Lifted textures in nematic shells Gaetano Napoli, 1, a) Oleksandr V. Pylypovskyi, 2,3, b) Denis D. Sheka, 4, c) and Luigi Vergori 5, d) 1) Dipartimento di Matematica e Fisica ”E. De Giorgi”, Universit` a del Salento, Lecce (Italy). 2) Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany 3) Kyiv Academic University, 03142 Kyiv, Ukraine 4) Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine 5) Dipartimento di Ingegneria, Universit` a di Perugia, Perugia (Italy) (Dated: 27 October 2021) Magnetic materials and liquid crystals are examples of materials with orientational order which give rise to textures whose complexity is as beautiful as challenging to study. Their confinement in curved layers causes the emergence of geometry-induced effects that are not usually observed in flat layers. In this paper we draw a parallel between ferromagnetic and nematic shells, which are both characterized by local interaction and anchoring potentials. Although similar curvature-induced effects (such as anisotropy and chirality) occur, the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, on a spherical shell the textures of ferromagnets are characterised by vortices with integer topological charge, while the textures of nematics may admit also half-integer charge vortices. Recent advances in the theory of curvilinear mag- netism have highlighted a range of fascinating geometry- induced effects in the magnetic properties of materials 1,2 . When confined in thin curved domains, effective physi- cal features arise from the interplay between the curved geometry and magnetic texture. According to continuum micromagnetic description of the ferromagnetic media, the magnetic textures can be well described by the vector order parameter m = M/|M|, which is the normalized magnetization vector. The energy of the ferromagnet typically includes: (i) a short-range exchange energy that penalizes the non- uniformity of the magnetization; (ii) an anisotropic term that models the existence of directions preferred by the magnetization; (iii) an non-local term describing the long-range magnetostatic interactions. When ferromagnets are confined in thin curvilinear layers, the magnetic energy can be decomposed 3 to reveal the emergence of geometry-induced anisotropy and geometry-induced chiral interaction with emergent Dzyaloshinskii–Moriya interaction (DMI) as characteris- tic example. With this decomposition, a number of new effects in ferromagnetic spherical shells have been stud- ied including topological patterning and magnetochiral effects, for the review see Refs. 1,2,4 Recent advances in experimental techniques have also made possible the manipulation of these effects for the design of new functional materials and applications for spintronics, shapeable magnetoelectronics, magnonics, biomedicine, and soft robotics 1,2,4,5 . a) Electronic mail: gaetano.napoli@unisalento.it b) Electronic mail: o.pylypovskyi@hzdr.de c) Electronic mail: sheka@knu.ua d) Electronic mail: luigi.vergori@unipg.it Soft matter also provides an area in which the inter- play between geometry of the substrate and the order parameter plays a crucial role. One example is provided by liquid crystal (LC) shells 6 . These are microscopic col- loidal particles coated with a thin layer of nematic LC, and have potential applications as the topological de- fects (which may occur on them) can be engineered to emulate the linear, trigonal and tetrahedral geometries of carbon atoms 7 . This feature opens up the possibil- ity to design meso-atoms with special optical properties whose valence and directional-binding can be controlled. Photonic lattices made of LC shells are the new frontier for the manufacture of a new-generation optical crypto- graphic devices 8 . Nematic LCs are aggregates of rodlike molecules. Within the classic theory of nematics 9 , the average mi- croscopic molecular orientation is described by a sole vec- tor order parameter n called the director. In the 1960s de Gennes introduced the order-tensor theory which bases on the orientational probability distribution and provides measures of the degree of orientation and biaxiality. In its simplest form, this theory uses as state variable a second- order symmetric traceless tensor Q = s(n ⊗ n − 1 3 I), with the scalar parameter s being the degree of orientation that vanishes at points where there is no privileged direc- tion. Contrarily to the director theory, de Gennes theory allows the study of nematic-isotropic phase transitions. Most theories of nematic shells are based on energy functionals defined on surfaces expressed in terms of vector 10,11 or tensor 12–14 order parameters. Theories in which the director field n is purely tangential have the flaw that topological defects, i.e. points where the direc- tor is not uniquely defined, inevitably arise on surfaces with the same topology as sphere. Unavoidably, the en- ergy blows up in a neighbourhood of a defect. This flaw can be overcome by introducing a theory in which the tensor order parameter Q is tangential, i.e. the normal arXiv:2102.13497v1 [cond-mat.soft] 26 Feb 2021