Studies in geometry and topology of translationally ordered materials.

从自发对称性破缺的范式来看，尽管有许多尝试，但仍然没有关于平动有序材料的缺陷和goldstones定理的连贯理论。该项目将以smectic-A液晶为原型，研究平移有序材料的几何和拓扑结构。该项目的第一部分将是通过理解系统的基态流形（GSM）中的路径分解及其与基态对称群的关系来检查二维平动有序材料中的缺陷。目的是表明在GSM上的一种新的量化现象解释了许多观察到的材料性质。这将与J.H. Hannay教授（布里斯托尔）和R.D. Kamien教授（宾夕法尼亚大学）合作。这一观点将被用于这些系统（最著名的是近晶液晶）的精确连续体模型的发展，这些模型成功地捕获了材料的完全对称性，这是目前的理论所不能做到的。博士项目的第二部分将致力于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，为一般齐次（对称）空间上的场开发Goldstone定理的一个版本，从而允许对平动有序系统的低能变形进行优雅和统一的描述。这项工作将尝试统一和扩展凝聚态和高能理论的结果。

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.