{"title":"三维向列相中拓扑电荷的张量密度测量方法","authors":"Cody D. Schimming, Jorge Vinals","doi":"10.1098/rspa.2023.0564","DOIUrl":null,"url":null,"abstract":"<p>A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor <span><math display=\"inline\">\n<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n<mrow>\n<mrow>\n<mi mathvariant=\"bold\">D</mi>\n</mrow>\n</mrow>\n</mstyle>\n</math></span><span></span>, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor <span><math display=\"inline\">\n<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n<mrow>\n<mrow>\n<mi mathvariant=\"bold\">D</mi>\n</mrow>\n</mrow>\n</mstyle>\n</math></span><span></span> may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from <span><math display=\"inline\">\n<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n<mrow>\n<mrow>\n<mi mathvariant=\"bold\">D</mi>\n</mrow>\n</mrow>\n</mstyle>\n</math></span><span></span>.</p>","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"30 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A tensor density measure of topological charge in three-dimensional nematic phases\",\"authors\":\"Cody D. Schimming, Jorge Vinals\",\"doi\":\"10.1098/rspa.2023.0564\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor <span><math display=\\\"inline\\\">\\n<mstyle displaystyle=\\\"true\\\" scriptlevel=\\\"0\\\">\\n<mrow>\\n<mrow>\\n<mi mathvariant=\\\"bold\\\">D</mi>\\n</mrow>\\n</mrow>\\n</mstyle>\\n</math></span><span></span>, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor <span><math display=\\\"inline\\\">\\n<mstyle displaystyle=\\\"true\\\" scriptlevel=\\\"0\\\">\\n<mrow>\\n<mrow>\\n<mi mathvariant=\\\"bold\\\">D</mi>\\n</mrow>\\n</mrow>\\n</mstyle>\\n</math></span><span></span> may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from <span><math display=\\\"inline\\\">\\n<mstyle displaystyle=\\\"true\\\" scriptlevel=\\\"0\\\">\\n<mrow>\\n<mrow>\\n<mi mathvariant=\\\"bold\\\">D</mi>\\n</mrow>\\n</mrow>\\n</mstyle>\\n</math></span><span></span>.</p>\",\"PeriodicalId\":20716,\"journal\":{\"name\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"volume\":\"30 1\",\"pages\":\"\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"FirstCategoryId\":\"103\",\"ListUrlMain\":\"https://doi.org/10.1098/rspa.2023.0564\",\"RegionNum\":3,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.1098/rspa.2023.0564","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
摘要
在阶次参数空间中引入了一种与路径无关的量度,当沿着三维向列相中的任何封闭轮廓积分时,就能得到轮廓所包围的任何线缺陷的拓扑电荷。当在封闭表面或开放表面上积分时,相关的测量结果与点缺陷(刺猬)或 Skyrmions 相关电荷的已知结果一致。我们进一步定义了一种张量密度,即离散密度张量 D,通过它可以确定离散线的位置。该张量密度在该线附近的二元分解为其切线和旋转矢量,从而可以方便地确定二者的位置。张量 D 在没有缺陷的特殊构型(双游动或双扭转构型)中可能不为零,我们将提供其在这些构型中的表现。此外,还研究了 Skyrmions 和刺猬缺陷的特殊情况,包括根据 D 计算它们的拓扑电荷。
A tensor density measure of topological charge in three-dimensional nematic phases
A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor , from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from .
期刊介绍:
Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.