstate manifold (GSM) of the system, and their relation to the symmetry group of the groundstate. The aim is to show that a novel quantisation phenomena on the GSM accounts for many observed material properties. This will be in collaboration with Prof. J.H. Hannay (Bristol) and Prof. R.D. Kamien (UPenn).This perspective will then be used to inform the development of accurate continuum models of these systems (most notably smectic liquid crystals) that successfully capture the fully symmetry of the material, which current theories do not.The second part of the PhD project will aim to develop a version of Goldstone's theorem for fields on general homogeneous (symmetric) spaces, by understanding the role of Lie algebra homorphisms between special symmetries and field symmetries, allowing for an elegant and unifying description of low-energy deformation of translationally ordered systems. This work will attempt to unify and extend results from condensed matter as well high energy theory.

尽管进行了许多尝试，但从自发对称性破缺的角度来看，有序材料的缺陷理论和Goldstones定理仍然没有统一的理论。本项目将研究有序材料的几何和拓扑结构，使用近晶-该项目的第一部分将是通过了解液晶的结构，在基态流形（GSM）的系统，以及它们的基态的对称群的路径分解。其目的是表明，一种新的量子化现象的GSM帐户许多观察到的材料特性。这将是在与J. H教授合作。Hannay（布里斯托）和R.这一观点将被用来为这些系统的精确连续模型的发展提供信息（最明显的近晶液晶），成功地捕捉了材料的完全对称性，目前的理论没有。博士项目的第二部分将旨在开发一个版本的戈德斯通定理的领域一般均匀（对称）空间，通过理解特殊对称性和场对称性之间的李代数同态的作用，允许对有序系统的低能变形进行优雅和统一的描述。这项工作将试图统一和扩展凝聚态以及高能理论的结果。