najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2026-17-1-46-58najo-1686Research ArticlePHYSICSФИЗИКАUmbilic surfaces as a robust topological probe of 3D solitons in cholesteric liquid crystalsПоверхности умбиликов как надежный метод топологического анализа трехмерных солитонов в холестерических жидких кристаллахhttps://orcid.org/0000-0002-4324-1491АксеноваЕ. В.AksenovaE. V.

Elena V.Aksenova

199034, 7-9 Universitetskaya Emb., Saint Petersburg

aksenova@spbu.ruhttps://orcid.org/0000-0001-8789-3267ЛобановИ. С.LobanovI. S.

Igor S. Lobanov

197101, Kronverksky Pr. 49, Saint Petersburg

Yerevan

lobanov@itmo.ruhttps://orcid.org/0000-0002-1594-291XОрловаТ.OrlovaT.

Tetiana Orlova

Alex Manoogian, Yerevan, 0025

tetiana.orlova@ysu.amhttps://orcid.org/0000-0002-9505-0996УздинВ. М.UzdinV. M.

Valery M. Uzdin

197101, Kronverksky Pr. 49, Saint Petersburg

vmuzdin@itmo.ruhttps://orcid.org/0000-0002-1023-3284КиселевА. Д.KiselevA. D.

Alexei D. Kiselev

197101, Kronverksky Pr. 49, Saint Petersburg

adkiselev@itmo.ruSaint Petersburg State UniversityRussian FederationITMO University; Institute of Physics, Yerevan State UniversityArmeniaInstitute of Physics, Yerevan State UniversityArmeniaSaint Petersburg State University; ITMO UniversityRussian FederationITMO UniversityRussian Federation2026040320261714658Copyright © Aksenova E.V., Lobanov I.S., Orlova T., Uzdin V.M., Kiselev A.D., 20262026Аксенова Е.В., Лобанов И.С., Орлова Т., Уздин В.М., Киселев А.Д.Aksenova E.V., Lobanov I.S., Orlova T., Uzdin V.M., Kiselev A.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1686

The method based on umbilics that expose line-like organization of complex director fields is used to introduce umbilic surfaces as a numerically robust probe of three-dimensional (3D) topological solitons in frustrated cholesteric liquid crystals. We present a coordinate-free analytical formulation of the umbilic-line approach that ensures reliable detection of umbilics on discrete simulation grids and thus avoids the problems caused by instabilities and sensitivity to coordinate choices. By using our method we introduce the laboratoryreferenced phase field giving a natural tool for intuitive surface colorization. In addition, we employ this field to define the two fundamental integer invariants of umbilic loops: the transverse index (the strength) and the longitudinal winding (the profile twist). These invariants directly link the umbilic geometry to the topological characteristics of textures, thus enabling soliton identification and a comparison of solitons by topological content. We apply the technique to the three canonical solitons obtained by the free-energy minimization: the toron and the looped cholesteric fingers of the first and second types with the Hopf indices equal to zero and unity, respectively. It is found that the umbilic-surface representation clearly exposes defect structures, discriminates between visually similar but topologically distinct textures and provides a tool for quantifying and visualizing 3D solitons from director field data.

Метод, основанный на умбиликах, которые выявляют линейную структуру сложного распределения полей директора, используется для введения поверхностей умбилика в качестве численно надежного зонда трехмерных (3D) топологических солитонов в фрустрированных холестерических жидких кристаллах. Мы представляем не зависящую от координат аналитическую формулировку подхода с использованием линий умбилика, которая обеспечивает надежное обнаружение умбиликов на дискретных сетках моделирования и, таким образом, позволяет избежать проблем, вызванных нестабильностью и чувствительностью к выбору координатной системы. В рамках используемого метода можно ввести фазовое поле с лабораторной привязкой, которое предоставляет естественный инструмент для интуитивной раскраски поверхности. Кроме того, это поле используется для определения двух фундаментальных целочисленных инвариантов петель умбилика: поперечного индекса и продольного закручивания. Эти инварианты напрямую связывают геометрию умбилика со стандартными топологическими характеристиками поля директора, позволяя напрямую идентифицировать и сравнивать солитоны по их топологическому содержанию. Мы применяем эту методику к трем каноническим солитонам, полученным путем минимизации свободной энергии: торону и петлеобразным холестерическим пальцам первого и второго типов с индексами Хопфа, равными нулю и единице соответственно. Показано, что анализ поверхности умбилика четко выявляет дефектные структуры, позволяет различить визуально похожие, но топологически различные текстуры, и предоставляет удобный инструмент для количественной оценки и визуализации 3D-солитонов на основе данных о поле директора.

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The authors declare that there are no conflicts of interest present.